Weighted Graph Algorithms Weighted shortest path problem Dijkstras

Weighted Graph Algorithms • Weighted shortest path problem – Dijkstra’s algorithm (single-source, non-negative edge weight) • All pairs shortest path – Warshall’s algorithm

Definition of a path • Consider a directed graph G=(V, E) with edgeweight function w: E R. – Weight of edge v 1 v 2: w(v 1, v 2) • The weight of a path p=v 1 v 2 … vk is defined to be w(p)= sum w(vi, vi+1), i=1. . k-1

Shortest path • A shortest path from u to v is a path of minimum weight from u to v – There might be multiple paths from u to v – The shortest path weight from u to v is defined as –

Single source shortest path • Problem: from a given source vertex s, find the shortest path weights from s to all vertices in V. – If all edge weights w(u, v) are non-negative, all shortest path weights must exist • Idea: greedy – Maintain a set S of vertices whose shortest path weights from s are known. – At each step, add to S the vertex v, whose distance estimate from s is minimal – Update the distance estimates of all vertices adjacent to v.

Dijkstra’s algorithm

Running time analysis

• Unweighted shortest path problem • Dijkstra with negative weights