Sorting is a fundamental problem with applications in all areas of computer science and engineering. In this work, we address the problem of sorting on mesh connected computers enhanced by endowing each row and each column with its own dedicated high-speed bus. This architecture, commonly referred to as a mesh with multiple broadcasting, is commercially available and has been adopted by the DAP family of multiprocessors. Somewhat surprisingly, the problem of sorting m, (m/spl les/n), elements on a mesh with multiple broadcasting of size /spl radic/n/spl times//spl radic/n has been studied, thus far, only in the sparse case, where m/spl isin//spl Theta/(/spl radic/n) and in the dense case, where m/spl isin//spl Theta/O(/spl radic/n). Yet, many applications require using an existing platform of size /spl radic/n/spl times//spl radic/n for sorting m elements, with /spl radic/n<m/spl les/n. Our main contribution is to present the first known adaptive time- and VLSI-optimal sorting algorithm for meshes with multiple broadcasting. Specifically we show that, for every choice of a constant 0

Recently it has been noticed that for semigroup computations and for selection, rectangular meshes with multiple broadcasting yield faster algorithms than their square counterparts. The contribution of the paper is to provide yet another example of a fundamental problem for which this phenomenon occurs. Specifically, we show that the problem of computing the convex hull of a set of n sorted points in the plane can be solved in O(n/sup 1/8/ log /sup 3/4/) time on a rectangular mesh with multiple broadcasting of size n/sup 3/8/ log/sup 1/4/ n/spl times/n/sup 5/8//log/sup 1/4/n. The fastest previously known algorithms on a square mesh of size /spl radic/n/spl times//spl radic/n run in O(n/sup 1/6/) time in case the n points are pixels in a binary image, and in O(n/sup 1/6/log/sup 3/2/ n) time for sorted points in the plane.

Given a collection of objects in the plane along with a viewpoint /spl omega/, the visibility problem involves determining the portion of each object that is visible to an observer positioned at /spl omega/. The visibility problem is central to various application areas including computer graphics, image processing, VLSI design, and robot navigation, among many others. The main contribution of this work is to provide time-optimal solutions to this problem for several classes of objects, namely ordered line segments, disks, and iso-oriented rectangles in the plane. In addition, our visibility algorithm for line segments is at the heart of time-optimal solutions for determining, for each element in a given sequence of real numbers, the position of the nearest larger element within that sequence, triangulating a set of points in the plane, determining the visibility pairs among a set of vertical line segments, and constructing the dominance and visibility graphs of a set of iso-oriented rectangles in the plane. All the algorithms in this paper involve an input of size n and run in O(log n) time on a mesh with multiple broadcasting of size n/spl times/n. This is the first instance of time-optimal solutions for these problems on this architecture.>

Consider arbitrary collections A = a_1,a_2,.. .,a_n of items and Q = q_1,q_2,...,q_m (1 leqslant mn leqslant n) of queries from a totally ordered universe. The multiple rank problem involves computing for every query qi the number of items in A that have a lesser value. Our contribution is to show that the problem at hand can be solved time-optimally on meshes with multiple broadcasting. More specifically, if the collection A is siored in some order one item per processor and if Q is stored one query per processor in the leftmost frac{m} {{sqrt n }} columns of a mesh with multiple broadcasting of size sqrt n x /sqrt n, the corresponding instance of the multiple rank problem can be solved in Theta left( {m^{frac{1} {3}} n^{frac{1} {6}} } right) time. As an application we present a time-optimal algorithm to compute the histogram of a m-level gray image of size sqrt n x sqrt n in Theta left( {m^{frac{1} {3}} n^{frac{1} {6}} } right) time.

Due to its attractive topological properties, the hypercube multiprocessor has emerged as one of the architectures of choice when it comes to implementing a large number of computational problems. In many such applications, Gray-code labelings of the hypercube are a crucial prerequisite for obtaining efficient algorithms. We propose a greedy algorithm that, given an n-dimensional hypercube H with N=22 nodes, returns a Gray-code labeling of H, that is, a labeling of the nodes with binary strings of length n such that two nodes are neighbors in the hypercube if, and only if, their labels differ in exactly one bit. Our algorithm is conceptually very simple and runs in O(N log N) time being, therefore, optimal. As it turns out, with a few modifications our labeling algorithm can be used to recognize hypercubes as well.

One of the fundamental algorithmic problems in computer science involves selecting the kth smallest element in a collection A of n elements. We propose an algorithm design methodology to solve the selection problem on meshes with multiple broadcasting. Our methodology leads to a selection algorithm that runs in O(n/sup 1/8/(log n)/sup 3/4/)) time on a mesh with multiple broadcasting of size n/sup 3/8/(log n)/sup 1/4//spl times/n/sup 5/8//(log n)/sup 1/4/. This result is optimal over a large class of selection algorithms. Our result shows that just as for semigroup computations, selection can be done faster on suitably chosen rectangular meshes than on square meshes.>

The compaction step of integrated circuit design motivates the study of various visibility problems among vertical segments in the plane. One popular variant is referred to as the Vertical Segment Visibility problem (VSV, for short) and is stated as follows. Given a collection S of n disjoint vertical line segments in the plane, for every endpoint of a segment in S determine the first line segment, if any, interacted by a horizontal ray to the right (resp. left) originating from that endpoint. The contribution of this paper is to propose a time-optimal algorithm for the VSP problem on meshes with multiple broadcasting. The authors then use this algorithm to derive time-optimal solutions for two related problems. All the algorithms run in O(log n) time on a mesh with multiple broadcasting of size n /spl times/ n. This is the first instance of time-optimal solutions for these problems known to us.>

Recently it has been noticed that for semigroup computations and for selection, rectangular meshes with multiple broadcasting yield faster algorithms than their square counterparts. The contribution of this paper is to provide yet another example of a fundamental problem for which this phenomenon occurs.

In this work, we present a time-and VLSI-optimal sorting algorithm for meshes with multiple broadcasting. Specifically, we show that for every choice of a positive integer constant c, m items \left( {n^{\frac{1} {2} + \frac{1} {{2c}}} \leqslant m \leqslant n} \right) stored in the first \left\lceil {\frac{m} {{\sqrt n }}} \right\rceil columns of a mesh with multiple broadcasting of size \sqrt {n} x \sqrt {n} can be sorted in O({\frac{m} {{\sqrt n }}}) time.