# Algorithms for finding optimal disjoint paths around a rectangle

## Abstract

We give algorithms to find the optimal disjoint paths around a rectangle. The set of disjoint paths connects a set of *sources* to a set of *sinks* (no fixed pairing between the sources and sinks) on the boundary of a rectangle where either the *longest path length* or the *total path length* is minimized. One algorithm finds the set of disjoint paths with the longest path length minimized in *O*(*n* log *n*) time and the other finds the set of disjoint paths with the total path length minimized in *O*(*n*^{2}) time. In particular, if the sets of sources and sinks lie on a straight line, the set of disjoint paths with the minimum longest path length or minimum total path length can be found in *O*(*n*) or *O*(*n*^{2}) time respectively.

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