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Logistics Quizzes Quiz 5: 74% Other quizzes: ~98% Ill drop lowest - - PowerPoint PPT Presentation
Logistics Quizzes Quiz 5: 74% Other quizzes: ~98% Ill drop lowest - - PowerPoint PPT Presentation
Logistics Quizzes Quiz 5: 74% Other quizzes: ~98% Ill 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
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Connected Components
Definitions, example, proof on board
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“Meta”-BFS algorithm
while there is an unexplored node s BFS(s) end Example Running time? argue O(m+n) running time on board
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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 2
1 2 3 4 5 2 4 5 1 3 4 2 5 1 2 1 3
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Running Time?
Set explored[u] to be false for all u A = { s } /
/ set of discovered but not explored nodes
while A is not empty Take a node u from A if explored[u] is false set explored[u] = true for each edge (u,v) incident to u add v to A end end end
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)
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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
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Application of BFS: Bipartite Testing
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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?
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Simple Observation: Odd Cycles
- Lemma. If G has a cycle of odd length, then G
is not bipartite Proof on board
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BFS and Bipartite Graphs
- 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).
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Layer 1 Layer 2 Layer 3 Layer 4 Layer 0
BFS and Bipartite Graphs
- 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).
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BFS and Bipartite Graphs
Layer 1 Layer 2 Layer 3 Layer 4 Layer 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).
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