Problem solving by search

Tomas Svoboda

Vision for Robots and Autonomous Systems, Center for Machine PerceptionDepartment of Cybernetics,

Faculty of Electrical Engineering, Czech Technical University in Prague

February 27, 2019

1 / 33

Outline

I Search problem.

I State space graphs.

I Search trees.

I Strategies, which tree branches to choose?

I Strategy/Algorithm properties?

I Programming instrastructure

2 / 33

Example: Romania

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

3 / 33

Ok, start with a simple one, almost everybody knows about the navigation- path planning problem. Waze, Garmin, . . .

Can you think about more problems?

Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4 / 33

Classical problem from the Book [1], we use it, too.

Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4 / 33

Classical problem from the Book [1], we use it, too.

Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4 / 33

Classical problem from the Book [1], we use it, too.

Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4 / 33

Classical problem from the Book [1], we use it, too.

Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4 / 33

Classical problem from the Book [1], we use it, too.

Example: The 8-puzzle

2

Start State Goal State

51 3

4 6

7 8

5

1

2

3

4

6

7

8

5

states?actions?solution?cost?

5 / 33

Also known as n − 1 puzzle.

Example: Vacuum cleaner

R

L

S S

S S

R

L

R

L

R

L

S

SS

S

L

L

LL R

R

R

R

states?actions?solution?cost?

6 / 33

A Search Problem

I State space (including Start/Initial state): position, boardconfiguration,

I Actions : drive to, Up, Down, Left . . .

I Transition model : Given state and action return state (and cost)

I Goal test : Are we done?

7 / 33

We will use the terminology throught the next 5-6 lectures; also for Markov

(Sequential) Decision Processes, Reinforcement Learning

A Search Problem

I State space (including Start/Initial state): position, boardconfiguration,

I Actions : drive to, Up, Down, Left . . .

I Transition model : Given state and action return state (and cost)

I Goal test : Are we done?

7 / 33

We will use the terminology throught the next 5-6 lectures; also for Markov

(Sequential) Decision Processes, Reinforcement Learning

A Search Problem

I State space (including Start/Initial state): position, boardconfiguration,

I Actions : drive to, Up, Down, Left . . .

I Transition model : Given state and action return state (and cost)

I Goal test : Are we done?

7 / 33

We will use the terminology throught the next 5-6 lectures; also for Markov

(Sequential) Decision Processes, Reinforcement Learning

A Search Problem

I State space (including Start/Initial state): position, boardconfiguration,

I Actions : drive to, Up, Down, Left . . .

I Transition model : Given state and action return state (and cost)

I Goal test : Are we done?

7 / 33

We will use the terminology throught the next 5-6 lectures; also for Markov

(Sequential) Decision Processes, Reinforcement Learning

State Space Graphs

State space graph: a representation of asearch problem

I Graph Nodes – states – are abstractedworld configurations

I Arcs represent action results

I Goal test – a set of goal nodes

Each state occurs only once in a state (search)space.

S b

c

d

G

a

8 / 33

Formalizing a real world problem – (creating) state space graph – could be

a problem of itself. I put creating into brackets is it may be also infinite.

Search Trees

Possible futures

Here we startS

a b c

I A “what if” tree of plans and their outcomes

I Start node is the root

I Children are successors

I Nodes show/contains states, but correspond to plans that achievethose states

9 / 33

• What if decision about an action, repeats . . .

• Nodes in the search tree are not the same as the nodes in the statespace graph.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

10 / 33

• There could be multiple search trees, depending ot the algorithm.

• A search tree can be much bigger than the state space.

• Both items will be discussed next.

Search tree for Romania(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

11 / 33

Search elements

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

I Expand plans - possible ways (tree nodes).

I Manage/Maintain fringe (or frontier ) of plans underconsideration.

I Expand new nodes wisely(?).

12 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

13 / 33

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

14 / 33

Before going to the next slide, think about algorithms. What properties of

an algorithm would you want?

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

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b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

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Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

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Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

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…………………

…

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Search (algorithm) properties

I Guaranteed to find a solution (if exists)? Complete?

