Parallel Algorithmic Techniques for Combinatorial Computation

Parallel computation offers the promise of great improvements in the solution of problems that, if we were restricted to sequential computation, would take so much time that solution would be impractical. There is a drawback to the use of parallel computers, however, and that is that they seem to be harder to program. For this reason, parallel algorithms in practice are often restricted to simple problems such as matrix multiplication. Certainly this is useful, and in fact we shall see later some non-obvious uses of matrix manipulation, but many of the large problems requiring solution are of a more complex nature. In particular, an instance of a problem may be structured as an arbitrary graph or tree, rather than in the regular order of a matrix. In this paper we describe a number of algorithmic techniques that have been developed for solving such combinatorial problems. The intent of the paper is to show how the algorithmic tools we present can be used as building blocks for higher level algorithms, and to present pointers to the literature for the reader to look up the specifics of these algorithms. We make no claim to completeness; a number of techniques have been omitted for brevity or because their chief application is not combinatorial in nature. In particular we give very little attention to parallel sorting, although sorting is used as a subroutine in a number of the algorithms we describe. We also only describe algorithms, and not lower bounds, for solving problems in parallel

Geometry-Oblivious FMM for Compressing Dense SPD Matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in $N \log N$ or even $N$ time, where $N$ is the matrix size. Compression requires $N \log N$ storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1

Adaptive Massively Parallel Constant-Round Tree Contraction

Miller and Reif's FOCS'85 classic and fundamental tree contraction algorithm is a broadly applicable technique for the parallel solution of a large number of tree problems. Additionally it is also used as an algorithmic design technique for a large number of parallel graph algorithms. In all previously explored models of computation, however, tree contractions have only been achieved in $\Omega(\log n)$ rounds of parallel run time. In this work, we not only introduce a generalized tree contraction method but also show it can be computed highly efficiently in $O(1/\epsilon^3)$ rounds in the Adaptive Massively Parallel Computing (AMPC) setting, where each machine has $O(n^\epsilon)$ local memory for some $0 < \epsilon < 1$. AMPC is a practical extension of Massively Parallel Computing (MPC) which utilizes distributed hash tables. In general, MPC is an abstract model for MapReduce, Hadoop, Spark, and Flume which are currently widely used across industry and has been studied extensively in the theory community in recent years. Last but not least, we show that our results extend to multiple problems on trees, including but not limited to maximum and maximal matching, maximum and maximal independent set, tree isomorphism testing, and more.Comment: 35 pages, 3 figures, to be published in Innovations in Theoretical Computer Science (ITCS

Efficient Linked List Ranking Algorithms and Parentheses Matching as a New Strategy for Parallel Algorithm Design

The goal of a parallel algorithm is to solve a single problem using multiple processors working together and to do so in an efficient manner. In this regard, there is a need to categorize strategies in order to solve broad classes of problems with similar structures and requirements. In this dissertation, two parallel algorithm design strategies are considered: linked list ranking and parentheses matching

Acceleration of Computational Geometry Algorithms for High Performance Computing Based Geo-Spatial Big Data Analysis

Geo-Spatial computing and data analysis is the branch of computer science that deals with real world location-based data. Computational geometry algorithms are algorithms that process geometry/shapes and is one of the pillars of geo-spatial computing. Real world map and location-based data can be huge in size and the data structures used to process them extremely big leading to huge computational costs. Furthermore, Geo-Spatial datasets are growing on all V’s (Volume, Variety, Value, etc.) and are becoming larger and more complex to process in-turn demanding more computational resources. High Performance Computing is a way to breakdown the problem in ways that it can run in parallel on big computers with massive processing power and hence reduce the computing time delivering the same results but much faster.This dissertation explores different techniques to accelerate the processing of computational geometry algorithms and geo-spatial computing like using Many-core Graphics Processing Units (GPU), Multi-core Central Processing Units (CPU), Multi-node setup with Message Passing Interface (MPI), Cache optimizations, Memory and Communication optimizations, load balancing, Algorithmic Modifications, Directive based parallelization with OpenMP or OpenACC and Vectorization with compiler intrinsic (AVX). This dissertation has applied at least one of the mentioned techniques to the following problems. Novel method to parallelize plane sweep based geometric intersection for GPU with directives is presented. Parallelization of plane sweep based Voronoi construction, parallelization of Segment tree construction, Segment tree queries and Segment tree-based operations has been presented. Spatial autocorrelation, computation of getis-ord hotspots are also presented. Acceleration performance and speedup results are presented in each corresponding chapter