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CS 161: Design and Analysis of Algorithms Mark Zhandry
CS 161: Design and Analysis of Algorithms
Mark Zhandry
Graphs 1: Basic Graphs/Undirected Connectivity • Representation • Graph Properties • BFS • DFS • Connectivity
Undirected Graphs
Directed Graphs
Graph Examples Example Nodes Edges
Social Network People Friendships
WWW Websites Links
Games Board positions Legal Moves
Road Maps Intersections Roads
Software Functions Function Calls
Chemicals Atoms Bonds
Scheduling Tasks Precedence Constraints
Electricity Grid Power Stations Power Lines
Graph Basics
• V = set of nodes/vertices • E = set of edges
– Denote an edge from u to v as (u,v) – Order matters in directed graph, but not in
undirected graph
• Usually write a graph G as the pair (V,E)
Representing Graphs
• Adjacency Matrix: – Matrix A with |V| rows and columns – Au,v = 1 if and only if the edge (u,v) exists – Undirected graph: A is symmetric
• Adjacency List: – Array with |V| entries – The entry for u is the list of edges (u,v)
Graph Concepts
• Undirected graphs: – degree(v) = number of edges incident on v
• Directed graphs: – Indegree(v) = number of edges into v – Outdegree(v) = number of edges out from v
Representing Graphs
• Space requirements: – Adjacency Matrix: O(|V|2) – Adjacency List: O(|V|+|E|)
• Time to see if (u,v) is in graph: – Adjacency Matrix: O(1) – Adjacency List: O(degree(v)) or O(outdegree(u))
• Time to get all (u,v) for specific u: – Adjacency Matrix: O(|V|) – Adjacency List: O(degree(v)) or O(outdegree(v))
Graph Concepts
• Path: sequence v1, v2, … vk, where (vi, vi+1) is an edge in the graph
• Length of path = number of edges in path – Infinity if no path
A B
C D
E F
A B C D
Graph Concepts
• Simple Path: path where all nodes are different
A
B
C
D
F E
Graph Concepts
• Cycle: path where first and last node are the same
A
B C
D
F
E
Graph Concepts
• Simple Cycle: cycle where all nodes are different (except the first and last)
A
B
C D
F
E
Graph Concepts
• Subgraph: A graph G’=(V’,E’) such that V’ is a subset of V and E’ is a subset of E
A
B
G
C
F
D
E
Types of Graphs • Complete Graph: all possible edges
A
B
G
C
F
D
E
Types of Undirected Graphs • Connected Graph: there is a path connecting
any two nodes
A B C
D F
E
G
H
Types of Undirected Graphs
• Tree: connected graph with no simple cycles
A B C
D F
E G
H
Tree Properties
• |E|=|V|-1 – Proof: If |V|=1, trivial – Assume true for |V|=n-1, we prove true for |V|=n – Pick a leaf node, and remove it. |V’|=|V|-1 and
|E’|=|E|-1 (is there always a leaf node?) – |E|=|E’|+1=|V’|-1+1=|V’|=|V|-1
Tree Properties
• Every two nodes u and v have exactly 1 path between them – Proof: Suppose they have 2 paths p1 and p2 – Let w be the first node in p1 that is also in p2 – Let p be the path obtained by following p1 from u
to w, and then p2 from w back to u – p is a simple cycle!
Shortest Paths
• The distance between nodes u and v is the minimum length of all paths from u to v – Shortest path is simple (why?)
• How do we compute the distance between u and v?
Breadth-First Search
• Idea: Let wi be the neighbors of v, and let pi be the shortest path from u to wi.
• Any path from u to v must have one of the wi be the last node before v.