I Guaranteed to find the least cost path? Optimal?

I How many steps - an operation with a node? Time complexity?

I How many nodes to remember? Space/Memory complexity?

How many nodes in a tree? What are tree parameters?

15 / 33

Draw a (symbolic–think about a triangle) sketch of a (search) tree. It maygrow upwards or downwards. How would you characterize/parametrize sizeof a tree.

• Depth of the tree d .

• Max-Depth of the treem. Can be ∞.

• Branching factor b.

• s denotes theshallowest Goal.

• How many nodes inthe whole tree?

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1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

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…………………

…

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Strategies

How to traverse/build a search tree?

16 / 33

It is perhaps worth to remember that the search tree is built as the algo-

rithm goes. Or better said, the tree is a human friendly representaion of

the machine run.

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

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f

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G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

What are the DFS properties (complete, optimal, time, space)?

17 / 33

• In animation, we will do the expansion step at once.

• What is the frontier - set of nodes, waiting to be expanded?

• When to stop the search?

• Thinking about optimality, what is the best solution we seek?

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

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…………………

…

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DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

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…………………

…

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DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

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1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

18 / 33

• Time, can process the whole tree: bm

• Space, only the path so far: bm

• Completness: m may be ∞ hence, not in general

• Optimality: No! It just takes the first solution found.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

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Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

What are the BFS properties?

19 / 33

BFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

20 / 33

• Time, can process the whole tree until s: bs , well actuallyb + b2 + b3 + · · ·+ bs but the last layer vastly dominates. Try somecalculations for various b.

• Space, all the frontier: bs

• Completness: Yes!

• Optimality, it does not miss the shallowest solution, hence if all thetransition costs are 1: Yes!

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…………………

…

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BFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

20 / 33

• Time, can process the whole tree until s: bs , well actuallyb + b2 + b3 + · · ·+ bs but the last layer vastly dominates. Try somecalculations for various b.

• Space, all the frontier: bs

• Completness: Yes!

• Optimality, it does not miss the shallowest solution, hence if all thetransition costs are 1: Yes!

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1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

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b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

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BFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

20 / 33

• Time, can process the whole tree until s: bs , well actuallyb + b2 + b3 + · · ·+ bs but the last layer vastly dominates. Try somecalculations for various b.

• Space, all the frontier: bs

• Completness: Yes!

• Optimality, it does not miss the shallowest solution, hence if all thetransition costs are 1: Yes!

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1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

BFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

20 / 33

• Time, can process the whole tree until s: bs , well actuallyb + b2 + b3 + · · ·+ bs but the last layer vastly dominates. Try somecalculations for various b.

• Space, all the frontier: bs

• Completness: Yes!

• Optimality, it does not miss the shallowest solution, hence if all thetransition costs are 1: Yes!

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

BFS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

a

S

b

c

d

e

f

G

S

a

b e

c

f b d

d

G

20 / 33

• Time, can process the whole tree until s: bs , well actuallyb + b2 + b3 + · · ·+ bs but the last layer vastly dominates. Try somecalculations for various b.

• Space, all the frontier: bs

• Completness: Yes!

• Optimality, it does not miss the shallowest solution, hence if all thetransition costs are 1: Yes!

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS vs BFS

What are (dis)advantages of the individual strategies?

S

a

b

e

f

f

e

f

c

f b

e

f

f

d

G

d

S

a

b e

c

f b d

d

G

21 / 33

• Do not overfit your thinking!

• Draw for yourself a different graph and contruct appropriate trees.

• Not everything is visible from the animations.

• Draw a comparison tables.

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

S

b

c

d

e

f

G

S

a

b

e f

e

f

c

f d

G

d

G

22 / 33

Iterative deepening DFS (ID-DFS)

I Start with maxdepth = 1

I Perform DFS with limited depth. Report success or failure.

I If failure, forget everything, increase maxdepth and repeat DFS

Is it not a terrible waste to forget everything between steps?

