Topology Control to Simultaneously Achieve Near-Optimal Node Degree and Low Path Stretch in Ad hoc Networks

Abstract

Our objective in this paper is to design topology control algorithms such that (i) nodes have low degree and (ii) paths in the network have few hops. Low node degree is desirable in networks equipped with smart antennas and to reduce access contention. Short paths are desirable for minimizing communication delays and for better robustness to channel impairments and to mobility. Given any arbitrary unit-disc graph G representing all feasible links, our algorithms find a sparse subgraph G' having a maximum node degree of six and, for each pair of vertices u, v, having hops <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G'</sub> (u, v) = O(hops <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (u,v) + logDelta), where Delta is the maximum node degree in G and hops <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (u, v) denotes the shortest path length from u to v in G. This result is near-optimal: (i) there is a connected UDG G in which no connected subgraph has degree less than five, and (ii) for any graph G, any bounded-degree subgraph G' must have hops <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G'</sub> (u, v) = Omega(hops <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (u, v) + logDelta) for some u, v. Our distributed algorithm scales, preserves link symmetry, does not need node synchronization, and requires only O(n) messages. We perform extensive simulations that quantify the performance of our algorithm in realistic scenarios