Finding Disjoint Paths on Directed Acyclic Graphs

Given k+1 pairs of vertices (s_1,s_2),(u_1,v_1),...,(u_k,v_k) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (u_l,v_l) with 1<=l<=k, a tuple (s_1,t_1,s_2,t_2) with {t_1,t_2}={u_l,v_l} in constant time such that there are two disjoint paths p_1, from s_1 to t_1, and p_2, from s_2 to t_2, 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(log(n)/log(2+m/n))+n*log³(n)). In this problem, given four vertices s_1, s_2, t_1, and t_2, we want to construct two disjoint paths p_1, from s_1 to t_1, and p_2, from s_2 to t_2, if such paths exist