Dharmavani Bhagavathi

A Time-Optimal Multiple Search Algorithm on Enhanced Meshes, with Applications

Given a sorted sequence A = a1, a2, ..., an of items from a totally ordered universe, along with an arbitrary sequence Q = q1, q2, ..., qm (1 ? m ? n) of queries, the multiple search problem involves computing for every qj (1 ? j ? m) the unique ai for which ai?1 ? qj < ai. In this paper we propose a time-optimal algorithm to solve the multiple search problem on meshes with multiple broadcasting. More specifically, on a formula] × formula] mesh with multiple broadcasting, our algorithm runs in formula] time which is shown to be time-optimal. We also present some surprising applications of the multiple search algorithm to computer graphics, image processing, robotics, and computational geometry.

A fast selection algorithm for meshes with multiple broadcasting

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. >

A Unifying Look at Semigroup Computations on Meshes with Multiple Broadcasting

Semigroup computations are a fundamental algorithmic tool finding applications in all areas of parallel processing. Given a sequence of m items a1, a2,..., am from a semigroup S with an associative operation ⊕, the semigroup computation problem involves computing a1 ⊕ a2 ⊕ ... ⊕ am. We consider the semigroup computation problem involving m (2 ≤ m ≤ n) items on a mesh with multiple broadcasting of size \(\sqrt n \times \sqrt n\). Our contribution is to present the first lower bound and the first time-optimal algorithm which apply to the entire range of m (2 ≤ m ≤ n). First, we show that any algorithm that solves the semigroup computation problem must take at least \(\Omega (max\{ min\{ log m, log\tfrac{{n^{\tfrac{2}{3}} }}{{m^{\tfrac{1}{3}} }}\} ,\tfrac{{m^{\tfrac{1}{3}} }}{{n^{\tfrac{1}{6}} }}\} )\) time. Second, we show that our bound is tight by designing an algorithm whose running time matches the lower bound. These results unify and generalize all semigroup lower bounds and algorithms known to the authors....

Convexity problems on meshes with multiple broadcasting

Abstract Our contribution is twofold. First, we show that Ω(log n) is a time lower bound on the CREW-PRAM and the mesh with multiple broadcasting for the tasks of computing the perimeter, the area, the diameter, the width, the modality, the smallest-area enclosing rectangle, and the largest-area inscribed triangle of a convex n-gon. We show that the same time lower bound holds for the tasks of detecting whether a convex n-gon lies inside another as well as for computing the maximum distance between two convex n-gons. We obtain our time lower bound results for the CREW-PRAM by using a novel technique involving geometric constructions. These constructions allow us to reduce the well-known OR problem to each of the geometric problems of interest. We then port these time lower bounds to the mesh with multiple broadcasting using simulation results. Our second contribution is to show that the Ω(log n) time lower bound is tight by providing O(log n) time algorithms to solve these problems on a mesh with multiple broadcasting of size n × n. Finally, we show that for two separable convex n-gons P and Q, the task of computing the minimum distance between P and Q can be performed in O(1) time on a mesh with multiple broadcasting of size n × n.

Convex Polygon Problems on Meshes with Multiple Broadcasting

We propose time-optimal algorithms for a number of convex polygon problems on meshes with multiple broadcasting. Specifically, we show that on a mesh with multiple broadcasting of size n × n, the t...

Square meshes are not optimal for convex hull computation

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.

Time- and VLSI-optimal Sorting on Meshes with Multiple Broadcasting

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.

Selection on rectangular meshes with multiple broadcasting

One of the fundamental algorithmic problems in computer science involves selecting thekth smallest element in a setS ofn elements. In this paper we present a fast selection algorithm which runs inO(n1/8 logn) time on a mesh with multiple broadcasting of sizen3/8 ×n5/8. Our result shows that, just like semigroup computations, selection can be done much faster on a suitably chosen rectangular mesh than on square meshes. We also show that if every processor can storen1/9 items, then our selection algorithm runs inO(n1/9 logn) time on a mesh with multiple broadcasting of sizen1/3 ×n5/9.

Square Meshes are not Optimal for Convex Hull Computation

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.

Time- and VLSI-optimal sorting on enhanced meshes

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</spl epsiv//spl les/ 1/2 , it is possible to sort m elements, n/sup 1/2 +/spl epsiv///spl les/m/spl les/n, stored in the leftmost [m//spl radic/n] columns of a mesh with multiple broadcasting of size /spl radic/n/spl times//spl radic/n in /spl Theta/(m//spl radic/n) time.