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.

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

  • Jordan Kuschner,
  • Yugarshi Shashwat,
  • Sarthak Yadav,
  • Marek Chrobak

摘要

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.