Pred 8 1. Dist. Pred

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1 CS Graph Algorithms, Page Shortest Path Problem Task: given a graph G, find the shortest path from a vertex u to a vertex v. ffl If all edges are of the same length, then BFS will do. ffl But some times edges are not equal. We associate a non-negative integer, called weight, with each edge. The weight of a path is the sum of the weights of all edges in the path. We want to find a minimum-weight path from u to v. CS Graph Algorithms, Page Let G =(V; E) bea graph where V comprises a set of cities interconnected by a set of roads (edges) in E. ffl If the weight of a road (u; v) represents the distance from u to v, then a minimum weight x-y path is the shortest way to get to y from x. ffl If the weight of a road (u; v) represents the fee for using the road (u; v), then a minimum weight x-y path is the cheapest way to get to y from x.

2 CS Graph Algorithms, Page Dijkstra's Shortest Path Algorithm ffl Actually, it is a one-to-all shortest paths algorithm given a starting vertex u, the algorithm finds the shortest paths from u to all the other nodes that are reachable from u. ffl Surprisingly, the one-to-one shortest path problem is no easier than the one-to-all problem, meaning that Dijkstra's algorithm is probably among the best we could find. ffl Dijkstra's algorithm is specified by many important networking standards for the computation of paths to deliver computer communication messages. in the Internet, the OSPF protocol. CS Graph Algorithms, Page Example

3 CS Graph Algorithms, Page Algorithm. Initialize a dist array such that dist[start] = and dist[i] = for any vertex i other than start.. Initialize a boolean array, marked, such that marked[i] =false, for every vertex i.. Repeat the following steps until all vertices are marked. (a) Let u be the vertex, among the unmarked, with the smallest distance; mark u. (b) For all neighbors v of u, dist[v] =minfdist[v]; dist[u]+weight(u; v)g Complexity: O(N ), where N is the number of vertices in the graph. CS Graph Algorithms, Page Implementation Notes ffl How to find the smallest, unmarked vertex? sequential search in the dist array heap (modified Dijkstra's algorithm) ffl But the algorithm produces only the shortest distances, not shortest paths!

4 CS Graph Algorithms, Page The Predecessors Array Pred Whenever dist[v] issettodist[u] + weight(u; v), Pred[v] issetto u. CS Graph Algorithms, Page A Second Example Dist Pred

5 CS Graph Algorithms, Page Minimum Spanning Tree Problem Let G =(V; E) bea graph where V is a set of cities and E a set of planned, but not yet constructed, roads to connect cities in V. Let the weight of a road/edge be the cost to construct the road. Problem: using minimum cost, construct a set of roads to connect all the cities. CS Graph Algorithms, Page ffl For the minimum spanning tree problem, we focus on undirected graphs. ffl An undirected graph with N vertices is a tree if it is connected and contains N edges.

6 CS Graph Algorithms, Page ffl A spanning tree of a graph G =(V; E) isa graph T =(V; E ) such that E E and T is a tree. In English, a spanning tree of a graph G uses as less edges as possible to connect all the vertices in G. ffl A minimum spanning tree of a weighted graph G is a spanning tree whose total weight is minimum among all spanning trees of G. Total cost = Total cost = CS Graph Algorithms, Page Prim's Algorithm. Initialize a cost array such that cost[] = and cost[i] = for any vertex i other than.. Initialize a pred array such that pred[] =.. Repeat the following steps until all vertices are marked. (a) Let u be the vertex, among the unmarked, with the smallest cost; mark u. (b) For each neighbor v of u, cost[v] = minfcost[v]; weight(u; v)g (c) If cost[v] is set to weight(u; v) in the previous step, then pred[v] =u.

7 CS Graph Algorithms, Page Example Cost Pred CS Graph Algorithms, Page Representing Weighted Graphs ffl In the adjacency table representation, edges[u][v] = indicates no edge from u to v, and edges[u][v] =w indicates an u to v edge with weight w. ffl In the adjacency list representation, add a weight field to each list node.

8 CS Graph Algorithms, Page neighbor weight next CS Graph Algorithms, Page Directed Acyclic Graph (DAG) A DAG is a directed graph containing no cycles. C A D B E V = {A,B,C,D,E} E = {(A,C),(A,D),(A,E) (B,E),(C,B),(C,D), (E,D)}

9 CS Graph Algorithms, Page Examples Consider the graphs G, G, G, and G. Are they DAGs? ffl G : ffl G : ffl G : ffl G : CS Graph Algorithms, Page Topological Order on DAGS Suppose we have tasks and a DAG where T i! T j means that task T i must be complete before task T j can start. Some possibilities:!!!!!!!!!!!!!!!!!!!!!!!!!!! Problem: find an ordering of tasks, called topological order, that satisfies the restricions presented by the DAG.

10 CS Graph Algorithms, Page Topological Sort. vertices v, compute pred v, the number of tasks that vertex depends on (directly).. Find a vertex w where pred w =. Output w.. vertices x where w! x, decrement pred x. repeat steps and until no more vertices left in the graph. CS Graph Algorithms, Page Example

Quiz 2 CS 3510 October 22, 2004

Quiz 2 CS 3510 October 22, 2004 NAME : (Remember to fill in your name!) For grading purposes, please leave blank: (1) Short Answer (40 points) (2) Cycle Length (20 points) (3) Bottleneck (20 points) (4)