Solving problems by

Searching

By

Dr. Budditha Hettige

Department of Computer Engineering

Outline

• Problem solving

• Search – an approach to problem solving

• Uninformed search

– Breadth-first (BF)

– Uniform cost (UC)

– Depth-first (DF)

– Depth limited(DL)

– Iterative deepening(ID)

– Bidirectional (BD)

• Informed search

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Problem solving

• Problem solving as a process of transforming a set of states

through actions/operations

• Components of a problem

– States

– Actions/operations/successor function

– Goal test

– Path cost

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State space

• State space consists of all possible states together with

actions

• States are represented by nodes, while actions are by arcs

• State space can be a tree or a graph

• A path from initial state to goal state in the state space is

called a solution

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Examples – state space

• Robot navigation

• Water jug problem

• Monkey–banana Problem

• Farmer, wolf, goat and cabbage

• 8-puzzel

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Assumptions

• World States

• Actions as transitions between states

• Goal Formulation: A set of states

• Problem Formulation:

– The sequence of required actions to move from current

state to a goal state

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Example

Search through the path

• States?

• Actions?

• Goal test?

• Path cost?

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Example

Farmer, Wolf, Goat and Cabbage problem

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Example: Romania

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Example: Romania

• On holiday in Romania; currently in Arad.

• Flight leaves tomorrow from Bucharest

• Formulate goal:– be in Bucharest

• Formulate problem:– states: various cities

– actions: drive between cities

• Find solution:– sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

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Vacuum World

States: The state is determined by both the agent location and the dirt locations. The agent is in one of two locations, each of which might or might not contain dirt. Thus there are 2 * 22 = 8 possible world states. A larger environment with n locations has n * 22 states.

Initial State: Any state can be designated as the initial state.

Actions: In this simple environment, each state has just three actions: Left, Right, and Suck. Larger environments might also include Up and Down.

Transition Model: The actions have their expected effects, except that moving Left in the leftmost square, moving Right in the rightmost square, and Sucking in a clean square have no effect.

Goal Test: This checks whether all the squares are clean.

Path Cost: Each steps costs 1, so the path cost is the number of steps in the path.

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Example: Vacuum World

• Single-state Problem:–You know all.

• Start in #5–Solution? [Right, Clean]

Example: vacuum world

• Multiple State Problem

– Sensorless

• Start in {1,2,3,4,5,6,7,8}

– Solution?

[Right, Clean, Left, Clean]

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Example: vacuum world

• Contingency

– Nondeterminism: Cleaning may

dirty a clean carpet.

– Partially observable: Location, dirt at current location.

– Percept: [L, Clean], i.e., start in #5 or #7

Solution?

[Right, if dirt then Clean]

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Vacuum world state space graph

• States? Dirt and robot location

• Actions? Left, Right, Clean

• Goal test? No dirt at all locations

• Path cost? 1 per action

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Example: The 8-puzzle

• States? Locations of tiles • Actions? Move blank left, right, up, down • Goal test? Given• Path cost? 1 per move

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Search - real world applications

• Route finding problem

• Traveling salesman problem

• VLSI layout design

• Automatic assembling

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Search algorithms

• A search algorithm defines how to find a path from

initial state to goal state

• All algorithms tell how to move from the current node to

next node

• Some algorithms provide just systematic ways to

explore the state space, while others tell how to

explore effectively

• In implementing search, it is easy to handle two list as

– OPEN – node to be visited (fringe)

– CLOSED – node already visited

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Evaluating Search algorithms

• Completeness – ability to find a solution when there is a

solution

• Optimality – ability to find the highest quality solution

• Time complexity – how long does it take

• Space complexity – how much memory is required

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Search strategies

• Uninformed search (blind search)

– Systematic way to explore a state space

– No information (uninformed) to explore effectively

– There are six algorithms

• Informed search (heuristic search)

– Preference of the use of operators

– Best-first search

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Uninformed search

• Breadth-first (BF)

• Uniform cost (UC)

• Depth-first (DF)

• Depth limited(DL)

• Iterative deepening(ID)

• Bidirectional (BD)

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Breadth-first

• Explore the state space layer by layer

• Explores all of the neighbor nodes at the present depth prior

to moving on to the nodes at the next depth level

• Algorithm using OPEN and CLOSE lists

• Example

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Breadth-first search algorithms

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Features-BFS

• Find the shallowest goal first

• Its Optimal and Complete

• Worst-case performance O(b {d})

• Worst-case space complexity O(b{d})

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Features – BFS contd..

• Take b=10, 10,000 nodes/second, 1000 bytes/node

• Depth Nodes Time Memory

2 1100 .11seconds 1MB

4 111,100 11seconds 106MB

6 107 19min 10GB

8 109 31hours 1TB

12 1013 35years 10petabytes

• Memory requirement is a bigger problem than timeE.g. for depth 8, 31 hours is not too big yet 1 terabytes is a huge memory requirement

• Time requirement is still importantE.g. For depth 12, 35 years

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Uniform Cost Search

• Version of BFS

• Expand the lowest cost node rather than lowest-depth node on the fringe

• Note that each time total cost must be considered

• Uniform-cost search is complete, such as if there is a solution, UCS will find it.

