On the parallel-decomposability of geometric problems

There is a large and growing body of literature concerning the solutions of geometric problems on mesh-connected arrays of processors. Most of these algorithms are optimal (i.e., run in timeO(n ^1/d ) on ad-dimensionaln-processor array), and they all assume that the parallel machine is trying to solve a problem of sizen on ann-processor array. Here we investigate the situation where we have a mesh of sizep and we are interested in using it to solve a problem of sizen >p. The goal we seek is to achieve, when solving a problem of sizen >p, the same speed up as when solving a problem of sizep. We show that for many geometric problems, the same speedup can be achieved when solving a problem of sizen >p as when solving a problem of sizep.The research of M. J. Atallah was supported by the Office of Naval Research under Contracts N00014-84-K-0502 and N00014-86-K-0689, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, the National Science Foundation under Grant DCR-8451393, and the National Library of Medicine under Grant R01-LM05118. Jyh-Jong Tsay's research was partially supported by the Office of Naval Research under Contract N00014-84-K-0502, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, and the National Science Foundation under Grant DCR-8451393.     

A faster parallel algorithm for a matrix searching problem

We give an improved parallel algorithm for the problem of computing the tube minima of a totally monotonen ×n ×n matrix, an important matrix searching problem that was formalized by Aggarwal and Park and has many applications. Our algorithm runs inO(log logn) time withO(n²/log logn) processors in theCRCW-PRAM model, whereas the previous best ran inO((log logn)²) time withO(n²/(log logn)² processors, also in theCRCW-PRAM model. Thus we improve the speed without any deterioration in thetime ×processors product. Our improved bound immediately translates into improvedCRCW-PRAM bounds for the numerous applications of this problem, including string editing, construction of Huffmann codes and other coding trees, and many other combinatorial and geometric problems.This research was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, the National Science Foundation under Grant DCR-8451393, and the National Library of Medicine under Grant R01-LM05118. Part of the research was done while the author was at Princeton University, visiting the DIMACS center.     

The accelerated centroid decomposition technique for optimal parallel tree evaluation in logarithmic time

A new general parallel algorithmic technique for computations on trees is presented. In particular, it provides the firstn/logn processor,O(logn)-time deterministic EREW PRAM algorithm for expression tree evaluation. The technique solves many other tree problems within the same complexity bounds.Richard Cole was supported in part by NSF Grants DCR-84-01633 and CCR-8702271, ONR Grant N00014-85-K-0046 and by an IBM faculty development award. Uzi Vishkin was supported in part by NSF Grants NSF-CCR-8615337 and NSF-DCR-8413359, ONR Grant N00014-85-K-0046, by the Applied Mathematical Science subprogram of the office of Energy Research, U.S. Department of Energy under Contract DE-AC02-76ER03077 and the Foundation for Research in Electronics, Computers and Communication, administered by the Israeli Academy of Sciences and Humanities.     

Topological numbering of features on a mesh

Assume we are given ann ×n binary image containing horizontally convex features; i.e., for each feature, each of its row's pixels form an interval on that row. In this paper we consider the problem of assigning topological numbers to such features, i.e., assign a number to every featuref so that all features to the left off in the image have a smaller number assigned to them. This problem arises in solutions to the stereo matching problem. We present a parallel algorithm to solve the topological numbering problem inO(n) time on ann ×n mesh of processors. The key idea of our solution is to create a tree from which the topological numbers can be obtained even though the tree does not uniquely represent the to the left of relationship of the features.The work of M. J. Atallah was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, and the National Science Foundation under Grant DCR-8451393, with matching funds from AT&T. Part of this work was done while he was a Visiting Scientist at the Center for Advanced Architectures project of the Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffett Field, CA 94035, USA. S. E. Hambrusch's work was supported by the Office of Naval Research under Contracts N00014-84-K-0502 and N00014-86K-0689, and by the National Science Foundation under Grant MIP-87-15652. Part of this work was done while she was visiting the International Computer Science Institute, Berkeley, CA 94704, USA. The work of L. E. TeWinkel was supported by the Office of Naval Research under Contract N00014-86K-0689.     

