Results in Computational Geometry: Geometric Embeddings and Query- Retrieval Problems

Abstract

Unclassified'el. PERFORMING ORGANIZATION RIET NUMBERo7n. MONITORING ORGANIZATION REPORT NUMBER(S)MIT TITLE (Include Security Classification)Results in Computational Geometry: Geometric Embeddings and Query-Retrieval Problems. PERSONAL AUTHOR(S)Mark David Hansen Part I examines the geometric embedding problem for many of the graphs which are important in the study of parallel computation 133]. Given an undirected graph G with n vertices, and a set P of n points in the plane, the geometric embedding problem consists of finding a bijection from the vertices of G to the points in the plane which minimizes the sum total of edge lengths of the embedded graph. We give fast approximation algorithms for embedding d-dimensional grids in the plane which are within a factor of O(log n) times optimal cost for d > 2 and O(log 2 n) for d = 2. We also show that any embedding of a hypercube, butterfly. or shuffle-exchange graph must be within an 0(log n) factor of optimal cost. When the points of P are randomly distributed, or arranged in a grid, we give a polynomial time algorithm which can embed arbitrary weighted graphs in these points with cost within an 0(log 2 n) factor of optimal. Many of these results extend to higher dimensions when P C Rd.Aside from the intrinsic mathematical interest of these problems, they also have applications in the held of parallel processing. For example. we show how the algorithms which we develop for geometric embeddings can be used to give solutions which are within an 0(log 2 N) factor of optimal to problems of performance optimization for array-based paralMe processors in the following areas: communication load balancing, dynarmic allocatiou of jobs to processors, reconfiguring around faults.and simulating other architectures.Part II of this thesis examines query-retrieval problems concerning distributions of points in euclidean space. In this part, we describe a new technique for solving a variety of query-retrieval problems in optimal time with optimal or near-optimal space steps.) Depending on the problem being solved, the space required for the data structure is either linear or O(nlogn). Hence, the time bounds are optimal and the space bounds are optimal or near-optimal. Previously known data structures for these problems required a factor of fl(logvn(loglogn) 2 ) or fl(lognloglogn) more space and/or more time to answer each query.Our compaction technique incorporates planar separators, filtering search, and the probabilistic method for discrepancy problems. The fundamental idea is that kth -order Voronoi diagrams (and other suitable proximity diagrams) can be compacted from k°( 1 )n space to 0(n) space and still retain all the information that is essential for solving query problems. Many fundamental questions in computational geometry arise from the consideration of distributions of points in euclidean space. This thesis explores two important areas of computational geometry in this setting: geometric embeddings anr queryretrieval problems. Each area is addressed in a separate part...