On Permutation Selectors and their Applications in Ad-Hoc Radio Networks Protocols
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| Format: | Preprint |
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2024
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| _version_ | 1866910334187470848 |
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| author | Kuschner, Jordan Shashwat, Yugarshi Yadav, Sarthak Chrobak, Marek |
| author_facet | Kuschner, Jordan Shashwat, Yugarshi Yadav, Sarthak Chrobak, Marek |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Permutation Selectors and their Applications in Ad-Hoc Radio Networks Protocols Kuschner, Jordan Shashwat, Yugarshi Yadav, Sarthak Chrobak, Marek Data Structures and Algorithms Computational Complexity 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. |
| title | On Permutation Selectors and their Applications in Ad-Hoc Radio Networks Protocols |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2402.10783 |