LogisticsQuizzes

Quiz 5: 74%Other quizzes: ~98%I’ll drop lowest 2 quizzes

HWHW 2 back: average 44.9/50, 5.2 hoursHW 3 dueHW 4 out tonight

No reading for Thursday

Questions?

Today

BFS/DFS

Connected components

Bipartite testing

Directed Graphs

Connected Components

Definitions, example, proof on board

“Meta”-BFS algorithm

while there is an unexplored node s

BFS(s)

end

Example

Running time? argue O(m+n) running time on board

Representing Graphs: Adjacency List

Adjacency list. Each node keeps a (linked) list of neighbors.

Find all edges incident to u: O(nu)

1

3

5 4

212345

2 4 51 3 42 51 21 3

Running Time?Set explored[u] to be false for all uA = { s } // set of discovered but not explored nodeswhile A is not empty

Take a node u from Aif explored[u] is false

set explored[u] = truefor each edge (u,v) incident to u

add v to Aend

endend

Same reasoning we just did: but now “charge” each line of code to either a node or an edge

O(n)

O(m)

O(m)O(m)

O(n)

Graph Traversal: Summary

BFS/DFS: O(n+m)Is G connected?Find connected components of GFind distance of every vertex from sourceGet BFS/DFS trees (useful in some other problems)

BFS: explore by distance, layers, queueDFS: explore deeply, recursive, stack

Application of BFS:Bipartite Testing

Bipartite GraphsA bipartite graph is an undirected graph G = (V, E) in which the nodes can be colored red or blue such that every edge has one red and one blue end.

is a bipartite graph

is NOT a bipartite graph

Examples? How can we check if a given graph is bipartite?

Simple Observation: Odd Cycles

Lemma. If G has a cycle of odd length, then G is not bipartite

Proof on board

BFS and Bipartite GraphsLemma. Let G be a connected graph, and let L0, …, Lk be the layers produced by BFS starting at node s. Exactly one of the following holds:(i) No edge of G joins two nodes of the same layer, and G is

bipartite.(ii) An edge of G joins two nodes of the same layer, and G

contains an odd-length cycle (and hence is not bipartite).

Layer 1 Layer 2 Layer 3 Layer 4Layer 0

BFS and Bipartite GraphsLemma. Let G be a connected graph, and let L0, …, Lk be the layers produced by BFS starting at node s. Exactly one of the following holds:(i) No edge of G joins two nodes of the same layer, and G is

bipartite.(ii) An edge of G joins two nodes of the same layer, and G

contains an odd-length cycle (and hence is not bipartite).

BFS and Bipartite Graphs

Layer 1 Layer 2 Layer 3 Layer 4Layer 0

Lemma. Let G be a connected graph, and let L0, …, Lk be the layers produced by BFS starting at node s. Exactly one of the following holds:(i) No edge of G joins two nodes of the same layer, and G is

bipartite.(ii) An edge of G joins two nodes of the same layer, and G

contains an odd-length cycle (and hence is not bipartite).

Algorithm for Bipartite-Testing

Run BFSCheck each non-tree edge

If any has endpoints in same layer, then G is not bipartiteOtherwise, G is bipartite

Running Time?