An efficient algorithm for maxdominance, with applications

Given a planar setS ofn points,maxdominance problems consist of computing, for everyp S, some function of the maxima of the subset ofS that is dominated byp. A number of geometric and graph-theoretic problems can be formulated as maxdominance problems, including the problem of computing a minimum independent dominating set in a permutation graph, the related problem of finding the shortest maximal increasing subsequence, the problem of enumerating restricted empty rectangles, and the related problem of computing the largest empty rectangle. We give an algorithm for optimally solving a class of maxdominance problems. A straightforward application of our algorithm yields improved time bounds for the above-mentioned problems. The techniques used in the algorithm are of independent interest, and include a linear-time tree computation that is likely to arise in other contexts.The research of this author was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, and the National Science Foundation under Grant DCR-8451393, with matching funds from AT&T.This author's research was supported by the National Science Foundation under Grant DCR-8506361.     

Programming simultaneous actions using common knowledge

This work applies the theory of knowledge in distributed systems to the design of efficient fault-tolerant protocols. We define a large class of problems requiring coordinated, simultaneous action in synchronous systems, and give a method of transforming specifications of such problems into protocols that areoptimal in all runs: these protocols are guaranteed to perform the simultaneous actions as soon as any other protocol could possibly perform them, given the input to the system and faulty processor behavior. This transformation is performed in two steps. In the first step we extract, directly from the problem specification, a high-level protocol programmed using explicit tests for common knowledge. In the second step we carefully analyze when facts become common knowledge, thereby providing a method of efficiently implementing these protocols in many variants of the omissions failure model. In the generalized omissions model, however, our analysis shows that testing for common knowledge is NP-hard. Given the close correspondence between common knowledge and simultaneous actions, we are able to show that no optimal protocol for any such problem can be computationally efficient in this model. The analysis in this paper exposes many subtle differences between the failure models, including the precise point at which this gap in complexity occurs.This research was supported by the Office of Naval Research under contract N00014-85-K-0168, by the Office of Army Research under contract DAAG29-84-K-0058, by the National Science Foundation under Grant DCR-8302391, and by the Defense Advanced Research Projects agency (DARPA) under contract N00014-83-K-0125, and was performed while both authors were at MIT. A preliminary version of this work appeared in theProceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science, Toronto, 1986.This author was primarily supported by an IBM postdoctoral fellowship.     

Augmented infinitesimal perturbation analysis: An alternate explanation

Augmented infinitesimal perturbation analysis (APA) was introduced by Gaivoronski [1991] to increase the purview of the theory of Infinitesimal Perturbation Analysis (IPA). In reference [Gaivoronski 1991] it is shown that an unbiased estimate for the gradient of a class of performance measures of DEDS represented bygeneralized semi-Markov processes (GSMPs) (cf. [Glynn 1989] can be expressed as a sum of an IPA-estimate and a term that takes into account the event order changes. In this paper we present an alternate approach to establishing the result of Gaivoronski, and from this we derive a necessary and sufficient condition for the validity of the IPA algorithm for this class of performance measures. Finally we validate our results by simulation examples.This research was supported by the National Science Foundation under grant number ECS-85-15449, Office of Naval Research Grants Nos. N00014-90-K-1093 and N00014-89-J-1023 and by Army Grant No. DAAL-03-86-K-0171.     

Convex hulls of objects bounded by algebraic curves

We present an0(n ·d ^o(1)) algorithm to compute the convex hull of a curved object bounded by0(n) algebraic curve segments of maximum degreed.Research supported in part by NSF Grant MIP-85 21356, ARO Contract DAA G29-85-C0018 under Cornell MSI, and ONR Contract N00014-88-K-0402. This paper is an updated version of a part of [6].     

