Arithmetic Circuit Complexity and Motion Planning

This dissertation presents the results of my research in two areas: parallel algorithms/circuit complexity, and algorithmic motion planning. The chapters on circuit complexity examine the parallel complexity of several fundamental problems (such as integer division) in the model of small depth circuits. In the later chapters on motion planning, we turn to the computationally intensive problem of planning efficient trajectories for robots in both cooperative and noncooperative environments. Specifically, we first examine the complexity of integer division with remainder under the standard model of constant fanin boolean circuits. We restrict our attention to circuits which are logspace uniform, and present a novel algorithm that has better asymptotic complexity bounds than any previously known algorithm. In fact, it matches the best previously known depth bound and the best previously known size bound simultaneously. Next, we examine circuits where each gate has arbitrary fanin, and can compute the MAJORITY function. Interestingly, while it is impossible to compute integer division with constant depth, unbounded fanin AND/OR circuits, we show that it is possible to compute it with only O(n1+ ) gates (for any constant > 0) and constant depth when MAJORITY gates are allowed. Unfortunately, to get a constant depth circuit, we allow the circuit to be only P-uniform (rather than logspace uniform). In the chapters on motion planning, we first give an approximation bound for optimal time motion planning, where the robot is given bounds on the L2 norms of velocity and acceleration. This (and concurrent, independent work by Donald and Xavier) was the first such approximation algorithm for robots with dynamics bounded in the L2 norm. The second chapter on motion planning addresses the following problem: what if a second, non-cooperating (or even adversarial) robot is added to the environment. This problem is referred to as a pursuit game, and we must make a plan that avoids collisions with the second robot. We present both an exponential time lower bound and several polynomial time approximation algorithms for this problem. The lower bound is the first truly intractable lower bound for a robotics problem with perfect information. Despite this lower bound, we present a polynomial time algorithm that gives approximately optimal solutions to an important class of pursuit games — namely, those where it is possible for our robot to keep a certain “safety margin” between it and its adversary.