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arXiv · 2208.06630

Short reachability networks

Carla Groenland·Tom Johnston·Jamie Radcliffe·Alex Scott
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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 $(1, \cdot)$.

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Carla Groenland, Tom Johnston, Jamie Radcliffe, Alex Scott. 2022-08-13. Short reachability networks. https://doi.org/10.46298/dmtcs.12454

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