Wireless Expanders

This paper introduces an extended notion of expansion suitable for radio networks. A graph G = (V,E) is said to be an (αw, βw)-wireless expander if for every subset S ⊆ V s.t. |S| ≤ αw · |V |, there exists a subset S′ ⊆ S s.t. there are at least βw · |S| vertices in V \S that are 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 (α, β)-expander with maximum degree ∆ and β ≥ 1/∆ is also a (αw, βw) wireless expander for βw = Ω(β/ log(2·min{∆/β,∆·β})). Thus the wireless expansion can be smaller than the ordinary expansion by at most a factor that is logarithmic in min{∆/β,∆ ·β}, which, in turn, depends on the average degree rather than the maximum degree of the graph. In particular, for low arboricity graphs (such as planar graphs), the wireless expansion matches the ordinary expansion up to a constant factor. We complement this positive result by presenting an explicit construction of a “bad” (α, β)-expander for which the wireless expansion is βw = O(β/ log(2 ·min{∆/β,∆ · β}). We also analyze the theoretical properties of wireless expanders and their connection to unique neighbor expanders, and then demonstrate their applicability: Our results (both the positive and the negative) yield improved bounds for the spokesmen election problem that was introduced in the seminal paper of Chlamtac and Weinstein [7] 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 [11] for a lower bound on the broadcast time in radio networks. ∗The Weizmann Institute of Science. Email: [email protected]. †The Weizmann Institute of Science. Email: [email protected]. ‡The Weizmann Institute of Science. Email: [email protected]. §IBM Research. Email: [email protected]. ar X iv :1 80 2. 07 17 7v 1 [ cs .D S] 2 0 Fe b 20 18