Efficient minimum spanning tree algorithms on the reconfigurable mesh

1 IntroductionLet G = (V, E) be a connected, undirected graph with a weight function W on the set Eof edges to the set of reals. A spanning tree is a subgraph T = (V, ET), ET G E, of C suchthat T is a tree. The weight W(T) of a spanning tree T is the sum of the weights of its edges.A spanning tree with the smallest possible'weight is called a minimum spanning tree (MST)of G. Computing an MST of a given weighted graph is an important problem that arisesin many applications. For this …     

Optimal parallel algorithms for multiple updates of minimum spanning trees

Parallel updates of minimum spanning trees (MSTs) have been studied in the past. These updates allowed a single change in the underlying graph, such as a change in the cost of an edge or an insertion of a new vertex. Multiple update problems for MSTs are concerned with handling more than one such change. In the sequential case multiple update problems may be solved using repeated applications of an efficient algorithm for a single update. However, for efficiency reasons, parallel algorithms for multiple update problems must consider all changes to the underlying graph simultaneously. In this paper we describe parallel algorithms for updating an MST whenk new vertices are inserted or deleted in the underlying graph, when the costs ofk edges are changed, or whenk edge insertions and deletions are performed. For multiple vertex insertion update, our algorithm achieves time and processor bounds ofO(log n·logk) and nk/(logn·logk), respectively, on a CREW parallel random access machine. These bounds are optimal for dense graphs. A novel feature of this algorithm is a transformation of the previous MST andk new vertices to a bipartite graph which enables us to obtain the above-mentioned bounds.     

Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log² n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.     

一种基于广度优先搜索的K-means初始化算法

K-means算法是在现实应用中非常广泛的聚类算法,K-means算法对初始中心的选择非常敏感,对已存在的有代表性的初始算法进行了研究,提出了一种基于广度优先搜索的K-means初始化算法。该算法综合考虑了密度与距离因素,选择初始点。分析表明该算法选择的初始点非常接近期望的中心点。     

基于改进遗传算法的最小生成树算法

以图论和改进遗传算法为基础,提出了一种求最小生成树的遗传算法。该算法采用二进制表示最小树问题,并设计出相应的适应度函数、算子以及几种控制策略,以提高执行速度和进化效率。传统算法一次只能得到一个候选解。用该算法对其求解,可以在较短的时间内以较高的概率获得多个候选解。应用实例表明该算法优于传统算法。     

Efficient parallel algorithms for graph problems

We present an efficient technique for parallel manipulation of data structures that avoids memory access conflicts. That is, this technique works on the Exclusive Read/Exclusive Write (EREW) model of computation, which is the weakest shared memory, MIMD machine model. It is used in a new parallel radix sort algorithm that is optimal for keys whose values are over a small range. Using the radix sort and known results for parallel prefix on linked lists, we develop parallel algorithms that efficiently solve various computations on trees and unicycular graphs. Finally, we develop parallel algorithms for connected components, spanning trees, minimum spanning trees, and other graph problems. All of the graph algorithms achieve linear speedup for all but the sparsest graphs.Part of this work was done while the first author was at the University of Illinois, Urbana-Champaign, the second author was at Carnegie-Mellon University, and the third author was at the Hebrew University and the Courant Institute of Mathematical Sciences, New York University. A preliminary version of this work was presented at the 1986 International Conference on Parallel Processing.     

A parallel algorithm for the enumeration of the spanning trees of a graph

As is well known, the strategy of divide-and-conquer is widely used in problem solving. The method of partitioning is also a fundamental strategy for the design of a parallel algorithm. The problem of enumerating the spanning trees of a graph arises in several contexts such as computer-aided design and computer networks. A parallel algorithm for solving the problem is presented in this paper. It is based on the principle of the inclusion and exclusion of sets, and not directly based on the partitioning of the graph itself. The results of the preliminary experiments on a MIMD system appear promising.     

Efficient Parallel Algorithms for Some Graph Theory Problems

In this paper,a sequential algorithm computing the aww vertex pair distance matrix D and the path matrix Pis given.On a PRAM EREW model with p,1≤p≤n^2,processors,a parallel version of the sequential algorithm is shown.This method can also be used to get a parallel algorithm to compute transitive closure array A^* of an undirected graph.The time complexity of the parallel algorithm is O(n^3/p).If D,P and A^* are known,it is shown that the problems to find all connected components,to compute the diameter of an undirected graph,to determine the center of a directed graph and to search for a directed cycle with the minimum(maximum)length in a directed graph can all be solved in O(n^2/p logp)time.     

