Optimal Geographic Routing for Wireless Networks with Near-Arbitrary Holes and Traffic Sundar Subramanian and Sanjay Shakkottai Department of ECE The University of Texas at Austin Email: {ssubrama, shakkott}@ece.utexas.edu Piyush Gupta Complex Systems Analysis & Optimization Department Bell Labs, Alcatel-Lucent Email: pgupta@research.bell-labs.com Abstract—We consider the problem of throughput-optimal routing over large-scale wireless ad-hoc networks. Gupta and Kumar (2000) showed that a throughput capacity (a uniform rate over all source-destination pairs) of Θ( 1 √ n log n ) is achievable in random planar networks, and the capacity is achieved by straight-line routes. In reality, both the network model and the traffic demands are likely to be highly non-uniform. In this paper, we first propose a randomized forwarding strategy based on geographic routing that achieves near-optimal throughput over random planar networks with an arbitrary number of routing holes (regions devoid of nodes) of varying sizes. Next, we study a random planar network with arbitrary source-destination pairs with arbitrary traffic demands. For such networks, we demon- strate a randomized local load-balancing algorithm that supports any traffic load that is within a poly-logarithmic factor of the throughput region. Our algorithms are based on geographic routing and hence inherit their advantageous properties of low- complexity, robustness and stability. I. I NTRODUCTION We study the problem of throughput-optimal routing in large wireless networks such as ad-hoc and sensor networks. In such large networks, there is a need for scalable, low-complexity and distributed routing algorithms that can provide good data rates for the traffic flows. The work in [8], [6] has shown that a throughput-capacity of Θ(  1 n ) is achievable in uniform networks with uniform traffic demands, and that the capacity achieving routes are straight-line paths. In many practical networks, both the network and the traffic distribution may be highly non uniform. Non-uniformities may arise due to factors such as network holes (regions devoid of any living nodes), the arbitrary locations of source-destination pairs or due to variations in the required data rates. In recent studies [9], [10], [17], [11], [5], geographic forwarding based protocols have been suggested as a stable routing (providing fixed routes that do not flip) technique over large non-uniform networks as they are scalable, low- complexity and highly distributed. However, in recent work [19], it was demonstrated that network non-uniformities can cause significant losses in throughput (rates could be as low as Θ(1/n)) while employing such schemes. A critical issue is that conventional “shortest-path” (such as straight-line) routes are oblivious to the distribution of other routes (between other source-destination pairs) and may cause heavy losses in throughput due to spatial congestion. Typically, throughput optimal routing schemes over non- uniform networks and traffic demands are based on (i) solving a global optimization problem [1] (setting up routes such that the traffic balanced over the wireless links) or (ii) adaptive schemes [21], [18] that converge to an optimal set of routes (or a per-packet route) over time. While global optimization requires co-ordination and heavy computation by the nodes, adaptive schemes may take a long time to converge to good paths and also have issues of stability. In this paper, we are interested in developing distributed routing algorithms that are “near-optimal” (close to the rates obtained by a global optimization) over non-uniform networks with arbitrary traffic demands, but are still low-complexity, distributed and stable. A. Main Contributions We consider a random planar network with n nodes arbi- trarily distributed over a unit region, with each node having a uniform circular radio range of M (n)= C  log n n , for any C> 1 √ π . This scaling ensures that the resultant graph is connected [7]. (a) We first consider non-uniform networks with large number of routing holes and n/2 uniformly randomly distributed source-destination pairs. In contrast to earlier work [19], with finite number of holes of constant area, we allow for an arbitrary number of holes of varying sizes. Over such networks, we demonstrate that a near-optimal throughput capacity of Θ(  1 n ) is achiev- able (up to poly-logarithmic factors) by our algorithm RandHT(n). Unlike the RANDOMWAY algorithm [19], the new algorithm does not overload the network with increasing number of holes, and is also oblivious to the number of holes in the network. (b) Next, we consider networks with an arbitrary number of source-destination pairs with arbitrary locations and varying rate requirements. We assume that the net- work however has no routing holes. Conventionally, cut- set bounds (amount of traffic that can enter/leave the boundary of any sub-region of the network) have been used to characterize upper bounds on network capacity [14], [12]. However, when sources and sinks can be arbitrarily close or far, and with widely varying traffic