• Then the shortest path from u to v must be one of the pi followed by v
Breadth-First Search
• What if we already knew the distances from u to all of the nodes with distance at most d
• We can find the distances to all nodes with distance at most d+1 as follows: – For every node v with distance d, look at every
outgoing edge (v,w) – If w does not have distance at most v, its distance
is d+1
• Base case: u is the only node with distance 0 from u
Breadth-First Search
Algorithm: Input graph G=(V,E), node u • Set distance(u) = 0, distance(v)=∞ for v≠u • q = new queue containing u • While(q is not empty)
– v = q.poll() – For every (v,w) in E with distance(w)=∞:
• Set distance(w)= distance(v)+1 • q.add(w)
Proof of Correctness
• Claim: For each d=0,1,…, there is some point at which – all nodes with distance at most d have their
distances correctly set, – all other nodes have distance set to infinity, – and q contains exactly the nodes at distance d
Proof of Correctness
• True for d = 0 at beginning. Inductively assume true for d-1 – At some point, q contains nodes of distance d-1,
all nodes with distance at most d-1 have correct distance, and all other nodes have distance set to infinity
Proof of Correctness
• Process all elements currently in q. – If a node v has distance at most d-1, distance(v)
not changed – If a node v has distance d, it has some neighbor
with distance d-1 that we will process • Thus we will set distance(v) to d and add it to q
– If a node v has distance > d, all of its neighbors have distance > d-1, so we do not change distance(v)
Proof of Correctness
• End result: – All nodes with distance d in queue – All nodes v with distance > d have distance(v)=∞ – All nodes v with distance at most d have correct
distance(v)
• Once d > distance of farthest node, all nodes have correct distance, queue is empty, and program stops
Running Time
• Every node added to queue only when its distance is changed from infinity to something finite – Each node added only once
• Each edge (u,v) only examined when u is removed from queue – Also when v is removed for undirected graphs – Therefore, each edge examined at most twice – If we use adjacency list, getting the edges from a node
takes constant time per edge • Running time: O(|V|+|E|)
BFS Example
BFS Example
d = 0
BFS Example
d = 0
BFS Example
d = 1
BFS Example
d = 1
BFS Example
d = 2
BFS Example
d = 2
BFS Example
d = 3
Abstract View of BFS • BFS(G,u) =
visited(u) = true q.add(u) While q is not empty: v = q.peak() for each edge (v,w) where not visited(w) previsit(w) visited(w) = true q.add(w) postvisit(v) q.poll()
DFS: Exchange Queues for Stacks • Explore(G,u) =
visited(u) = true s.push(u) While q is not empty: v = s.peak() for each edge (v,w) where not visited(w) previsit(w) visited(w) = true s.push(w) postvisit(v) s.pop()
Recursion: Implicit Stack
• Explore(G,u) = visited(u) = true previsit(u) For each edge (u,v) where not visited(v): explore(v) postvisit(u)
Running Time
• Same as BFS, so O(|V|+|E|)
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
DFS Example
What does DFS do?
• Depends on what previsit and postvisit do!
Undirected Connectivity
• Say two nodes u, v are connected if there is a path from u to v
• Equivalence relation: – u is connected to u – If u is connected to v, v is also connected to u – If u is connected to v and v is connected to w, then
u is connected to w
Undirected Connectivity
• The equivalence relation induces equivalence classes on graph. Called connected components
A B C
D E F
G H
What does Explore Do?
• Visits exactly the nodes in the connected component containing v
• How do we get all of the connected components? – Run explore from all nodes
DFS
• DFS(G) = initialize() visited(v) = false for all v For all v,
If not visited(v): update() explore(G,v)
• How do we set initialize(), update(), previsit(v), and postvisit(v) to get connected components?
Running Time
• Assume initialize is O(|E|+|V|), update() is O(1)
• Call explore once per node. • Each call to explore visits all the edges from
that node • Each node visited once, each edge twice • Constant time per visit to node/edge • Total running time O(|V|+|E|)
How to Find Connected Components?
• initialize(): cc = 0 • update(): cc = cc + 1 • previsit(v): ccnum(v) = cc • postvisit(v): do nothing
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