Short reachability networks

TL;DR

This paper explores the minimal length of t-reachability networks, generalizing permutation networks, and provides exact and asymptotic bounds for different values of t, including a special case for t=2.

Contribution

It establishes the shortest 2-reachability network length as approximately 1.5n and offers a randomized construction for t ≥ 3 with about 2n transpositions, extending understanding of reachability networks.

Findings

Shortest 2-reachability network length is approximately 1.5n.

For fixed t ≥ 3, t-reachability networks exist with about 2n transpositions.

Analysis of star-transpositions restrictions on reachability networks.

Abstract

We investigate the following generalisation of permutation networks. We say a sequence $T=(T_1,\dots,T_\ell)$ of transpositions in $S_n$ forms a $t$-reachability network if, for every choice of $t$ distinct points $x_1, \dots, x_t\in \{1,\dots,n\}$, there is a subsequence of $T$ whose composition maps $j$ to $x_j$ for every $1\leq j\leq t$. When $t=n$, any permutation in $S_n$ can be created and $T$ is a permutation network. Waksman [JACM, 1968] showed that the shortest permutation networks have length about $n \log_2(n)$. In this paper, we investigate the shortest $t$-reachability networks for other values of $t$. Our main result settles the case of $t=2$: the shortest $2$-reachability network has length $\lceil 3n/2\rceil-2 $. For fixed $t \geq 3$, we give a simple randomised construction which shows that there exist $t$-reachability networks with $(2+o_t(1))n$ transpositions. We also study the effect of restricting to star-transpositions, i.e. restricting all transpositions to have the form $(1, \cdot)$.