Complexity Results in VLSI and Distributed Computing
Abstract
Two classes of complexity problems are investigated in this thesis. For the first class, namely, VLSI Complexity, the main results are bounds on combinations of the area A, computation time T and pipelined period P expended by a VLSI circuit in solving a problem. Closely related to VLSI Complexity is the problem of Communication Complexity. It concerns the amount of communication required to compute a function whose inputs are distributed between two processors. Based on the Grid Model for VLSI circuits, we develop in the first part of the thesis a unified framework for formalizing circuit partitions. Multiple circuit partitions in two and threedimensions are introduced. In both cases, we show that there must exist a block in the partition that satisfies bounds on the number of inputs and outputs it contains, as well as on its boundary size. The information transfer requirement of this block then gives AT('2) and AT lower bounds for two and threedimensional partitions respectively. Employing these proof methods, matching (or the strongest known) lower and upper bounds are derived for a class of functions with localized information flow. These include: generalized discrete Fourier transform, generalized barrel shift and rectangular matrix multiplication. In the second part of the thesis, we give formal definitions to and derive general complexity bounds for four communication models. They are the deterministic, (epsilon)error, (epsilon)randomized and (delta)distortion models. The communication complexity of computing the Hamming Distance function is then investigated in detail. Lower and upper bounds differing by no more than one bit are proved for the deterministic model. A number of independently interesting extremal combinatorial results in Hamming Space are also established. For the (epsilon)error and (epsilon) randomized models, we prove asymptotically tight bounds, thereby resolving an open problem in the literature. Asymptotically tight bounds are also proved for the (delta)distortion model.
 Publication:

Ph.D. Thesis
 Pub Date:
 December 1985
 Bibcode:
 1985PhDT........25P
 Keywords:

 VLSI;
 COMMUNICATION;
 COMBINATORICS;
 Physics: Electricity and Magnetism;
 Complexity;
 Distributed Processing;
 Integrated Circuits;
 Large Scale Integration;
 Architecture (Computers);
 Asymptotic Methods;
 Computation;
 Fourier Transformation;
 Electronics and Electrical Engineering