Finding Disjoint Paths in Expanders Deterministically and Online

We describe a deterministic, polynomial time algorithm for finding edge-disjoint paths connecting given pairs of vertices in an expander. Specifically, the input of the algorithm is a sufficiently strong d-regular expander G on n vertices, and a sequence of pairs s_i, t_i (1lesilesr) of vertices, where, r=Theta(nd log d/log n), and no vertex appears more than d/3 times in the list of all endpoints s1, t1,... ,s_r,t_r. The algorithm outputs edge-disjoint paths Q₁,...,Q_r, where Q_i connects s_i and t_i. The paths are constructed online, that is, the algorithm produces Q_i as soon as it gets s_i, t_i and before the next requests in the sequence are revealed. This improves in several respects a long list of previous algorithms for the above problem, whose study is motivated by the investigation of communication networks. An analogous result is established for vertex disjoint paths in blowups of strong expanders.