An algorithm to find the \(k\) shortest paths from node \(s\) to node \(t\) is presented.

In a digraph \(G\), we derive a subgraph \(T\) which is the shortest path tree (spanning tree) from every node to node \(t\). Then any edge \(e\) not in \(T\) is called the sidetrack edges. A path from \(s\) to \(t\), not necessarily the shortest path, can be uniquely identified by the set of sidetracked edges it contains: Given a set of sidetracked edges, a path is derived as follows: We begin from node \(s\), traverse along the path in \(T\). Whenever a sidetracked edge is encountered, we traverse it and switched to another branch, until node \(t\) is reached.

For every edge \(e\), we define a quantity \(\delta(e)=\ell(e)+d(e_{\textrm{tail}},t)-d(e_{\textrm{head}},t)\) to denote the extra distance traversed by sidetracking to edge \(e\) [note: the paper has this equation wrong]. Here, \(\ell(e)\) is the length of \(e\), and \(d(a,b)\) is the shortest distance between nodes \(a\) and \(b\) according to tree \(T\). If edge \(e\) is in \(T\), \(\delta(e)=0\). Then, the path \(p\) joining \(s\) to \(t\) has the distance of \(\ell(p)=d(s,t)+\sum_{e\in p}\delta(e)\)

The focus of this paper is to derive the complexity for enumeration of the k paths of minimum distance, by using a heap data structure.

## Bibliographic data

```
@article{
title = "Finding the k shortest paths",
author = "David Eppstein",
journal = "SIAM J. Computing",
volume = "28",
number = "2",
pages = "652--673",
year = "1998",
}
```