Approximation algorithms for distributed systems

Distributed Approximation is a new and rapidly developing discipline that lies at the crossroads of various well-established areas of Computer Science - Distributed Computing, Approximation Algorithms, Graph Theory and often, Computational Geometry. This thesis focuses on the design and analysis of distributed algorithms to solve optimization problems that usually arise in large-scale, heavily dynamic, resource constrained networks, e.g. wireless ad-hoc and sensor networks, P2P systems, mobile networks etc. These problems can often be abstracted by variations of well-known combinatorial optimization problems, such as topology control, clustering etc. Many of these problems are known to be hard (NP-complete). But we need fast and light-weight distributed algorithms for these problems, that yield near-optimal solutions.

The results presented in this thesis can be broadly divided in two parts. The first part contains a set of results that obtain improved solutions to the classic problem of computing a sparse "backbone" for Wireless Sensor Networks (WSNs). In graph-theoretic terms, the goal is to compute a spanning subgraph of the input graph, that is sparse, lightweight and has low stretch. The term "low stretch" indicates that in spite of dropping many edges, the distance between any two nodes in the graph is not increased by much. We model WSNs as geometric graphs - unit ball graphs, quasi-unit ball graphs etc. in Euclidean spaces, as well as in more general metric spaces of low doubling dimension. We identify and exploit a variety of geometric features of those models to obtain our results.

In the second part of the thesis we focus on distributed algorithms for clustering problems. We present several distributed approximation algorithms for clustering problems (e.g., minimum dominating set, facility location problems) that improve on best known results so far. The main contribution here is the design of distributed algorithms where the running time is a "tunable" parameter. The advent of distributed systems of unprecedented scale and complexity motivates the question of whether it is possible to design algorithms that can provide non-trivial approximation guarantees even after very few rounds of computation and message exchanges. We call these algorithms "k-round algorithms". We design k-round algorithms for various clustering problems that yield non-trivial approximation factors even if k is a constant. Additionally, if k assumes poly-logarithmic values, our algorithms match or improve on the best-known approximation factors for these problems.