Tradeo between stretch factor and load balancing ratio in wireless network routing

A wireless multi-hop network consists of a set of nodes in the plane where two nodes can directly communicate to each other if their distance is at most 1. We consider two quality measures for routing in wireless networks. One is the stretch factor of the paths used by a routing algorithm, and the other is the load balancing ratio, which measures how evenly the traffic is distributed. In this paper, we show a trade-off between the two measures dependent on the density of the point set. When the maximum density of the point set is ρ, the optimal algorithm by using only paths with stretch factor no more than c generates the maximum load on the nodes at most O(min( √ ρn/c, n/c)) times that of the optimal algorithm without path length restriction. In particular, when the density is bounded by a constant, the shortest path routing has an approximation ratio of O( √ n) compared to the optimal algorithm. This bound is tight in the worst case. The result can be extended to k-dimensional unit-ball graphs and graphs with “growth rate” k. We also present a tradeoff when considering the average density ρ̄. We show a bound of O( √ ρ̄n log n) and Ω( √ ρ̄n/ log c) for this case. We also discuss issues such as algorithms for computing load-balanced short path routing and load-balanced routing in spanner graphs.