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Weighted Graph Algorithms

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 edge-weight function w: E R.

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Weighted Graph Algorithms

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

  2. Definition of a path • Consider a directed graph G=(V, E) with edge-weight function w: ER. • Weight of edge v1v2: w(v1,v2) • The weight of a path p=v1v2… vk is defined to be w(p)= sum w(vi, vi+1), i=1..k-1

  3. 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

  4. 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.

  5. Dijkstra’s algorithm

  6. Running time analysis

  7. Running time analysis

  8. Unweighted shortest path problem • Dijkstra with negative weights

  9. Unweighted shortest path problem • Dijkstra with negative weights

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