A modular drinking philosophers algorithm

Summary A variant of the drinking philosophers algorithm of Chandy and Misra is described and proved correct in a modular way. The algorithm of Chandy and Misra is based on a particular dining philosophers algorithm and relies on certain properties of its implementation. The drinking philosophers algorithm presented in this paper is able to use an arbitrary dining philosophers algorithm as a subroutine; nothing about the implementation needs to be known, only that it solves the dining philosophers problem. An important advantage of this modularity is that by substituting a more time-efficient dining philosophers algorithm than the one used by Chandy and Misra, a drinking philosophers algorithm withO(1) worst-case waiting time is obtained, whereas the drinking philosophers algorithm of Chandy and Misra hasO(n) worst-case waiting time (forn philosophers). Careful definitions are given to distinguish the drinking and dining philosophers problems and to specify varying degrees of concurrency. Jennifer L. Welch received her B.A. in 1979 from the University of Texas at Austin, and her S.M. and Ph.D. from the Massachusetts Institute of Technology in 1984 and 1988 respectively. She has been a member of technical staff at GTE Laboratories Incorporated in Waltham, Massachusetts and an assistant professor at the University of North Carolina at Chapel Hill. She is currently an assistant professor at Texas A&M University. Her research interests include algorithms and lower bounds for distributed computing.Much of this work was performed while this author was at the Laboratory for Computer Science, Massachusetts Institute of Technology, supported by the Advanced Research Projects Agency of the Department of Defense under contract N00014-83-K-0125, the National Science Foundation under grants DCR-83-02391 and CCR-86-11442, the Office of Army Research under contract DAAG29-84-K-0058, and the Office of Naval Research under contract N00014-85-K-0168. This author was also supported in part by NSF grant CCR-9010730, an IBM Faculty Development Award, and NSF Presidential Young Investigator Award CCR-9158478This author was supported by the Office of Naval Research under contract N00014-91-J-1046, the Advanced Research Projects Agency of the Department of Defense under contract N00014-89-J-1988, and the National Science Foundation under grant CCR-89-15206. The photograph and autobiography of Professor N.A. Lynch were published in Volume 6, No. 2, 1992 on page 121     

Decentralized detection by a large number of sensors

We consider the decentralized detection problem, in whichN independent, identical sensors transmit a finite-valued function of their observations to a fusion center which then decides which one ofM hypotheses is true. For the case where the number of sensors tends to infinity, we show that it is asymptotically optimal to divide the sensors intoM(M-1)/2 groups, with all sensors in each group using the same decision rule in deciding what to transmit. We also show how the optimal number of sensors in each group may be determined by solving a mathematical programming problem. For the special case of two hypotheses and binary messages the solution simplifies considerably: it is optimal (asymptotically, asN→∞) to have all sensors perform an identical likelihood ratio test, and the optimal threshold is very easy to determine numerically. Research supported by the Army Research Office under Contract DAAG-29-84-K-0005 and by the Office of Naval Research under Contract N00014-84-K-0519.     

A parallel algorithm for approximating the minimum cycle cover

We address the problem of approximating aminimum cycle cover in parallel. We give the first efficient parallel algorithm for finding an approximation to aminimum cycle cover. Our algorithm finds a cycle cover whose size is within a factor of 0(1 +n logn/(m + n) of the minimum-sized cover usingO(log² n) time on (m + n)/logn processors.Research supported by ONR Grant N00014-88-K-0243 and DARPA Grant N00039-88-C0113 at Harvard University.Research supported by a graduate fellowship from GE. Additional support provided by Air Force Contract AFOSR-86-0078, and by an NSF PYI awarded to David Shmoys, with matching funds from IBM, Sun Microsystems, and UPS.     

The complexity of induced minors and related problems

The computational complexity of a number of problems concerning induced structures in graphs is studied, and compared with the complexity of corresponding problems concerning non-induced structures. The effect on these problems of restricting the input to planar graphs is also considered. The principal results include: (1) Induced Maximum Matching and Induced Directed Path are NP-complete for planar graphs, (2) for every fixed graphH, InducedH-Minor Testing can be accomplished for planar graphs in time0(n), and (3) there are graphsH for which InducedH-Minor Testing is NP-complete for unrestricted input. Some useful structural theorems concerning induced minors are presented, including a bound on the treewidth of planar graphs that exclude a planar induced minor.The research of the first author was supported by the U.S. Office of Naval Research under Contract N00014-88-K-0456, by the U.S. National Science Foundation under Grant MIP-8603879, and by the National Science and Engineering Research Council of Canada. The second author acknowledges the support of the U.S. Office of Naval Research when visiting the University of Idaho in spring 1990. Some results were also obtained during a visit to the University of Cologne in fall 1990.     

