Shortest-path tree

A shortest-path tree rooted at a source vertex is a spanning tree whose unique tree route from the source to every vertex solves the shortest path problem in the original weighted graph. Dijkstra's algorithm constructs one for a finite connected graph with nonnegative edge lengths, recording a predecessor whenever a distance improves. It need not be a minimum spanning tree: one minimizes each source-route length, the other the total weight of the tree's edges.

In a finite connected graph with strictly positive edge weights, choose any shortest path from a root $v$ to every other vertex $u$, and let $p(u)$ be its penultimate vertex. Every prefix is shortest: a better prefix followed by the final edge would give a cheaper walk, whose positive-weight cycles can be erased to give a cheaper path. Thus the displayed distance identity holds.

Following predecessors strictly decreases the shortest-path distance, so no predecessor cycle is possible. Since every nonroot vertex has a predecessor, every chain reaches the root. Opposite predecessor choices cannot select the same undirected edge, again by strict distance decrease. The selected graph is connected and has exactly one edge per nonroot vertex, so is a spanning tree. Along its root paths the distance differences telescope, proving that it is a shortest-path tree even if the original shortest paths were chosen independently.

In a triangle with root $v$, assign $w(va)=w(vb)=2$ and $w(ab)=-1$. Among graph paths, the unique shortest path from $v$ to $a$ is $v,b,a$, and that from $v$ to $b$ is $v,a,b$, both of weight one. A shortest-path tree would have to contain all three edges, impossible for a tree. The example uses shortest simple paths; arbitrary walks can have unboundedly negative weight by repeatedly traversing the negative edge.