# A Faster Algorithm for Finding Disjoint Paths in Grids

## Abstract

Given a set of sources and a set of sinks in the two dimensional grid of size n, the *disjoint paths (DP) problem* is to connect every source to a distinct sink by a set of *edge-disjoint paths*. Let *v* be the total number of sources and sinks. In [3], Chan and Chin showed that without loss of generality we can assume *v* ≤ *n* ≤ 4*v* ^{2}. They also showed how to compress the grid optimally to a *dynamic network* (structure of the network may change depending on the paths found currently) of size \(
O(\sqrt {nv} )
\)
, and solve the problem in \(
O(\sqrt n v^{3/2} )
\)
time using augmenting path method in maximum flow. In this paper, we improve the time complexity of solving the DP problem to *O*(*n* ^{3/4} *v* ^{3/4}). The factor of improvement is as large as \(
\sqrt v
\)
when *n* is *⊝*(itv), while it is at least \(
\sqrt[4]{v}
\)
for *n* is *⊝*(*v* ^{2}).

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