Suurballe's algorithm

Let G be a weighted directed graph with vertex set V and edge set E (figure A); let s be a designated source vertex in G, and let t be a designated destination vertex. Let each edge (u,v) in E, from vertex u to vertex v, have a non-negative cost w(u,v).

Define d(s,u) to be the cost of the shortest path to vertex u from vertex s in the shortest path tree rooted at s (figure C).

Note: Node and Vertex are often used interchangeably.

Suurballe's algorithm performs the following steps:

1. Find the shortest path tree T rooted at node s by running Dijkstra's algorithm (figure C). This tree contains for every vertex u, a shortest path from s to u. Let P₁ be the shortest cost path from s to t (figure B). The edges in T are called tree edges and the remaining edges (the edges missing from figure C) are called non-tree edges.
2. Modify the cost of each edge in the graph by replacing the cost w(u,v) of every edge (u,v) by w′(u,v) = w(u,v) − d(s,v) + d(s,u). According to the resulting modified cost function, all tree edges have a cost of 0, and non-tree edges have a non-negative cost. For example:
   If u=B, v=E, then w′(u,v) = w(B,E) − d(A,E) + d(A,B) = 2 − 3 + 1 = 0
   If u=E, v=B, then w′(u,v) = w(E,B) − d(A,B) + d(A,E) = 2 − 1 + 3 = 4
3. Create a residual graph G_t formed from G by removing the edges of G on path P₁ that are directed into s and then reverse the direction of the zero length edges along path P₁ (figure D).
4. Find the shortest path P₂ in the residual graph G_t by running Dijkstra's algorithm (figure E).
5. Discard the reversed edges of P₂ from both paths. The remaining edges of P₁ and P₂ form a subgraph with two outgoing edges at s, two incoming edges at t, and one incoming and one outgoing edge at each remaining vertex. Therefore, this subgraph consists of two edge-disjoint paths from s to t and possibly some additional (zero-length) cycles. Return the two disjoint paths from the subgraph.

The following example shows how Suurballe's algorithm finds the shortest pair of disjoint paths from A to F.

Figure A illustrates a weighted graph G.

Figure B calculates the shortest path P₁ from A to F (A–B–D–F).

Figure C illustrates the shortest path tree T rooted at A, and the computed distances from A to every vertex (u).

Figure D shows the residual graph G_t with the updated cost of each edge and the edges of path P₁ reversed.

Figure E calculates path P₂ in the residual graph G_t (A–C–D–B–E–F).

Figure F illustrates both path P₁ and path P₂.

Figure G finds the shortest pair of disjoint paths by combining the edges of paths P₁ and P₂ and then discarding the common reversed edges between both paths (B–D). As a result, we get the two shortest pair of disjoint paths (A–B–E–F) and (A–C–D–F).

The weight of any path from s to t in the modified system of weights equals the weight in the original graph, minus d(s,t). Therefore, the shortest two disjoint paths under the modified weights are the same paths as the shortest two paths in the original graph, although they have different weights.

Suurballe's algorithm may be seen as a special case of the successive shortest paths method for finding a minimum cost flow with total flow amount two from s to t. The modification to the weights does not affect the sequence of paths found by this method, only their weights. Therefore, the correctness of the algorithm follows from the correctness of the successive shortest paths method.

This algorithm requires two iterations of Dijkstra's algorithm. Using Fibonacci heaps, both iterations can be performed in time ${\displaystyle O(|E|+|V|\log |V|)}$ where ${\displaystyle |V|}$ and ${\displaystyle |E|}$ are the number of vertices and edges respectively. Therefore, the same time bound applies to Suurballe's algorithm.

The version of Suurballe's algorithm as described above finds paths that have disjoint edges, but that may share vertices. It is possible to use the same algorithm to find vertex-disjoint paths, by replacing each vertex by a pair of adjacent vertices, one with all of the incoming adjacencies u-in of the original vertex, and one with all of the outgoing adjacencies u-out. Two edge-disjoint paths in this modified graph necessarily correspond to two vertex-disjoint paths in the original graph, and vice versa, so applying Suurballe's algorithm to the modified graph results in the construction of two vertex-disjoint paths in the original graph. Suurballe's original 1974 algorithm was for the vertex-disjoint version of the problem, and was extended in 1984 by Suurballe and Tarjan to the edge-disjoint version.[3]

By using a modified version of Dijkstra's algorithm that simultaneously computes the distances to each vertex t in the graphs G_t, it is also possible to find the total lengths of the shortest pairs of paths from a given source vertex s to every other vertex in the graph, in an amount of time that is proportional to a single instance of Dijkstra's algorithm.

Note: The pair of adjacent vertices resulting from the split are connected by a zero cost uni-directional edge from the incoming to outgoing vertex. The source vertex becomes s-out and the destination vertex becomes t-in.

• Edge disjoint shortest pair algorithm