• Uniform-cost search is always optimal as it only selects a path with the lowest path cost.

• UCS is also

– Complete

– guarantee to find the minimum cost pat

– Space complexity - O(bd+1)

– Time complexity - O(bd)

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Depth-first search

• Explore a branch deep in

• The algorithm starts at the root node and explores as far as

possible along each branch before backtracking.

• Algorithm using OPEN and CLOSE

• Example

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Depth-first search algorithm

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Features of DFS

• Not complete

• Not optimal

• Space complexity O(bd)

• Modest memory requirement

– Consider a tree with branching factor 3

• Time complexity O(bd)

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Comparison

Criteria BFS UCS DFS DLS IDS BDS

Complete? Yes Yes No No Yes Yes

Optimal Yes Yes No No Yes Yes

Time O(bd+1) O(bd) O(bm) O(bl) O(bd) O(bd/2)

Space O(bd+1) O(bd+1) O(bm) O(bl) O(bm) O(bd/2)

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Informed Search

• Informed search is also called heuristic search

• General approach is best-first search

• Node is selected for expansion on the basis of

evaluation function f(n).

• Essential component of evaluation function is a

heuristic function h(n)

– h(n) = estimated cost of the cheapest path from

node n to a goal node

– If n is the goal node, then h(n) =0

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Greedy best-first search

• Expand the node that is closest to the goal

• Thus f(n)=h(n)

• In a route finding problem h(n) can be the straight-line

distance heuristic.

• Consider the following route finding problem

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Map

Arad

Zerind

Rimnicu

Sibiu Fagaras

Timisora

Lugoj

Oradea

DobretaCraiova

Oradea

Petesti

Bucharest

Urziceni

Vaslui

IasiNeamt

Hirsova

75

71

118

111

70

75

120

146 138

97

101

90

151

9921180

85

142

9287

98

140

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Straight-line distances to Bucharest

• Arad 366, Bucharest-0, Craiova-160

• Dobreta-242, Fagaras-176, Giurgui-77

• Iasi-226, Neamt-234, Oradea-380

• Petesti-100, Rimnicu-193

• Sibui-253, Timosoara-329, Zerind-374

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Example – Greedy search

Arad

366

Arad

Timisoara ZerindSibiu

253 329 374

(a) The initial state

(b) After expanding Arad

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Greedy search- Cont’d

Arad

Timisoara ZerindSibiu

329 374

Arad Fagaras Orade

a

Rimnicu

Vilcea

366 176 380 193

(c) After expanding Sibiu

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Greedy search- Cont’d

Arad

Timisoara Zerin

d

Sibi

u

329 374

Arad Fagara

s

Orade

aRimnicu

Vilcea366 380

193

Buchares

t

Sibiu

253 0

(d) After expanding Fagaras

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Comments on Greedy search

• Greedy best-first search is not optimal, path via sibiu,

Fagaras to Bucharest is 32Km longer than Path through

Rimnica and Pitesti.

• Further problems:

– Consider getting from Iasi to Fagaras

– Heuristic suggest that Neamt be expanded first, since it

is the closet to Fagaras

– But Neamt is a dead end

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A* Search

• f(n) = g(n) + h(n)

– f(n) – estimated cost of the cheapest solution through n

– g(n) – cost to reach to the node

– h(n) – cost to get from the node to the goal

• This is in fact a combination Greedy search (due to h(n)),

and uniform cost search (due to g(n)).

• Worst-case performance O(b{d})

• Worst-case space O(b{d})

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Example - A*

Arad

366=0+366

Arad

Timisoara ZerindSibiu

393=140+253447=118+329 449=75+374

(a) The initial state

(b) After expanding Arad

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A*- Cont’d

Arad

Timisoara ZerindSibi

u

449=75+374

Arad Fagaras Orade

a

Rimnicu

Vilcea

646=280+366415=239+176

671=291+380413=220+193

(c) After expanding Sibiu

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A*- Cont’d

(d) After expanding Rimnicu Vilcea

Sibiu

Arad Fagaras Oradea Rimnicu

Vilcea646=280+366 671=291+380

417=317+100

SibiuCraiova

526=366+160 553=300+253

Pitesti

415=239+176

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A*- Cont’d

(d) After expanding Fagaras

Sibiu

Arad Fagara

s

Oradea Rimnicu

Vilcea

646=280+366 671=291+380

BucharestSibiu

591=338+253450=450+0 417=317+100

SibiuCraiov

a526=366+160 553=300+253

Pitesti

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A*- Cont’d

(f) After expanding Pitesti

Sibiu

Arad Fagara

s

Orade

aRimnicu

Vilcea646=280+366 671=291+380

Buchares

t

Sibiu

591=338+253450=450+0

SibiuCraiova

526=366+160 553=300+253

Pitest

i

Buchares

t

Rimnicu

VilceaCraiov

a607=414+193

615=455+160418=418+0

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Comments on A*

• Note how A* find the path through Rimnica and Pitesti.

• So A* is complete and optimal

• Note: A* can be improved with iterative deepening search,

in which depth limit is the f-cost

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