Routing permutations on spectral expanders via matchings

TL;DR

This paper demonstrates that on strong spectral expanders, any permutation routing can be achieved in logarithmic rounds using matchings, solving a longstanding problem in graph routing efficiency.

Contribution

It proves that spectral expanders enable permutation routing in O(log n) rounds, resolving a problem posed in 1994 and establishing optimal bounds for constant degree graphs.

Findings

Permutation routing on spectral expanders takes O(log n) rounds.

This routing efficiency is optimal for constant degree graphs.

The result applies to a broad class of regular graphs with spectral expansion properties.

Abstract

We consider the following matching-based routing problem. Initially, each vertex $v$ of a connected graph $G$ is occupied by a pebble which has a unique destination $\pi(v)$. In each round the pebbles across the edges of a selected matching in $G$ are swapped, and the goal is to route each pebble to its destination vertex in as few rounds as possible. We show that if $G$ is a sufficiently strong $d$-regular spectral expander then any permutation $\pi$ can be achieved in $O(\log n)$ rounds. This is optimal for constant $d$ and resolves a problem of Alon, Chung, and Graham [SIAM J. Discrete Math., 7 (1994), pp. 516--530].