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Graphs

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Data structures that connect a set of objects to form a kind of a network Objects are called “Nodes” or “Vertices” Connections are called “Edges” 2

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Types of Graphs 3 Directed vs. undirected Undirected (Edges have no direction) Directed (Edges have directions) Weighted vs. unweighted Unweighted (Edges have no cost/weight) Weighted (Edges have associated cost/weight)

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Types of Graphs (Cont’d) 4 Dense vs. Sparse Dense graphs (many edges between nodes) Sparse graphs (few edges between nodes) If the graph has n vertices (nodes) Maximum # of edges is n 2 In dense graphs number of edges is close to n 2 In sparse graphs number of edges is close to n

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5 Some Graph Applications transportation Graph street intersections NodesEdges highways communicationcomputersfiber optic cables World Wide Webweb pageshyperlinks socialpeoplerelationships food webspeciespredator-prey software systemsfunctionsfunction calls schedulingtasksprecedence constraints circuitsgateswires

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6 World Wide Web Web graph. n Node: web page. n Edge: hyperlink from one page to another. cnn.com cnnsi.com novell.comnetscape.com timewarner.com hbo.com sorpranos.com

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7 Social Network Social network graph. n Node: people. n Edge: relationship between two people. Reference: Valdis Krebs, http://www.firstmonday.org/issues/issue7_4/krebs

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Protein Networks 8 Nodes are proteins Edges are connections (interaction between proteins)

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More Formalization 9

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10 Undirected Graphs Undirected graph. G = (V, E) n V = set of nodes or vertices n E = edges between pairs of nodes. n Graph size parameters: n = |V|, m = |E|. V = { 1, 2, 3, 4, 5, 6, 7, 8 } E = { 1-2, 1-3, 2-3, 2-4, 2-5, 3-5, 3-7, 3-8, 4-5, 5-6 } n = |V| = 8 m = |E| = 11

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Graph Representation Two main methods 11 Adjacency MatrixAdjacency List

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12 Graph Representation: Adjacency Matrix Adjacency matrix. V-by-V matrix (A) n A[i, j] = 1 if exists edge between node i and node j n Space proportional to V 2 n Checking if (u, v) is an edge takes (1) time. n Identifying all edges takes (V 2 ) time. n For undirected graph matrix is symmetric across the diagonal 1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 1 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0 Vertices

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13 Graph Representation: Adjacency List Adjacency list. Node indexed array of lists. n Two representations of each edge. n Space proportional to O(E + V). n Checking if (u, v) is an edge takes O(deg(u)) time. n Identifying all edges takes (E + V) time. 1 23 2 3 4 25 5 6 7 38 8 1345 125 8 7 2346 5 degree = number of neighbors of u 37

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Degree of a Node In-degree(v): Number of edges coming to (entering) node v Out-degree(v): Number of edges getting out (leaving) node v For Undirected graphs In-degree = Out-degree 14 In-degree(V3) = Out-degree(3) = 5 In-degree(E) = 2 Out-degree(E) = 3

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Graph Traversal 15

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Graph Traversal Graph Traversal means visiting each node in the graph There is a starting node (s) Two main types of traversal Breadth-First-Search (BFS) Depth-First-Search (DFS) Both are applicable for directed and undirected graphs 16 BFSDFS

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17 Breadth First Search s 2 5 4 7 8 369 s L1L1 L2L2 L n-1 Visit the nodes one-level at a time Requires a queue (First-come-first-served)

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18 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: s Top of queue 2 1 Shortest path from s

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19 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: s 2 Top of queue 3 1 1

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20 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: s 2 3 Top of queue 5 1 1 1

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21 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 2 3 5 Top of queue 1 1 1

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22 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 2 3 5 Top of queue 4 1 1 1 2

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23 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 2 3 5 4 Top of queue 1 1 1 2 5 already discovered: don't enqueue

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24 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 2 3 5 4 Top of queue 1 1 1 2

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25 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 3 5 4 Top of queue 1 1 1 2

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26 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 3 5 4 Top of queue 1 1 1 2 6 2

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27 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 3 5 4 6 Top of queue 1 1 1 2 2

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28 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 5 4 6 Top of queue 1 1 1 2 2

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29 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 5 4 6 Top of queue 1 1 1 2 2

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30 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 4 6 Top of queue 1 1 1 2 2

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31 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 4 6 Top of queue 1 1 1 2 2 8 3

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32 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 4 6 8 Top of queue 1 1 1 2 2 3

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33 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 6 8 Top of queue 1 1 1 2 2 3 7 3

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34 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 6 8 7 Top of queue 1 1 1 2 2 3 9 3 3

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35 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 6 8 7 9 Top of queue 1 1 1 2 2 3 3 3

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36 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 8 7 9 Top of queue 1 1 1 2 2 3 3 3

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37 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 7 9 Top of queue 1 1 1 2 2 3 3 3

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38 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 7 9 Top of queue 1 1 1 2 2 3 3 3

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39 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 7 9 Top of queue 1 1 1 2 2 3 3 3

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40 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 7 9 Top of queue 1 1 1 2 2 3 3 3

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41 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 9 Top of queue 1 1 1 2 2 3 3 3

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42 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 9 Top of queue 1 1 1 2 2 3 3 3

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43 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: 9 Top of queue 1 1 1 2 2 3 3 3

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44 Breadth First Search s 2 5 4 7 8 369 0 Undiscovered Discovered Finished Queue: Top of queue 1 1 1 2 2 3 3 3

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45 Breadth First Search s 2 5 4 7 8 369 0 Starting from s, we visited all (reachable) nodes BFS forms a tree rooted at s (BFS Tree) For each node x reachable from s we created a shortest path from s to x 1 1 1 2 2 3 3 3

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46 Breadth First Search Example problems in which we use BFS Find if node x is reachable from node y Start from node y and do BFS Find the shortest path from node x to node y Start from node x and perform BFS Search for a value v in the graph Start from any node and perform BFS Always keep in mind whether we talk about undirected graph or directed graph

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47 Breadth First Search: Algorithm Start at node v Can do any processing on t

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48 Breadth First Search: Analysis Theorem. The above implementation of BFS runs in O(V + E) time if the graph is given by its adjacency list. Proof n Each node will be queued only once O(V) n For each node, we visit all its out edges O(E) – In fact each edge (u,v) is visited twice once from u’s side and once form v’s side n Total is O(V + E)

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Lecture 14: Graph Algorithms Shang-Hua Teng. Undirected Graphs A graph G = (V, E) –V: vertices –E : edges, unordered pairs of vertices from V V –(u,v)

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