The k-shortest paths computes the k first shortest path between two vertices in a directed graph. If you put this in the context of route planning, it gives you k alternative routes in case the shortest path is blocked by snow :) While the single source shortest path is well-known and implemented in many languages, there are not many implementations available for this problem, although it has been extensively studied. After looking around, I stumbled on a nice article comparing various approaches. The authors pointed out an algorithm from 1959 from Hoffman and Pavley that solved this problem (there are actually many others). This algorithm looked like a good fit:

• it requires a single call to a single-source shortest path algorithm. Other approaches will require as much as kn calls to a shortest path algorithm.
• it sounded simple and did not require new specialized data structures.

I want to try it!

The algorithm is available in QuickGraph 3.1.40104.00 and up. You can take a look at it at http://www.codeplex.com/quickgraph/SourceControl/changeset/view/29858#377982. To use it on a BidirectionalGraph,

IBidirectionalGraph<TVertex, TEdge> g = …;
foreach(IEnumerable<TEdge> path in g.RankedShortestPathHoffmanPavley(weights, source, goal, 4))
     …

A glimpse at the Hoffmann-Pavley algorithm

The algorithm works in 2 phases.

On the first phase, we build a minimum ‘successor’ tree towards the goal vertex. This tree can be used to build a shortest path from any (reachable) vertex to the goal vertex. To build this tree, we can simply apply the Dijkstra shortest path algorithm on the reversed graph. This can be done in a couple lines with QuickGraph.

As for many other k-shortest path algorithm, phase works by building deviation path and picking the best one. In the case of the Hoffman-Pavley algorithm, it works as follows: pick the latest shortest path, for each vertex of this path, build a deviation path (more later) for each out-edge and add it to a priority queue. Then start again:

var queue = new PriorityQueue<Path>();
queue.Enqueue(shortestpath);
while(queue.Count > 0) {
    var path = queue.Dequeue();
    foreach(var vertex in path)
        foreach(var edge in graph.OutEdges(vertex))
             queue.Enqueue(CreateDeviation(path, vertex, edge));
}

A deviation path is composed of three parts:

1. the initial part of the ‘seeding’ path, i.e. the edges before the deviation edge,
2. the deviation edge
3. the remaining shortest path to the goal. When we build the deviation path, we also compute it’s weight. A nice property of the deviation paths is that they can be ‘built’ when needed. This saves a lot of space and computation as most deviation paths will probably not end up in the winning set of paths – instead of storing a full path, we store a path index and an edge index.

That’s it! The details contain more code to deal with self-edges and loops but the main idea is there. This is definitely a very elegant algorithm!

What about testing!

Algorithm authors usually illustrate their approach with an example. This is a good way to get started on a small graph example and ensure that the algorithm works as the original author expected. This is the kind of unit test I get started with.

The next step is to apply the algorithm to a large number of graph instances. Unfortunately, I do not have any other k-shortest path algorithm, so the oracle is harder to build here. Nevertheless, the result of the algorithm, i.e. the shortest path collection, has a couple of properties that should always be true:

• the algorithm produces loopless paths,
• path k is lower or equal to path k + 1.

The problem with this test is that it does not guarantee that some shortest path have been missed. At this point, I’m a bit puzzled on how to test that.