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Article . 2018
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Wireless Expanders

Authors: Shirel Attali; Merav Parter; David Peleg; Shay Solomon;
Abstract

This paper introduces an extended notion of expansion suitable for radio networks. A graph $G=(V,E)$ is called an $(��_w, ��_w)$-{wireless expander} if for every subset $S \subseteq V$ s.t. $|S|\leq ��_w \cdot |V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $��_w \cdot |S|$ vertices in $V\backslash S$ adjacent in $G$ to exactly one vertex in $S'$. The main question we ask is the following: to what extent are ordinary expanders also good {wireless} expanders? We answer this question in a nearly tight manner. On the positive side, we show that any $(��, ��)$-expander with maximum degree $��$ and $��\geq 1/��$ is also a $(��_w, ��_w)$ wireless expander for $��_w = ��(��/ \log (2 \cdot \min\{��/ ��, ��\cdot ��\}))$. Thus the wireless expansion is smaller than the ordinary expansion by at most a factor logarithmic in $\min\{��/ ��, ��\cdot ��\}$, which depends on the graph \emph{average degree} rather than maximum degree; e.g., for low arboricity graphs, the wireless expansion matches the ordinary expansion up to a constant. We complement this positive result by presenting an explicit construction of a "bad" $(��, ��)$-expander for which the wireless expansion is $��_w = O(��/ \log (2 \cdot \min\{��/ ��, ��\cdot ��\})$. We also analyze the theoretical properties of wireless expanders and their connection to unique neighbor expanders, and demonstrate their applicability: Our results yield improved bounds for the {spokesmen election problem} that was introduced in the seminal paper of Chlamtac and Weinstein (1991) to devise efficient broadcasting for multihop radio networks. Our negative result yields a significantly simpler proof than that from the seminal paper of Kushilevitz and Mansour (1998) for a lower bound on the broadcast time in radio networks.

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FOS: Computer and information sciences, Computer Science - Data Structures and Algorithms, Data Structures and Algorithms (cs.DS)

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
Average
Average
Average
Green