Edson Norberto Cáceres

Efficient Parallel Graph Algorithms For Coarse Grained Multicomputers and BSP

In this paper, we present deterministic parallel algorithms for the coarse grained multicomputer (CGM) and bulk-synchronous parallel computer (BSP) models which solve the following well known graph problems: (1) list ranking, (2) Euler tour construction, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, (7) 2-edge connectivity and biconnectivity (testing and component computation), and (8) cordai graph recognition (finding a perfect elimination ordering). The algorithms for Problems 1–7 require O(log p) communication rounds and linear sequential work per round. Our results for Problems 1 and 2 hold for arbitrary ratios \(\frac{n}{p}\), i.e. they are fully scalable, and for Problems 3–8 it is assumed that \(\frac{n}{p} \geqslant p^ \in ,{\mathbf{ }} \in {\mathbf{ }} > 0\), which is true for all commercially available multiprocessors. We view the algorithms presented as an important step towards the final goal of O(1) communication rounds. Note that, the number of communication rounds obtained in this paper is independent of n and grows only very slowly with respect to p. Hence, for most practical purposes, the number of communication rounds can be considered as constant. The result for Problem 1 is a considerable improvement over those previously reported. The algorithms for Problems 2–7 are the first practically relevant deterministic parallel algorithms for these problems to be used for commercially available coarse grained parallel machines.

Parallel dynamic programming for solving the string editing problem on a CGM/BSP

In this paper we present a coarse-grained parallel algorithm for solving the string edit distance problem for a string A and all substrings of a string C. Our method is based on a novel CGM/BSP parallel dynamic programming technique for computing all highest scoring paths in a weighted grid graph. The algorithm requires \log p rounds/supersteps and O(\fracn^2p\log m) local computation, where

A parallel wavefront algorithm for efficient biological sequence comparison

p

An all-substrings common subsequence algorithm

is the number of processors, p^2 \leq m \leq n. To our knowledge, this is the first efficient CGM/BSP algorithm for the alignment of all substrings of C with A. Furthermore, the CGM/BSP parallel dynamic programming technique presented is of interest in its own right and we expect it to lead to other parallel dynamic programming methods for the CGM/BSP.

A CUDA based Solution to the Multidimensional Knapsack Problem Using the Ant Colony Optimization

In this paper we present a parallel wavefront algorithm for computing an alignment between two strings A and C, with |A| = m and |C| = n. On a distributed memory parallel computer of p processors each with O((m + n)/p) memory, the proposed algorithm requires O(p) communication rounds and O(mn/p) local computing time. The novelty of this algorithm is based on a compromise between the workload of each processor and the number of communication rounds required, expressed by a parameter called α. The proposed algorithm is expressed in terms of this parameter that can be tuned to obtain the best overall parallel time in a given implementation. We show very promising experimental results obtained on a 64-node Beowulf machine. A characteristic of the wavefront communication requirement is that each processor communicates with few other processors. This makes it very suitable as a potential application for grid computing.

BSP/CGM Algorithms for Maximum Subsequence and Maximum Subarray

Abstract Given two strings A and B of lengths n a and n b , n a ⩽ n b , respectively, the all-substrings longest common subsequence (ALCS) problem obtains, for every substring B ′ of B, the length of the longest string that is a subsequence of both A and B ′ . The ALCS problem has many applications, such as finding approximate tandem repeats in strings, solving the circular alignment of two strings and finding the alignment of one string with several others that have a common substring. We present an algorithm to prepare the basic data structure for ALCS queries that takes O ( n a n b ) time and O ( n a + n b ) space. After this preparation, it is possible to build a matrix of size O ( n b 2 ) that allows any LCS length to be retrieved in constant time. Some trade-offs between the space required and the querying time are discussed. To our knowledge, this is the first algorithm in the literature for the ALCS problem.

A BSP/CGM algorithm for the all-substrings longest common subsequence problem

Abstract The Multidimensional Knapsack Problem (MKP) is a generalization of the basic Knapsack Problem, with two or more constraints. It is an important optimization problem with many real-life applications. To solve this NP-hard problem we use a metaheuristic algorithm based on ant colony optimization (ACO). Since several steps of the algorithm can be carried out concurrently, we propose a parallel implementation under the GPGPU paradigm (General Purpose Graphics Processing Units) using CUDA. To use the algorithm presented in this paper, it is necessary to balance the number of ants, number of rounds used, and whether local search is used or not, depending on the quality of the solution desired. In other words, there is a compromise between time and quality of solution. We obtained very promising experimental results and we compared our implementation with those in the literature. The results obtained show that ant colony optimization is a viable approach to solve MKP efficiently, even for large instances, with the parallel approach.

ON FINDING EULER TOURS IN PARALLEL

The maximum subsequence problem finds the contiguous subsequence of n real numbers with the highest sum. This problem appears in the analysis of DNA or protein sequences. It can be solved sequentially in O(n) time. In the 2-D version, given an n × n array A, the maximum subarray of A is the contiguous subarray that has the maximum sum. The sequential algorithm for the maximum subarray problem takes O(n 3) time. We present efficient BSP/CGM parallel algorithms that require a constant number of communication rounds for both problems. In the first algorithm, the sequence stored on each processor is reduced to only five numbers, so that the resulting values can be concentrated on a single processor which runs an adaptation of the sequential algorithm to obtain the result. The parallel algorithm requires O(n/p) computing time. In the second algorithm, the input array is partitioned equally among the processors and we first reduce each subarray to a sequence, and then apply the first algorithm to solve it. The parallel algorithm takes O(n 3/p) computing time. The good performance of the parallel algorithms is confirmed by experimental results run on a 64-node Beowulf parallel computer.

A BSP/CGM algorithm for finding all maximal contiguous subsequences of a sequence of numbers

Given two strings X and Y of lengths m and n, respectively, the all-substrings longest common subsequence (ALCS) problem obtains the lengths of the subsequences common to X and any substring of Y. The sequential algorithm takes O(mn) time and O(n) space. We present a parallel algorithm for ALCS on a coarse-grained multicomputer (BSP/CGM) model with p < /spl radic/m processors that takes O(mn/p) time and O(n/spl radic/m) space per processor, with O(log p) communication rounds. The proposed parallel algorithm also solves the well-known LCS problem. To our knowledge this is the best BSP/CGM algorithm for the ALCS problem in the literature.

Coarse Grained Parallel Algorithms for Detecting Convex Bipartite Graphs

We describe an alternative implementation of Atallah and Vishkin’s parallel algorithm for finding an Euler Tour of a graph. Instead of finding a spanning tree as an intermediate step, this algorithm is based on identifying a strut which is easier to compute. Using the strut, vertices which have more than one circuit passing through them are identified directly. Stitching at such vertices reduces the number of circuits in the Euler Partition.