Short reachability networks

TL;DR

The paper addresses the problem of minimizing transposition sequences that realize t-reachability in the symmetric group, generalizing permutation networks. It proves the exact minimum length for $t=2$ as $\lceil 3n/2\rceil-2$ and presents a simple randomized construction achieving $(2+o_t(1))n$ transpositions for fixed $t\ge 3$, highlighting a near-linear regime distinct from full permutation networks. It also analyzes star-transposition restrictions, showing tight results for $2$-reachability up to parity and establishing a lower bound for $2$-uniformity proving a separation between reachability and uniformity in this setting, with extensions to general $t$. These results illuminate how randomization and structural constraints affect the efficiency of reachability in permutation networks, with implications for network design and combinatorial generation problems.

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 $(1, \cdot)$.

Figures (1)

• Figure 1: The multigraph corresponding to the minimum 2-reachable sequence given in the proof of Theorem \ref{['thm:2reach']}. All vertices except $1$ and $9$ have deficiency 0.

Theorems & Definitions (14)

• Theorem 1: waksman1968permutationgoldstein1967synthesisbeauquier2002arbitrary
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof : of Theorem \ref{['thm:2reach']}
• Claim 6
• proof
• proof : of Theorem \ref{['thm:treach']}
• proof : of Theorem \ref{['thm:2reachstar']}