Shortest-path tree

Given a connected, undirected graph G, a shortest-path tree rooted at vertex v is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G.

In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm:

The above algorithm guarantees the existence of shortest-path trees. Like minimum spanning trees, shortest-path trees in general are not unique.

In graphs for which all edges weights equal one, shortest path trees coincide with breadth-first search trees.

In graphs that have negative cycles, the set of shortest simple paths from v to all other vertices do not necessarily form a tree.

Cahn, Robert S. Wide Area Network Design. 

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