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Slides06 - Connected Components, Bipartite Testingsheldon/teaching/mhc... · Logistics Quizzes Quiz...

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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?
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  • 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?


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