In this paper we study the problem of scheduling wireless links in the geometric SIN R model, which explicitly uses the fact that nodes are distributed in the Euclidean plane. We present the first NP-completeness proofs in such a model. In particular, we prove two problems to be NP-complete: Scheduling and One-Shot Scheduling. The first problem consists in finding a minimum-length schedule for a given set of links. The second problem receives a weighted set of links as input and consists in finding a maximum-weight subset of links to be scheduled simultaneously in one shot. In addition to the complexity proofs, we devise an approximation algorithm for each problem.

The topology of a wireless multi-hop network can be controlled by varying the transmission power at each node. In this paper, we give a detailed analysis of a cone-based distributed topology control algorithm. This algorithm, introduced in [16], does not assume that nodes have GPS information available; rather it depends only on directional information. Roughly speaking, the basic idea of the algorithm is that a node u transmits with the minimum power pu,α required to ensure that in every cone of degree α around u, there is some node that u can reach with power pu,α. We show that taking α = 5π/6 is a necessary and sufficient condition to guarantee that network connectivity is preserved. More precisely, if there is a path from s to t when every node communicates at maximum power then, if α ≤ 5π/6, there is still a path in the smallest symmetric graph Gα containing all edges (u, v) such that u can communicate with v using power pu,α. On the other hand, if α > 5π/6, connectivity is not necessarily preserved. We also propose a set of optimizations that further reduce power consumption and prove that they retain network connectivity. Dynamic reconfiguration in the presence of failures and mobility is also discussed. Simulation results are presented to demonstrate the effectiveness of the algorithm and the optimizations.

Topology control in ad-hoc networks tries to lower node energy consumption by reducing transmission power and by confining interference, collisions and consequently retransmissions. Commonly low interference is claimed to be a consequence to sparseness of the resulting topology. In this paper we disprove this implication. In contrast to most of the related work-claiming to solve the interference issue by graph sparseness without providing clear argumentation or proofs-, we provide a concise and intuitive definition of interference. Based on this definition we show that most currently proposed topology control algorithms do not effectively constrain interference. Furthermore we propose connectivity-preserving and spanner constructions that are interference-minimal.

In this paper we present GOAFR, a new geometric ad-hoc routing algorithm combining greedy and face routing. We evaluate this algorithm by both rigorous analysis and comprehensive simulation. GOAFR is the first ad-hoc algorithm to be both asymptotically optimal and average-case efficient. For our simulations we identify a network density range critical for any routing algorithm. We study a dozen of routing algorithms and show that GOAFR outperforms other prominent algorithms, such as GPSR or AFR.

We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(n c/k 2 /k) and Ω(∆ 1/k /k) for some constant c, where n and ∆ denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at least Ω(log n/ log log n) and Ω(log ∆/ log log ∆). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

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This paper examines the role inter-domain topology and routing policy play in the process of delayed Internet routing convergence. In recent work, we showed that the Internet lacks effective inter-domain path fail-over. Unlike circuit-switched networks which exhibit fail-over on the order of milliseconds, we found Internet backbone routers may take tens of minutes to reach a consistent view of the network topology after a fault. In this paper, we expand on our earlier work by exploring the impact of specific Internet provider policies and topologies on the speed of routing convergence. Based on data from the experimental injection and measurement of several hundred thousand inter-domain routing faults, we show that the time for end-to-end Internet convergence depends on the length of the longest possible backup autonomous system path between a source and destination node. We also demonstrate significant variation in the convergence behaviors of Internet service providers, with the larger providers exhibiting the fastest convergence latencies. Finally, we discuss possible modifications to BGP and provider routing policies which if deployed, would improve interdomain routing convergence.

Finding a small dominating set is one of the most fundamental problems of classical graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary, possibly constant parameter k and maximum node degree ∆, our algorithm computes a dominating set of expected size O k∆ 2/k log(∆)|DS OPT | in O k 2 rounds. Each node has to send O k 2 ∆ messages of size O(log ∆). This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.