 
Summary: Finding disjoint paths in expanders deterministically and online
Noga Alon
Michael Capalbo
October 14, 2007
Abstract
We describe a deterministic, polynomial time algorithm for finding edgedisjoint paths connect
ing given pairs of vertices in an expander. Specifically, the input of the algorithm is a sufficiently
strong dregular expander G on n vertices, and a sequence of pairs si, ti, (1 i r) of vertices,
where r = (nd log d
log n ), and no vertex appears more than d/3 times in the list of all endpoints
s1, t1, . . . , sr, tr. The algorithm outputs edgedisjoint paths Q1, . . . , Qr, where Qi connects si and
ti. The paths are constructed online, that is, the algorithm produces Qi as soon as it gets si, ti
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.
1 Introduction
Given an undirected graph G = (V, E) and a set of pairs si, ti, 1 i r, we are interested in finding
r edgedisjoint paths Q1, . . . , Qr, where Qi connects si and ti. This problem received a consider
able amount of attention, motivated by a variety of communication contexts including the study of
