External Memory Algorithms for Finding Disjoint Paths in Undirected Graphs

Abstract

Consider the following well-known combinatorial problem: given an undirected graph $G = (V,E)$, terminals $s, t \in V$, and an integer $k \ge 1$, find k edge-disjoint s–t paths in G or report that such paths do not exist.

We study this problem in the external memory (EM) model of Agrawal and Vitter, i.e. assume that only M words of random access memory (RAM) are available while the graph resides in EM, which enables reading and writing contiguous blocks of B words per single I/O. The latter external memory is also used for storing the output and some intermediate data.

For $k = 1$, the problem consists in finding a single s–t path in an undirected graph and can be solved in $Conn(V,E) = O\left( \frac{V+E}{V}\right. \left. Sort(V) \log \log \frac{VB}{E}\right)$ I/Os, where $Sort(N) = O\left( \frac{N}{B}\log _{M/B} \frac{N}{B}\right)$ is the complexity of sorting N words in external memory.

Our contribution is two novel EM algorithms that solve the problem for $k \le \frac{M}{B}$. The first takes $O(k \cdot Conn(V,E))$ I/Os. The second one applies the ideas of Ibaraki–Nagamochi sparse connectivity certificates and takes $O\left( (Sort(V+E) + k \cdot Conn(V,kV)) \cdot \log \frac{V}{M}\right)$ I/Os, which improves upon the first bound for sufficiently dense graphs.

Both algorithms outperform the naive approach based on successive BFS- or DFS-augmentations for a wide range of parameters $| V |$, $| E |$, M, B.