# Graphs. 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”

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Graphs

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

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)

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

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

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

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

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

More Formalization 9

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

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

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

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

Graph Traversal 15

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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