On Permutation Selectors and their Applications in Ad-Hoc Radio Networks 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 ${ \left\{ {0,1,...,N-1} \right\} }$ 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.

Research supported by NSF grant CCF-2153723.