A parallel algorithm for computing minimum spanning trees

2004

Dehne et al. present a BSP/CGM algorithm for computing a spanning tree and the connected components of a graph, that requires O(log p) communication rounds, where p is the number of processors. It requires the solution of the Euler tour problem which in turn is based on the solution of the list ranking problem. In this paper we present experimental results of a parallel algorithm that does not depend on the solution of the Euler tour or the list ranking problem. The proposed algorithm has the practical advantage of avoiding the list ranking computation and is based on the integer sorting algorithm which can be implemented efficiently on the BSP/CGM model. We implemented the proposed algorithm on a Beowulf cluster and on a grid running the InteGrade middleware. We obtained encouraging albeit modest speedup on a small Beowulf cluster and expect good speedups on the grid for larger size graphs and clusters.

Journal of Algorithms, 1982

SIAM Journal on Computing, 1984

In this paper, we present efficient parallel algorithms for the following graph problems: finding the lowest common ancestors for vertex pairs of a directed tree; finding all fundamental cycles, a directed spanning forest, all bridges, all bridge-connected components, all separation vertices, all biconnected components, and testing the biconnectivity of an undirected graph. All these algorithms achieve the O(lg n) time bound, with the first two algorithms using n[n/lg n] processors and the remaining algorithms using n[n/lg n] processors. In all cases, our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n. Moreover, our algorithms are optimal with respect to the time-processor product for dense graphs, with the exception of the first two algorithms. The machine model we use is the PRAM which is a SIMD model allowing simultaneous reads but not simultaneous writes to the same memory location.

While forming reliable communication networks, we must guarantee that, after failure of a node or links, the surviving network still allows communication between all other nodes by choosing alternate path which gives strict requirement on the connectivity of the corresponding graph. A general network design problem which requires the underlying network to be resilient to link failures is known as the edge-connectivity survivable network design problem. In this work we present a new approach to find a Minimum Spanning Tree (NAFMST) used to parallelize efficiently Kruskal's Minimum Spanning Tree algorithm. This algorithm is known for exhibiting inherently sequential Characteristics. More specifically, the strict order by which the algorithm checks the edges of a given graph is the main reason behind the lack of explicit parallelism. Our proposed scheme attempts to overcome the imposed restrictions and improve the performance of the algorithm.

Parallel Processing Symposium, 1992

The vertex updating problem for a minimum spanning tree (MST) is de ned as follows: Given a graph G = (V; E G ) and its MST T, update T when a new vertex z is introduced along with weighted edges that connect z with the vertices of G. We present a set of rules that, together with a valid tree-contraction schedule, are used to produce simple optimal parallel algorithms that run in O(lgn) parallel time using n= lgn EREW PRAMs where n = jV j. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best known parallel result was a rather complicated algorithm that used n processors of the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O(lg k lgn) parallel time using k n lg k lgn EREW PRAM processors.

Algorithmica, 1996

The vertex updating problem for a minimum spanning tree (MST) is de ned as follows: Given a graph G = (V; E G ) and an MST T for G, nd a new MST for G to which a new vertex z has been added along with weighted edges that connect z with the vertices of G. We present a set of rules that produce simple optimal parallel algorithms that run in O(lg n) time using n= lg n EREW PRAM processors, where n = jV j. These algorithms employ any valid tree-contraction schedule that can be produced within the stated resource bounds. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best known parallel result was a rather complicated algorithm that used n processors in the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O(lg k lg n) parallel time using k n lg k lg n EREW PRAM processors. This is optimal for graphs having (kn) edges.

Dartmouth College, Hanover, NH, 1992

In this thesis we examine three problems in graph theory and propose e cient parallel algorithms for solving them. We also introduce a number of parallel algorithmic techniques.

Sukkur IBA Journal of Computing and Mathematical Sciences, 2017

In last decade, application developers attained improved performances by merely employing the machines based on higher-clocked processors. However, in 2003 multi-core processors emerged and eradicated the old processor manufacturing technology based on increasing processor’s clock frequencies. After emergence of new parallel processor architectures, serial applications must be re-engineered into parallel versions to exploit the computing power of the existing hardware. In this paper, we present an efficient parallel implementation of minimum spanning tree algorithm to take advantage of the computing power of multi-core machines. Computer network routing, civil infrastructure planning and cluster analysis are typically use-cases of spanning tree problem. The experimental results show that the proposed algorithm is scalable for different machine and graph sizes. The methodology is simple and can easily be implemented using different shared-memory parallel programming models.

Algorithmica, 1990

In this thesis we examine three problems in graph theory and propose e cient parallel algorithms for solving them. We also introduce a number of parallel algorithmic techniques.

Discrete Applied Mathematics, 2007

In this paper, we consider the problems of co-biconnectivity and strong co-connectivity, i.e., computing the biconnected components and the strongly connected components of the complement of a given graph. We describe simple sequential algorithms for these problems, which work on the input graph and not on its complement, and which for a graph on n vertices and m edges both run in optimal O(n + m) time. Our algorithms are not data structure-based and they employ neither breadth-first-search nor depth-first-search. Unlike previous linear co-biconnectivity and strong co-connectivity sequential algorithms, both algorithms admit efficient parallelization. The co-biconnectivity algorithm can be parallelized resulting in an optimal parallel algorithm that runs in O(log 2 n) time using O((n + m)/log 2 n) processors. The strong co-connectivity algorithm can also be parallelized to yield an O(log 2 n)-time and O(m 1.188 / log n)-processor solution. As a byproduct, we obtain a simple optimal O(log n)-time parallel co-connectivity algorithm. Our results show that, in a parallel process environment, the problems of computing the biconnected components and the strongly connected components can be solved with better time-processor complexity on the complement of a graph rather than on the graph itself.