23 / 33

Really, how much do we repeat/waste? Compare number fo nodes gener-ated ID-DFS vs. BFS:

N(ID-DFS) = (s)b + (s − 1)b2 + (s − 2)b3 + · · ·+ (1)bs

N(BFS) = b + b2 + b3 + · · ·+ bs

Try some calculations for various s and b.

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b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

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…………………

…

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Iterative deepening DFS (ID-DFS)

I Start with maxdepth = 1

I Perform DFS with limited depth. Report success or failure.

I If failure, forget everything, increase maxdepth and repeat DFS

Is it not a terrible waste to forget everything between steps?

23 / 33

Really, how much do we repeat/waste? Compare number fo nodes gener-ated ID-DFS vs. BFS:

N(ID-DFS) = (s)b + (s − 1)b2 + (s − 2)b3 + · · ·+ (1)bs

N(BFS) = b + b2 + b3 + · · ·+ bs

Try some calculations for various s and b.

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b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Iterative deepening DFS (ID-DFS)

I Start with maxdepth = 1

I Perform DFS with limited depth. Report success or failure.

I If failure, forget everything, increase maxdepth and repeat DFS

Is it not a terrible waste to forget everything between steps?

23 / 33

Really, how much do we repeat/waste? Compare number fo nodes gener-ated ID-DFS vs. BFS:

N(ID-DFS) = (s)b + (s − 1)b2 + (s − 2)b3 + · · ·+ (1)bs

N(BFS) = b + b2 + b3 + · · ·+ bs

Try some calculations for various s and b.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Iterative deepening DFS (ID-DFS)

I Start with maxdepth = 1

I Perform DFS with limited depth. Report success or failure.

I If failure, forget everything, increase maxdepth and repeat DFS

Is it not a terrible waste to forget everything between steps?

23 / 33

Really, how much do we repeat/waste? Compare number fo nodes gener-ated ID-DFS vs. BFS:

N(ID-DFS) = (s)b + (s − 1)b2 + (s − 2)b3 + · · ·+ (1)bs

N(BFS) = b + b2 + b3 + · · ·+ bs

Try some calculations for various s and b.

m<latexit sha1_base64="4Gt0MY3yLD/OalBm/rL7VXeHCpc=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVTda6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCWrApDw=</latexit>

1 node<latexit sha1_base64="jHZBbUzr1Fh9bD4SkqNLpPYDO5A=">AAACKXicZVDLSgNBEJz1GeMr0aOXwSh4Crt60GPQi8cIxgSyQXpne3VwHsvMrBKW/IRX/QO/xpt69UecxIivhoaiqhuqKskFty4MX4OZ2bn5hcXKUnV5ZXVtvVbfuLC6MAw7TAtteglYFFxhx3EnsJcbBJkI7CY3J2O9e4vGcq3O3TDHgYQrxTPOwHmqtxPtUKVTvKw1wmY4GfofRFPQINNpX9aD5TjVrJCoHBNgbT8KczcowTjOBI6qcWExB3YDV9j3UIFEOygnhkd01zMpzbTxqxydsD8/SpDWDmXiLyW4a/tXG5PfmkGFd0xLCSot4wwkF8MUMyiEG5Wxzb7wb08uOxqUXOWFQ8U+LWWFoE7TcU805QaZE0MPgBnuU1F2DQaY821W4xwMV6kPT33sqq8v+lvWf3Cx34wOmuHZfqN1PC2yQrbINtkjETkkLXJK2qRDGBHknjyQx+ApeA5egrfP05lg+rNJfk3w/gHYRKX4</latexit>

