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?