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Probabilistic and Topological methods in Computational Geometry

We consider four problems connected by the common thread of geometry. The first three arise in applications that apriori do not involve geometry but this turns out to be the right language for visualizing and analyzing them. In the fourth, we generalize some well known results in geometry to the topological plane. The techniques we use come from probability and topology. First, we consider two algorithms that work well in practice but the theoretical mechanism behind whose success is not very well understood. Greedy routing is a routing mechanism that is commonly used in wireless sensor networks. While routing on the Internet uses standard established protocols, routing in ad-hoc networks with little structure (like sensor networks) is more difficult. Practitioners have devised algorithms that work well in practice, however there were no known theoretical guarantees. We provide the first such result in this area by showing that greedy routing can be made to work on Planar triangulations. Linear Programming is a technique for optimizing a linear function subject to linear constraints. Simplex Algorithms are a family of algorithms that have proven quite successful in solving Linear Programs in practice. However, examples of Linear Programs on which these algorithms are very inefficient have been obtained by researchers. In or-