On Permutation Selectors and their Applications in Ad-Hoc Radio Networks Protocols

TL;DR

This paper introduces $(k,N)$-permutation selectors, a order-aware variant of strong $(k,N)$-selectors, and shows they can be constructed probabilistically with size $m = O(k^2\log N)$. This matches the asymptotic bound for strong selectors while enabling isolation of each $k$-subset in any specified order, which is particularly advantageous for multi-hop information dissemination in ad-hoc radio networks. The authors prove a main result via a randomized construction and coupon-collection–style analysis, culminating in a practical improvement: gossiping in ad-hoc networks can be achieved in time $O(n^{4/3}\log^2 n(\log\log n)^{2/3})$ when $N$ is polynomial in $n$. They also extend the framework to $(k,q,N)$-permutation selectors with size $O(kq\log N)$, and discuss broader connections to related selector families and network protocols.

Abstract

Selective families of sets, or selectors, are combinatorial tools used to "isolate" individual members of sets from some set family. Given a set $X$ and an element $x\in X$, to isolate $x$ from $X$, at least one of the sets in the selector must intersect $X$ on exactly $x$. We study (k,N)-permutation selectors which have the property that they can isolate each element of each $k$-element subset of $\{0,1,...,N-1\}$ in each possible order. These selectors can be used in protocols for ad-hoc radio networks to more efficiently disseminate information along multiple hops. In 2004, Gasieniec, Radzik and Xin gave a construction of a (k,N)-permutation selector of size $O(k^2\log^3 N)$. This paper improves this by providing a probabilistic construction of a (k,N)-permutation selector of size $O(k^2\log N)$. Remarkably, this matches the asymptotic bound for standard strong (k,N)-selectors, that isolate each element of each set of size $k$, but with no restriction on the order. We then show that the use of our (k,N)-permutation selector improves the best running time for gossiping in ad-hoc radio networks by a poly-logarithmic factor.

Figures (2)

• Figure 1: On the left, an illustration of MAC contention resolution, with $N=9$ and $k=4$. Active devices are marked with darker colors. On the right, a node $v$ in a radio network, with label $9$ and four in-neighbors.
• Figure 2: An example of a path and its in-neighborhood, with nodes identified by their labels. The path is $P = 30,11,22,42$ and its neighborhood (shaded) has $9$ nodes.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• proof