并行任务图的优化调度算法

科学与工程计算中的很多复杂应用问题需要使用科学工作流技术,超算领域中的科学工作流常以并行任务图建模,并行任务图的有效调度对应用的高效执行有重要意义。给出了资源限制条件下并行任务图的调度模型;针对Fork-Join类并行任务图给出了若干最优化调度结论;针对一般并行任务图提出了一种新的调度算法,该算法考虑了数据通信开销对资源分配和调度性能的影响,并对已有的CPA算法在特定情况下进行了改进。通过实验与常用的CPR和CPA算法做比较,验证了提出的新算法能够获得很好的调度效果。本文提出的调度算法和得到的最优调度结论对工作流应用系统的高性能调度功能开发具有借鉴意义。     

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

Let S be a set ofn points in the plane. For an arbitrary positive rationalr, we construct a planar straight-line graph onS that approximates the complete Euclidean graph onS within the factor (1 + 1/r)[2/3 cos(/6)], and it has length bounded by 2r + 1 times the length of a minimum Euclidean spanning tree onS. Given the Deiaunay triangulation ofS, the graph can be constructed in linear time.     

Efficient algorithms for (3, 1) graphs

In this paper, we define a class of graphs which are referred to as (3, 1) graphs. A graph is a member of this class if it has the property that within each set of three vertices, there is at least one edge. We derive a lower bound for the size of a maximum clique in a (3, 1) graph as well as an upper bound for the size of a minimum clique covering. In addition, we show that there exists a linear algorithm for constructing a Hamiltonian circuit in a connected (3, 1) graph and an n⁴-algorithm for finding a minimum coloring in a (3, 1) graph.     

Parallel algorithms for minimal spanning trees of directed graphs

The main results of this paper are efficient parallel algorithms, MSP and LOCATE, for computing minimal spanning trees and locating minimal paths in directed graphs, respectively. Algorithm MSP has time complexityO(log³ n) usingO(n ³/logn) processors, while LOCATE has time complexityO(logn) usingO(n ²) processors. Algorithm MSP is derived from sequential algorithms, when the unbounded parallelism model is used.     

内部节点受限的最小生成树问题算法研究

研究内部节点受限的最小生成树问题：给定一个赋权无向完全图[G=V,E],假定[w:E→R+]为边集[E]的权重函数且满足三角不等式,给定点集[V]的一个子集[RR?V],目标是寻找图[G]的一个满足[R]中的点皆为内部顶点的权重最小的生成树。由于该问题是[NP-]困难的,提出了一个伪多项式时间最优算法,设计了一个近似比为2的多项式时间近似算法,并且给出例子以说明该近似比是紧的。     

Efficient approximation algorithms for shortest cycles in undirected graphs

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time . Thus, in general, it yields a approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,…,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n²logn(logn+logM)).     

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

Parallel breadth-first search algorithms for trees and graphs

Parallel Breadth-First Search (BFS) algorithms for ordered trees and graphs on a shared memory model of a Single Instruction-stream Multiple Data-stream computer are proposed. The parallel BFS algorithm for trees computes the BFS rank of eachnode of an ordered tree consisting of n nodes in time of 0(β log n) when 0(n ^1+1/β) processors are used, β being an integer greater than or equal to 2. The parallel BFS algorithm for graphs produces Breadth-First Spanning Trees (BFSTs) of a directedgraph G having n nodes in time 0(log d.log n) using 0(n ³) processors, where d is the diameter of G If G is a strongly connected graph or a connected undirected graph the BFS algorithm produces n BFSTs, each BFST having a different start node.     

最小比率生成树的竞争决策算法

最小比率生成树是找出目标函数形式为两个线性函数比值最小的生成树,例如总代价与总收益比值最小的生成树。当不限制分母的符号时,这是一个NP-hard问题。在分析最小比率生成树数学性质的基础上,提出了最小比率生成树的竞争决策算法。为了防止算法陷入局部最优,采用edge_exchange操作来增加算法的搜索范围。为了验证算法的有效性,采用无关和相关两种策略产生测试数据,并使用Delphi 7.0实现了算法的具体步骤。