Finding Disjoint Paths on Directed Acyclic Graphs

Given k + 1 pairs of vertices (s1,s2),(u1,v1),...,(uk,vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul,vl) with 1 ≤ l≤ k, a tuple (s1,t1,s2,t2) with {t1,t2} = {ul,vl} in constant time such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. Disjoint can mean vertex- as well as edge-disjoint. As an application we show that the presented data structure can be used to improve the previous best known running time O(mn) for the so called 2-disjoint paths problem on directed acyclic graphs to O(m(log2+m/nn) + nlog3n). In this problem, given a tuple (s1,s2,t1,t2) of four vertices, we want to construct two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such paths exist.