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CS 253: Algorithms Chapter 22 Graphs Credit: Dr. George Bebis

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2 Graphs Definition = a set of nodes (vertices) with edges (links) between them. G = (V, E) - graph V = set of vertices V = n E = set of edges E = m ◦ Subset of V x V ={(u,v): u V, v V} 1 2 34

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Applications Applications that involve not only a set of items, but also the connections between them Computer networks Circuits Hypertext Maps

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Terminology Complete graph ◦ A graph with an edge between each pair of vertices Subgraph ◦ A graph (V ’, E ’ ) such that V ’ V and E ’ E Path from v to w ◦ A sequence of vertices such that v 0 =v and v k =w Length of a path ◦ Number of edges in the path 1 2 34 path from v 1 to v 4

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Terminology (cont’d) w is reachable from v ◦ If there is a path from v to w Simple path ◦ All the vertices in the path are distinct Cycles ◦ A path forms a cycle if v 0 =v k and k≥2 Acyclic graph ◦ A graph without any cycles 1 2 34 cycle from v 1 to v 1

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6 Terminology (cont’d)

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7 A bipartite graph is an undirected graph G = (V, E) in which V = V 1 + V 2 and there are edges only between vertices in V 1 and V 2 1 2 3 5 4 9 7 6 8 V1V1 V2V2

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8 Graph Representation Adjacency list representation of G = (V, E) ◦ An array of V lists, one for each vertex in V ◦ Each list Adj[u] contains all the vertices v that are adjacent to u (i.e., there is an edge from u to v) ◦ Can be used for both directed and undirected graphs 1 2 54 3 2 5/ 1 5 3 4/ 1 2 3 4 5 2 4 2 5 3/ 4 1 2 Undirected graph

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9 Properties of Adjacency-List Representation Memory required = (V + E) Preferred when ◦ The graph is sparse: E << V 2 ◦ We need to quickly determine the nodes adjacent to a given node. Disadvantage ◦ No quick way to determine whether there is an edge between node u and v Time to determine if (u, v) E: ◦ O(degree(u)) Time to list all vertices adjacent to u: ◦ (degree(u)) 1 2 54 3 Undirected graph 1 2 34 Directed graph

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Graph Representation Adjacency matrix representation of G = (V, E) ◦ Assume vertices are numbered 1, 2, … V ◦ The representation consists of a matrix A V x V : ◦ a ij = 1 if (i, j) E 0 otherwise 1 2 54 3 Undirected graph 1 2 3 4 5 12345 011 0 0 11 110 11000 1 11 0 0 11 1 00 For undirected graphs, matrix A is symmetric: a ij = a ji A = A T

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Properties of Adjacency Matrix Representation Memory required ◦ (V 2 ), independent on the number of edges in G Preferred when ◦ The graph is dense: E is close to V 2 ◦ We need to quickly determine if there is an edge between two vertices Time to determine if (u, v) E (1) Disadvantage ◦ No quick way to list all of the vertices adjacent to a vertex Time to list all vertices adjacent to u (V) 1 2 54 3 Undirected graph 1 2 34 Directed graph

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Problem 1 Given an adjacency-list representation, how long does it take to compute the out-degree of every vertex? ◦ For each vertex u, search Adj[u] Θ(V+E) How about using an adjacency-matrix representation? Θ(V 2 ) 2 5/ 1 5 3 4/ 1 2 3 4 5 2 4 2 5 3/ 4 1 2

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Problem 2 How long does it take to compute the in-degree of every vertex? For each vertex u, search entire list of edges Θ(V+E) How long does it take to compute the in-degree of only one vertex? Θ(V+E) (unless a special data structure is used) 2 5/ 1 5 3 4/ 1 2 3 4 5 2 4 2 5 3/ 4 1 2

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14 Problem 3 The transpose of a graph G=(V,E) is the graph G T =(V,E T ), where E T ={(v,u) є V x V: (u,v) є E}. Thus, G T is G with all edges reversed. (a) Describe an efficient algorithm for computing G T from G, both for the adjacency-list and adjacency-matrix representations of G. (b) Analyze the running time of each algorithm.

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Problem 3 (cont’d) Adjacency matrix for (i=1; i ≤ V; i++) for(j=i+1; j ≤ V; j++) if(A[i][j] && !A[j][i]) { A[i][j]=0; A[j][i]=1; } O(V 2 ) complexity 011 0 0 00 110 10000 1 11 0 0 00 1 00 1 2 3 45 2 3 4 5 1

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Problem 3 (cont’d) Adjacency list Allocate V list pointers for G T (Adj’[]) for(i=1; i ≤ V, i++) for every vertex v in Adj[i] add vertex i to Adj’[v] O(V) O(E) Total time: O(V+E ) 2 5/ 1 5 3 4/ 1 2 3 4 5 2 4 2 5 3/ 4 1 2

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Problem 4 When adjacency-matrix representation is used, most graph algorithms require time Ω(V 2 ), but there are some exceptions. Show that determining whether a directed graph G contains a universal sink – a vertex of in-degree |V|-1 and out-degree 0 – can be determined in time O(V). Example: 000 0 1 10 100 01100 0 00 0 0 00 1 00 1 2 3 45 2 3 4 5 1

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Problem 4 (cont.) How many universal sinks could a graph have? ◦ 0 or 1 How can we determine whether a vertex i is a universal sink? ◦ The i th - row must contain 0’s only ◦ The i th - column must contain 1’s only (except at A[i][i]=0) Observations ◦ If A[i][j]=1, then i cannot be a universal sink ◦ If A[i][j]=0, and i j then j cannot be a universal sink Can you come up with an O(V) algorithm that checks if a universal sink exists in a graph ?

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Problem 4 (cont.) How long would it take to determine whether a given graph contains a universal sink if you were to check every single vertex in the graph? O(V 2 ) A SIMPLE ALGORITHM to check if vertex k is a UNIVERSAL SINK:

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Problem 4 (cont.) v1v1 v2v2 v5v5 v4v4 v3v3 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 v 1 v 2 v 3 v 4 v 5 v1v2v3v4v5v1v2v3v4v5 Loop terminates when i > |V| or j > |V| Upon termination, the only vertex that has potential to be a univ.sink is i Any vertex k < i can not be a sink Why? With the same reasoning, if i > |V|, there is no sink If i i can not be a universal sink Why? i

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Problem 4 (cont.) Why do we need this check? (see the last slide)

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Problem 4 (supl.) v1v1 v2v2 v5v5 v4v4 v3v3 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 v 1 v 2 v 3 v 4 v 5 v1v2v3v4v5v1v2v3v4v5 v1v1 v2v2 v5v5 v4v4 v3v3 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 v 1 v 2 v 3 v 4 v 5 v1v2v3v4v5v1v2v3v4v5

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