b nodes<latexit sha1_base64="X/txyhIIuDx49R6RKr7nMT/tLWE=">AAACKnicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eFQwJpINoXe2VwfnsczMKmHJV3jVP/BrvIlXP8RJjKhJQ0NR1Q1VleSCWxeG78HM7Nz8wmJlqbq8srq2vlHbvLa6MAybTAtt2glYFFxh03EnsJ0bBJkIbCX3Z0O99YDGcq2uXD/HroRbxTPOwHnqZjfZpUqnaHsb9bARjoZOg2gM6mQ8F71asBynmhUSlWMCrO1EYe66JRjHmcBBNS4s5sDu4RY7HiqQaLvlyPGA7nkmpZk2fpWjI/bvRwnS2r5M/KUEd2cntSH5qxlU+Mi0lKDSMs5ActFPMYNCuEEZ2+wH//fksuNuyVVeOFTs21JWCOo0HRZFU26QOdH3AJjhPhVld2CAOV9nNc7BcJX68NTHrvr6osmypsH1QSM6bISXB/WT03GRFbJNdsg+icgROSHn5II0CSOSPJFn8hK8Bm/Be/DxfToTjH+2yL8JPr8AHtympg==</latexit>

b2 nodes<latexit sha1_base64="ZGTW0YYeXQcbuGbIvHUVnYaFmjc=">AAACLHicZVDLSgNBEJyN7/iKevQyGAVPYTce9Ch68ahgVMhG6Z3t1SHzWGZmlbDkM7zqH/g1XkS8+h1OYkQTGxqKqm6oqiQX3LowfAsqU9Mzs3PzC9XFpeWV1dra+oXVhWHYYlpoc5WARcEVthx3Aq9ygyATgZdJ93igX96jsVyrc9fLsSPhVvGMM3Ceam8n181tqnSK9qZWDxvhcOh/EI1AnYzm9GYtWIxTzQqJyjEB1rajMHedEozjTGC/GhcWc2BduMW2hwok2k459NynO55JaaaNX+XokP37UYK0ticTfynB3dlJbUD+agYVPjAtJai0jDOQXPRSzKAQrl/GNvvB455cdtApucoLh4p9W8oKQZ2mg6poyg0yJ3oeADPcp6LsDgww5wutxjkYrlIfnvrYVV9fNFnWf3DRbER7jfCsWT88GhU5TzbJFtklEdknh+SEnJIWYUSTR/JEnoOX4DV4Dz6+TyvB6GeDjE3w+QVpg6dK</latexit>

bm nodes<latexit sha1_base64="psCuRV1mt1LvxYwyneuj19UIUhU=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kSm5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9Hqp4U=</latexit>

…………………

…

s<latexit sha1_base64="r8gbB3BUNfto6YMSFfYvarzt09Y=">AAACJHicZVDLSgNBEJz1GeMzevQymAiewq4e9Ch68RjRJIIbQu9sbzJkZnaZmVXCkk/wqn/g13gTD178FicP8dXQUFR1Q1VFmeDG+v67Nze/sLi0XFopr66tb2xuVbZbJs01wyZLRapvIjAouMKm5VbgTaYRZCSwHQ3Ox3r7DrXhqbq2www7EnqKJ5yBddRVzdS6W1W/7k+G/gfBDFTJbBrdircaxinLJSrLBBhzG/iZ7RSgLWcCR+UwN5gBG0APbx1UINF0ionXEd13TEyTVLtVlk7Ynx8FSGOGMnKXEmzf/NXG5LemUeE9S6UEFRdhApKLYYwJ5MKOitAkX/i3J5ucdAqustyiYlNLSS6oTem4IhpzjcyKoQPANHepKOuDBmZdkeUwA81V7MJTF7vs6gv+lvUftA7rwVHdvzysnp7NiiyRXbJHDkhAjskpuSAN0iSM9MgDeSRP3rP34r16b9PTOW/2s0N+jffxCXU6pEI=</latexit> bs nodes<latexit sha1_base64="kR75eygmGA/ulNW+P3CFg39Zu38=">AAACLHicZVDLSgNBEJz1GeMzevQyGAVPYVcPehS9eIxgNJCN0jvbq4PzWGZmlbDkM7zqH/g1XkS8+h1OHqLRhoaiqhuqKskFty4M34Kp6ZnZufnKQnVxaXllda22fmF1YRi2mBbatBOwKLjCluNOYDs3CDIReJncnQz0y3s0lmt17no5diXcKJ5xBs5Tne3kym5TpVO012v1sBEOh/4H0RjUyXia17VgMU41KyQqxwRY24nC3HVLMI4zgf1qXFjMgd3BDXY8VCDRdsuh5z7d8UxKM238KkeH7O+PEqS1PZn4Swnu1v7VBuSPZlDhA9NSgkrLOAPJRS/FDArh+mVss2886cllh92Sq7xwqNjIUlYI6jQdVEVTbpA50fMAmOE+FWW3YIA5X2g1zsFwlfrw1Meu+vqiv2X9Bxd7jWi/EZ7t1Y+Ox0VWyCbZIrskIgfkiJySJmkRRjR5JE/kOXgJXoP34GN0OhWMfzbIxASfX9yIp4s=</latexit>

