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A Cartesian product network is obtained by applying the cross operation on two graphs. We study the problem of constructing the maximum number of edge-disjoint spanning trees (abbreviated to EDSTs) in Cartesian product networks. Let G=(V/sub G/, E/sub G/) be a graph having n/sub 1/ EDSTs and F=(V/sub F/, E/sub F/) be a graph having n/sub 2/ EDSTs. Two methods are proposed for constructing EDSTs in the Cartesian product of G and F, denoted by G/spl times/F. The graph G has t/sub 1/=|E/sub G/|/spl middot/n/sub 1/(|V/sub G/|-1) more edges than that are necessary for constructing n/sub 1/ EDSTs in it, and the graph F has t2=|E/sub F/'-n/sub 2/(|V/sub F/|-1) more edges than that are necessary for constructing n/sub 2/ EDSTs in it. By assuming that t/sub 1//spl ges/n/sub 1/ and t/sub 2//spl ges/n/sub 2/, our first construction shows that n/sub 1/+n/sub 2/ EDSTS can be constructed in G/spl times/F. Our second construction does not need any assumption and it constructs n/sub 1/+n/sub 2/-1 EDSTs in G/spl times/F. By applying the proposed methods, it is easy to construct the maximum numbers of EDSTs in many important Cartesian product networks, such as hypercubes, tori, generalized hypercubes, mesh connected trees, and hyper Petersen networks.

An edge is a bisector of a simple path if it contains the middle point of the path. Let T=(V,E) be a tree. Given a source vertex s /spl isin/ V, the single-source tree bisector problem is to find, for every vertex /spl upsi/ /spl isin/ V, a bisector of the simple path from s to /spl upsi/. The all-pairs tree bisector problem is to find for, every pair of vertices u, /spl upsi/ /spl isin/ V, a bisector of the simple path from u to /spl upsi/. In this paper, it is first shown that solving the single-source tree bisector problem of a weighted tree has a time lower bound /spl Omega/(n log n) in the sequential case. Then, efficient parallel algorithms are proposed on the EREW PRAM for the single-source and all-pairs tree bisector problems. Two O(log n) time single-source algorithms are proposed. One uses O(n) work and is for unweighted trees. The other uses O(n log n) work and is for weighted trees. Previous algorithms for the single-source problem could achieve the same time O(log n) and the same optimal work, O(n) for unweighted trees and O(n log n) for weighted trees, on the CRCW PRAM. The contribution of our single-source algorithms is the improvement from CRCW to EREW. One all-pairs parallel algorithm is proposed. It requires O(log n) time using O(n/sup 2/) work. All the proposed algorithms are cost-optimal. Efficient tree bisector algorithms have practical applications to several location problems on trees. Using the proposed algorithms, efficient parallel solutions for those problems are also presented.

Let G be an undirected graph and T be a spanning tree of G. In this paper, an efficient parallel algorithm is proposed for determining whether T is an unordered depth-first search tree of G. The proposed algorithm runs in O(m/p+log m) time using p processors on the EREW PRAM, where m is the number of edges contained in G. It is cost-optimal and achieves linear speedup.

The first main contribution of this work is to propose an efficient VLSI architecture obtained by augmenting the Mesh with Multiple Broadcasting (MMB) with precharged 1-bit row and column buses. The new architecture, which we call Mesh with Hybrid Buses (MHB for short), is realizable in VLSI with no increase in the area or the wiring complexity of the MMB chip. Our second main contribution is to show that the MHB is extremely well-suited for solving an entire slew of digital geometry tasks. The MHB is not a reconfigurable architecture. Yet, quite remarkably, for a large number of fundamental digital geometry tasks, the MHB offers a level of performance previously attained only by reconfigurable architectures. Specifically, with a digital image pretiled onto a MHB of size /spl radic/n/spl times//spl radic/n one pixel per processor, we show that the problems of computing the convex hull of the image, computing the diameter and the width of the image, deciding whether a set of digital points is a digital line, computing the maximum distance between two images, deciding whether two images are linearly separable, computing several moments and low-level descriptors of the image, including the perimeter, area, center, and median row of its convex hull, can be solved in O(log n) time. By contrast, the fastest possible algorithms for the problems above on the MMB run in /spl Theta/(n/sup 1/6/) time. Finally, we go on to show that, with minor changes, our algorithms can be implemented to run within cost-optimality on a MHB of size /spl radic/n/log n/spl times//spl radic/n/log n.

A k-tree core of a tree network is a subtree with exactly k leaves that minimizes the total distance from vertices to the subtree. A k-tree center of a tree network is a subtree with exactly k leaves that minimizes the distance from the farthest vertex to the subtree. In this paper, two efficient parallel algorithms are proposed for finding a k-tree core and a k-tree center of a tree network, respectively. Both the proposed algorithms perform on the EREW PRAM in O(log n log n) time using O(n) work (time-processor product). Besides being efficient on the EREW PRAM, in the sequential case, our algorithm for finding a k-tree core of a tree network improves the two algorithms previously proposed.

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O(1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n*n*n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n/sup 2/*n/sup 2/ processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs. >