Logistics Quizzes Quiz 5: 74% Other quizzes: ~98% I’ll drop lowest 2 quizzes HW HW 2 back: average 44.9/50, 5.2 hours HW 3 due HW 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(n u ) 1 2 4 5 1 2 2 1 3 4 3 3 2 5 4 1 2 5 4 5 1 3
Running Time? Set explored[u] to be false for all u O(n) A = { s } / / set of discovered but not explored nodes while A is not empty O(m) Take a node u from A O(m) if explored[u] is false O(n) set explored[u] = true for each edge (u,v) incident to u add v to A O(m) end end end Same reasoning we just did: but now “charge” each line of code to either a node or an edge
Graph Traversal: Summary BFS/DFS: O(n+m) Is G connected? Find connected components of G Find distance of every vertex from source Get BFS/DFS trees (useful in some other problems) BFS: explore by distance, layers, queue DFS: explore deeply, recursive, stack
Application of BFS: Bipartite Testing
Bipartite Graphs A 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 Graphs Lemma. Let G be a connected graph, and let L 0 , …, L k 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 Lemma. Let G be a connected graph, and let L 0 , …, L k 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 4 Layer 3 Layer 2 Layer 1 Layer 0
BFS and Bipartite Graphs Lemma. Let G be a connected graph, and let L 0 , …, L k 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 4 Layer 3 Layer 2 Layer 1 Layer 0
Algorithm for Bipartite-Testing Run BFS Check each non-tree edge If any has endpoints in same layer, then G is not bipartite Otherwise, G is bipartite Running Time?
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