Koichi Wada

Parallel Algorithms for Partitioning Sorted Sets and Related Problems

We consider the following partition problem: Given a set S of n elements that is organized as k sorted subsets of size n/k each and given a parameter h with 1/k≤h≤n/k, partition S into g=O(n/(hk)) subsets D1D2,..., D g of size Θ(hk) each, such that for any two indices i and j with 1≤i≤j≤g, no element in D1i is bigger than any element in D j . Note that with various combinations of the values of parameters h and k, several fundamental problems, such as merging, sorting,and finding an approximate median, can be formulated as or be reduced to this partition problem. The partition problem also finds applications in solving problems of parallel computing and computational geometry. In this paper, we present efficient parallel algorithms for solving the partition problem and its applications. Our parallel partition algorithm runs in O(log n) time using O(min{(n/h)*max{log h 1},n*max{log(1/h),1}}/log n) processors in the EREW PRAM model.The complexity bounds of our parallel partition algorithm on the respective special cases match those of the optimal EREW PRAM algorithms for merging, sorting, and finding an approximate median. Using our parallel partition algorithm, we are also able to obtain better complexity bounds (even possibly on a weaker parallel model) than the previously best known parallel algorithms for several important problems, including parallel multi-selection, parallel multi-ranking, and parallel sorting of k sorted subsets.

Efficient algorithms for a mixed K -partition problem of graphs without specifying bases

Abstract This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements (vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases (called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called k-PART-WOB). In this paper, we show that k-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1) for any k⩾2, k-PART-WOB can be solved in O(¦V¦ √¦V¦ log 2 ¦V¦ + ¦E¦) time for every 4-edge-connected graph G = (V,E), (2) 3-PART-WOB can be solved in O(¦V¦ 2 ) for every 2-edge-connected graph G = (V,E) and (3) 4-PART-WOB can be solved in O(¦E¦ 2 ) for every 3-edge-connected graph G = (V,E).

Parallel robust algorithms for constructing strongly convex hulls

Parallel Robust Algorithms for Constructing Strongly Convex Hulls* Wei Chen~ Koichi Wadat Kimio Kawaguchit Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than d outside P and such that even if the vertices of .F’ arc perturbed by as much as e, P remains convex (e.g., the convex hull of S is a O-strongly convex O-hull of S). This paper presents the first parallel robust method for the generalized convex hull problem. Wc show that an c-strongly convex O (e + ~)-hull of S can be constructed in 0(log3 n) time using n processors with imprecise computations, where /3 is the error unit of the primitive operations. The rmult implies an improved sequential algorithm for t hc problem. Our method consists of three algorithms: finding an approximate common tangent of two sets of n points each in O (log2 n) time lining n processors, computing a convex O(e + /?)-hull of n points in 0(log3 n) time using n processors, and constructing an e-strongly convex O(e + ~)-hull of a convex n-gon in O(log n) time with n processors using imprecise computations. Wc use the CREW PRAM computational model in the paper. * This work is partly supported by The Hori Information Science Promotion Foundation(1995). t Department of Electrica.land Computer Engineering, Nagoya Institute of Technology, Showa, Nagoya 466, .Japan. E-mail: (then, wada, kawaguchi)@elcom. nitecll.ac.jp Permission to makedtgital/hard copies of all or part of thk material for personal or classrmm use is grantad without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication and its data appear, and notice is given that copyright ia by permission of the ACM, Jnc. To copy otherwise, to republish, to post on servers or to dkibutc to lists, requires specific permiaaion andlor fee. Computational Geometry’96, Philadelphia PA, USA

Efficient Algorithms for a Mixed k-Partition Problem of Graphs without Specifying Bases

This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements(vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases(called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called k-PART-WOB). In this paper, we show that k-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1)for any k≥2, k-PART-WOB can be solved in O(∥V∥√∥V∥log2∥V∥+∥E∥) time for every 4-edge-connected graph G=(V,E), (2)3-PART-WOB can be solved in O(∥V∥2) for every 2-edge-connected graph G=(V,E) and (3)4-PART-WOB can be solved in O(∥E∥2) for every 3-edge-connected graph G=(V,E).

On computing the upper envelope of segments in parallel

Given a collection of segments in the plane that intersect pairwise at most k times, regarding the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point (0,+/spl infin/). We give efficient parallel methods for finding the upper envelope of k-intersecting segments for any integer k/spl ges/0, in the weakest shared memory model, the EREW PRAM. We show that the upper envelope of n k-intersecting segments can be found in 0(log/sup 1+/spl epsiv//n) time using 0(/spl lambda//sub k+1/(n)/log/sup /spl epsiv//n) processors for any /spl epsiv/>0, where /spl lambda//sub k+2/(n)/sup 1/ is the size of the upper envelope. In particular, for line segments we show the following optimal algorithms: the upper envelope of n line segments can be found in O(log n) time using O(n) processors, and if the line segments are nonintersecting and sorted, the envelope can be found in O(log n) time using O(n/log n) processors. We also show that our methods imply a fast sequential result: the upper envelope of n sorted line segments can be found in O(n log log n) time sequentially, which improves the known lowest upper bound O(n log n).

Finding the Convex Hull of Discs in Parallel

We present a parallel method for finding the convex hull of planar discs in the EREW PRAM model. We show that the convex hull of n discs in the plane can be computed in O(log1+e n) time using O(n/l...

A Hardware Implementation of PRAM and Its Performance Evaluation

A PRAM (Parallel Random Access Machine) [4] is the parallel computational model most notable for supporting the parallel algorithmic theory. It consists of a number of processors sharing a common memory. The processors communicate by exchanging data through a shared memory cell. Each processor can access any memory cell at one unit of time and all processors operate synchronously under the control of a common clock. These facts make the model a very advantageous platform for considering the inherent parallelism of problems. How ever, the development of parallel computers which fit this model has not quite matched the theoretical requests. The researches focusing on the reduction of this gap has been carried out [2, 5, 6, 7, 8, 9]. However, most of them give only theoretical analysis; the implementation of the PRAM on hardware level is seldom seen.

Parallel algorithms for partitioning sorted sets and related problems

Abstract. We consider the following partition problem: Given a set S of n elements that is organized as k sorted subsets of size n/k each and given a parameter h with 1/k ≤ h ≤ n/k , partition S into g = O(n/(hk)) subsets D1, D2, . . . , Dg of size Θ(hk) each, such that, for any two indices i and j with 1 ≤ i < j ≤ g , no element in Di is bigger than any element in Dj . Note that with various combinations of the values of parameters h and k , several fundamental problems, such as merging, sorting, and finding an approximate median, can be formulated as or be reduced to this partition problem. The partition problem also finds many applications in solving problems of parallel computing and computational geometry. In this paper we present efficient parallel algorithms for solving the partition problem and a number of its applications. Our parallel partition algorithm runs in O( log n) time using

An Optimal Fault-Tolerant Routing for Triconnected Planar Graphs

Oleft(frac{min{(n/h)*max{log h,1}, n*max{log(1/h),1}}}{log n}right)