An O(log n)algorithm for parallel update of minimum spanning trees

Traditionally, interest in parallel computation centered around the speedup provided by parallel algorithms over their sequential counterparts. In this paper, we ask a different type of question: Can parallel computers, due to their speed, do more than simply speed up the solution to a problem? We show that for real-time optimization problems, a parallel computer can obtain a solution that is better than that obtained by a sequential one. Specifically, a sequential and a parallel algorithm are exhibited for the problem of computing the best-possible approximation to the minimum-weight spanning tree of a connected, undirected and weighted graph whose vertices and edges are not all available at the outset, but instead arrive in real time. While the parallel algorithm succeeds in computing the exact minimum-weight spanning tree, the sequential algorithm can only manage to obtain an approximate solution. In the worst case, the ratio of the weight of the solution obtained sequentially to that of the solution computed in parallel can be arbitrarily large.

Parallel real-time optimization: beyond speedup

Parallel algorithms for computing maximal independent sets in trees and for updating minimum spanning trees

The primary purpose of parallel computation is the fast execution of computational tasks that require an inordinate amount of time to perform sequentially. As a consequence, interest in parallel computation to date has naturally focused on the speedup provided by parallel algorithms over their sequential counterparts. The thesis of this paper is that a second equally important motivation for using parallel computers exists. Specifically, the following question is posed: Can parallel computers, thanks to their multiple processors, do more than simply speed up the solution to a problem? We show that within the paradigm of real-time computation, some classes of problems have the property that a solution to a problem in the class, when computed in parallel, is far superior in quality than the best one obtained on a sequential computer. What constitutes a better solution depends on the problem under consideration. Thus, `better' means `closer to optimal' for optimization problems, `more secure' for cryptographic problems, and `more accurate' for numerical problems. Examples from these classes are presented. In each case, the solution obtained in parallel is significantly, provably, and consistently better than a sequential one. It is important to note that the purpose of this paper is not to demonstrate merely that a parallel computer can obtain a better solution to a computational problem than one derived sequentially. The latter is an interesting (and often surprising) observation in its own right, but we wish to go further. It is shown here that the improvement in quality can be arbitrarily high (and certainly superlinear in the number of processors used by the parallel computer). This result is akin to superlinear speedup --- a phenomenon itself originally thought to be impossible.

Parallel real-time computation: Sometimes quantity means quality

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.

Optimal parallel algorithms for multiple updates of minimum spanning trees

A Parallel Algorithm for Multiple Updates of Minimum Spanning Trees.

Parallel Updates of Graph Properties in Logarithmic Time.

Parallel Algorithm for Maximum Weight Matching in Trees.

University of Chicago, Chicago, Illinois [August 2002 present] Associate Professor, Department of Computer Science. AT&T Labs – Research, Florham Park, New Jersey [May 2001 ] Technology Consultant, Software Systems Research Department. [July 1996 April 2001] Principal Technical Staff Member, Software Systems Research Department. Princeton University, Princeton, New Jersey [September 1990 June 1997] Assistant Professor, Department of Computer Science.

Shaunak Pawagi

Papers

A Parallel Algorithm for Multiple Updates of Minimum Spanning Trees.

Parallel Updates of Graph Properties in Logarithmic Time.

Parallel Algorithm for Maximum Weight Matching in Trees.

A Parallel Algorithm for Dominators.