Graphs. Graph G=(V,E): representation

Transcription

1 Graphs G = (V,E) V the vertices of the graph {v 1, v 2,..., v n } E the edges; E a subset of V x V A cost function cij is the cost/ weight of the edge (v i, v j ) Graph G=(V,E): representation 1. Adjacency Matrix a V x V matrix, with the [i,j] th entry representing the edge from the i th to the j th vertex

2 Graph G=(V,E): representation 1. Adjacency Matrix a V x V matrix, with the [i,j] th entry representing the edge from the i th to the j th vertex 2. Adjacency List an array of linked lists of length V, with the i th entry denoting the edges from the i th vertex Topological Sort Given directed acyclic graph (DAG) G=(V,E). List the vertices in V such that for each edge (i,j) in E, vertex i is listed before vertex j 2

3 Topological Sort Given directed acyclic graph (DAG) G=(V,E). List the vertices in V such that for each edge (i,j) in E, vertex i is listed before vertex j Shortest-path problems on graphs Input: a weighted graph G = (V,E), with cost function c Single-pair SP For specified vertices s and t, determine the shortest path the path of minimum total cost from s to t Example application: determining best route in traffic E.g., s = v 2, t = v 1 Path = v 2 à v 4 à v 3 à v 1 Path cost = 9 3

4 Shortest-path problems on graphs Input: a weighted graph G = (V,E), with cost function c Single-pair SP Single-source SP For specified vertices s and t, determine the shortest path the path of minimum total cost from s to t For a specified vertex s, determine shortest paths from s to all vertices Example application: a router on a network (In the worst case, single-pair SP requires the solution of single-source SP) E.g., s = v 2 Paths form a tree (why?) Path cost to each vertex should be computed Shortest-path problems on graphs Input: a weighted graph G = (V,E), with cost function c Single-pair SP Single-source SP All Pairs SP For specified vertices s and t, determine the shortest path the path of minimum total cost from s to t For a specified vertex s, determine shortest paths from s to all vertices Determine shortest paths between each pair of vertices Example application: what the airlines/ navigation services pre-compute We ll study the Warshall-Floyd Algorithm for the All-Pairs SP (APSP) problem 4

5 Shortest-path problems on graphs Do negative-cost edges matter? Shortest-path problems on graphs Do negative-cost edges matter? If there is a cycle with negative cost, shortest path may not be defined Otherwise, the problem is well-defined 5

6 APSP: The Warshall-Floyd algorithm Input: the graph in adjacency-matrix form: adj[ V x V ] Output: Two matrices dist[ V x V ]: dist[i][j] is the length of the SP from v i to v j path[ V x V ]: path[i][j] is some vertex on the SP from v i to v j APSP: The Warshall-Floyd algorithm Input: the graph in adjacency-matrix form: adj[ V x V ] Output: Two matrices dist[ V x V ]: dist[i][j] is the length of the SP from v i to v j path[ V x V ]: path[i][j] is some vertex on the SP from v i to v j dist path 6

7 APSP: The Warshall-Floyd algorithm Input: the graph in adjacency-matrix form: adj[ V x V ] Output: Two matrices dist[ V x V ]: dist[i][j] is the length of the SP from v i to v j path[ V x V ]: path[i][j] is some vertex on the SP from v i to v j Big-idea definition: dist (k) [i][j]: length of SP from v i to v j that only uses {v 1,v 2,, v k } as intermediate vertices dist (0) [i][j] = adj[i][j], we seek dist ( V ) [i][j] dist (k) [i][j] = min dist (k-1) [i][j] dist (k-1) [i][k] + dist (k-1) [k][j] APSP: The Warshall-Floyd algorithm public static void apsp(int[][] adj, int[][] dist, int[][] path){ } //Initialization int n = adj.length; //Notation for V for (int i=0; i<n; i++) for (int j=0; j<n; j++){ dist[i][j] = adj[i][j]; path[i][j] = i; //A default value } for (int k=0; k<n; k++) //Considering the k th vertex for (int i=0; i<n; i++) for (int j=0; j<n; j++) if (dist[i][k] + dist[k][j] < dist[i][j]) { dist[i][j] = dist[i][k] + dist[k][j]; path[i][j] = k; } 7

8 APSP: The Warshall-Floyd algorithm public static void apsp(int[][] adj, int[][] dist, int[][] path){ } //Initialization int n = adj.length; //Notation for V for (int i=0; i<n; i++) for (int j=0; j<n; j++){ dist[i][j] = adj[i][j]; path[i][j] = i; //A default value } for (int k=0; k<n; k++) //Considering the k th vertex for (int i=0; i<n; i++) for (int j=0; j<n; j++) if (dist[i][k] + dist[k][j] < dist[i][j]) { dist[i][j] = dist[i][k] + dist[k][j]; path[i][j] = k; } O( V 2 ) O( V 3 ) 8