Cost sensitive search

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2 e,2

c,1

f,2 b,2 d,2

d,1

G,2

I In BFS, DFS, node ±depth was thenode-value.

I How was the depth actually computed?

I How to evaluate nodes with path cost?

24 / 33

Cost sensitive search

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2 e,2

c,1

f,2 b,2 d,2

d,1

G,2

I In BFS, DFS, node ±depth was thenode-value.

I How was the depth actually computed?

I How to evaluate nodes with path cost?

24 / 33

Cost sensitive search

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2 e,2

c,1

f,2 b,2 d,2

d,1

G,2

I In BFS, DFS, node ±depth was thenode-value.

I How was the depth actually computed?

I How to evaluate nodes with path cost?

24 / 33

Cost sensitive search

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2 e,2

c,1

f,2 b,2 d,2

d,1

G,2

I In BFS, DFS, node ±depth was thenode-value.

I How was the depth actually computed?

I How to evaluate nodes with path cost?

24 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

Uniform Cost Search (UCS)

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

1

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 b,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

25 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

When to stop, when visiting or expanding?

a

S

b

c

d

e

f

G

1

3

1

1

2

1

1

11.2

0.5

1

1

1

1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

26 / 33

UCS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

27 / 33

Solution cost C∗, transition cost at least ε. Effective depth, roughly C∗/ε.

• Time: bC∗/ε

• Space: bC∗/ε

• Completness: Yes!

• Optimality: Yes! Why?

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…………………

…

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UCS properties

I Time complexity?

I Space complexity?

I Complete?

I Optimal?

S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

b,2

e,3 f,4

d,3

27 / 33

Solution cost C∗, transition cost at least ε. Effective depth, roughly C∗/ε.

• Time: bC∗/ε

• Space: bC∗/ε

• Completness: Yes!

• Optimality: Yes! Why?

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…………………

…

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Example: Graph with costs

Uniform#Cost#Search#

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

q e

p

h

f

r

q

q c G

a

Strategy: expand a cheapest node first:

Fringe is a priority queue (priority: cumulative cost) S

G

d

b

p q

c

e

h

a

f

r

3 9 1

16 4 11

5

7 13

8

10 11

17 11

0

6

3 9

1

1

2

8

8 2

15

1

2

Cost contours

2

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Try it on paper, mark which nodes are in frontier, mark lines of equal cost.

Programming a Tree Search

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Infrastructure for (tree) search algorithms

What should a tree node n now?

I n.state

I n.parent

I n.pathcost

Perhaps we may add something later, if needed . . .

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Infrastructure for (tree) search algorithms

What should a tree node n now?

I n.state

I n.parent

I n.pathcost

Perhaps we may add something later, if needed . . .

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How to organize nodes?

The Python examples are just suggestions, . . .

I A dynamically linked structure (list()).

I Add a node (list.insert(node)).

I Take a node and remove from the structure (node=list.pop()).

I Check the Python modules heapq1 and queue2 for inspiration.

1https://docs.python.org/3.5/library/heapq.html2https://docs.python.org/3.5/library/queue.html

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What is the solution?

I We stop when Goal is reached.

I How do we contruct the path?

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References

Some figures if from [1].

[1] Stuart Russell and Peter Norvig.Artificial Intelligence: A Modern Approach.Prentice Hall, 3rd edition, 2010.http://aima.cs.berkeley.edu/.

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