Shortest-Paths Trees Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

Shortest-Paths Trees The objective is to find the set of edges connecting all nodes such that the sum of the edge lengths from the source to each node is minimized. In order to minimize the total path lengths, the path from the source to each node must be a shortest path connecting them.

Shortest-Paths Trees

Negative edges in an undirected graph

Directed graphs

Dijkstra's Algorithm

Choose a

Relax (a, b) and (a, g)

Choose b; Add (a, b) to T

Relax (b, c) and (b, d)

Choose g; Add (a, g) to T

Relax (g, e) and (g, h)

Choose d; Add (b, d) to T

Relax (d, e)

Choose h; Add (g, h) to T

Choose e; Add (d, e) to T

Relax (e, f)

Choose c; Add (b, c) to T

Relax (c, h)

Choose f; Add (e, f) to T

Relax (f, d) and (f, h)

The resulting SPT

Negative edge

Choose a

Choose b; Add (a, b) to T

Choose d; Add (b, d) to T

Choose c; Add (b, c) to T

Choose e; Add (d, e) to T

Something went wrong

A wrong SPT

A correct SPT

The Bellman-Ford Algorithm

δ[b] and δ[g] modified

δ[c], δ[d], δ[e] and δ[h] modified

δ[f] modified

δ[h] modified

A correct SPT

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