# Disjoint Path Finding

The algorithms to find a shortest path in a graph are well-documented, but the algorithms for finding multiple disjoint (non-overlapping) paths are not so well documented. This is an attempt to explain two of them, Suurballe's algorithm (for finding node-disjoint paths) Bhandari's algorithm (for finding edge-disjoint paths) by means of an example.

I assume that the reader is familiar with algorithms to find a single path, such as:

- Edsgar W. Dijkstra's algorithm (or the A*/Astar algorithm which speeds it up by taking the topological distance to the destination in consideration while sorting the nodes in the queue).
- Bellman-Ford algorithm, which is a variant of Dijkstra's algorithm to allow for edges with negative weight.

## Contents

## Bhandari

Ramesh Bhandari published the following algorithm in his book “Survivable Networks: Algorithms for Diverse Routing” (Kluwer Academic Publishers, 1999). It's a simple variant of Suurballe's algorithm and find 2 or more edge-disjoint paths through a graph. The sum of the costs of all paths is minimum.

### Example

Problem: find the two shortest edge-disjoints paths between A and H in this graph:

#### Step 1

Find the (directed) shorest path p1. (e.g. with Dijkstra's algorithm)

#### Step 2

Make the graph directed (if it wasn't) and replace the edges from the found path with inverse edges with negative costs.

#### Step 3

Find the shortest path p2 in the new graph. Use a modified Dijkstra algorithm such as Bellman-Ford to allow for negative edge weights.

Repeat steps 2 and 3 as necessary (for *k*-shortest paths)

#### Step 4

Add all shortest paths in the graph.

#### Step 5

Remove all edges which are used in both directions. (here: edge E-D)

You have now 2 (or k) disjoint shortest paths.