On Locality and Related Problems

Locality is a peculiarity which is exhibited by many problems appearing in the most disparate domains of theoretical computer science and mathematics. In this thesis, we demonstrate this assertion by studying three classes of problems that are related to the task of communicating, computing and exploring, respectively. By a local problem, we understand a problem which is easily solved if the problem solver(s) would have access to the whole instance at once. However, the difficulty is that the problem solver is allowed to initially look only at a small portion of the instance, and at every point of the resolution process she gets a small piece of new information that depends on her choices and on the model she is operating with. The problem solver incurs a unit cost when she gets a new information, but can work with the information she already has for free. In the first part of this thesis we focus on the process of communication. We introduce and study the so called algebraic communication complexity of a multivariate polynomial or rational function. In this model two or more players aim to evaluate a rational function according to some input. However the input is distributed among the players. Hence, they exchange messages in such a way that eventually a player is able to correctly determine the value of the function at the given input. Our goal is to determine the least number of messages that are required to solve the task for every given input. We propose different techniques that establish lower bounds for this complexity measure. We apply these results to show that a broad class of multicast cost sharing mechanisms – namely truthful budget-balanced mechanisms – need to send many messages in a network in the worst case during the resolution process. In the second part we deal with the realm of computation. We focus on distributed algorithms for wireless sensor networks. In such a network, every node is a computational unit that performs some actions. However, its view is limited to the set of nodes that can be reached through direct communication with its wireless equipment. As an algorithm progresses, each node gathers more information about the network topology. We start with the observation that many such algorithms require to solve some graph-theoretical problem during their execution. Hence a good understanding of the locality of such problems is much needed. In this thesis we design