A bisection method for computing the H∞ norm of a transfer matrix and related problems

We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H_∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H_∞ Armijo line-search problem with no more computation than is required to compute a single H_∞ norm. Research supported in part by NSF under Grant ECS-85-52465, ONR under Grant N00014-86-K-0112, an IBM faculty development award, and Bell Communications Research.     

Distributed match-making

In many distributed computing environments, processes are concurrently executed by nodes in a store- and-forward communication network. Distributed control issues as diverse as name server, mutual exclusion, and replicated data management involve making matches between such processes. We propose a formal problem called distributed match-making as the generic paradigm. Algorithms for distributed match-making are developed and the complexity is investigated in terms of messages and in terms of storage needed. Lower bounds on the complexity of distributed match-making are established. Optimal algorithms, or nearly optimal algorithms, are given for particular network topologies.The work of the second author was supported in part by the Office of Naval Research under Contract N00014-85-K-0168, by the Office of Army Research under Contract DAAG29-84-K-0058, by the National Science Foundation under Grant DCR-83-02391, and by the Defence Advanced Research Projects Agency (DARPA) under Contract N00014-83-K-0125. Current address of both authors: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.     

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.Work on this paper by this author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, the IBM Corporation, and from the U.S.-Israel Binational Science Foundation.Work on this paper by this author has been supported by National Science Foundation Grant DCR-86-05962.     

Implementing a Scheme-based parallel processing system

The Scheme language can be converted into a parallel processing language by adding two new data types (placeholders andweak pairs), two processor synchronization primitives, and a task distribution mechanism. The mechanisms that support task creation, scheduling, and task synchronization are built using these extensions and features already present in the sequential language. Implementing the core of the parallel processing component in Scheme itself provides testbed for a variety of experiments and extensions.MultiScheme, the system resulting from these extensions, supports Halstead's future construct as the simple model for parallelism. By revealing the underlying placeholders on top of which this construct is built, Multischeme supports a variety of additional parallel programming techniques. It supports speculative computation through a simple procedural interface and the automatic garbage collection of tasks. The qlet and qlambda constructs of the QLisp language are also easily implemented in MultiScheme, as are the more familiar fork and join constructs of imperative programming.This research was supported in part by the Defense Advanced Research Projects Agency and was monitored by the Office of Naval Research under contract numbers N00014-83-K-0125, N00014-84-K0099, N00014-86-K-0180, and MDA903-84-C-0033. Additional funds and resources were provided by BBN Advanced Computers Inc., and the Hewlett-Packard Corporation. The work was performed as part of the author's dissertation research at the Massachusetts Institute of Technology.     

An efficient parallel algorithm for shortest paths in planar layered digraphs

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs.We show that these graphs admit special kinds of separators calledone- way separators which allow the paths in the graph to cross it only once. We use these separators to give divide- and -conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in ann-node planar layered digraph in timeO(log² n) usingn/logn processors. We then use results of Aggarwal and Park [1] and Atallah [4] to improve the time bound toO(log² n) in the CREW model andO(logn log logn) in the CREW model. The processor bounds still remain asn/logn for the CREW model andn/log logn for the CRCW model.Support for the first and third authors was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052, ARPA, Order 8225. Support for the second author was provided in part by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052 and ARPA Order 8225.     

A note on weighted distributed match-making

In many distributed computing environments, processes are concurrently executed by nodes in a store-and-forward network. Distributed control issues as diverse as name-server, mutual exclusion, and replicated data management, involve making matches between processes. The generic paradigm is a formal problem called “distributed match-making.” We define multidimensional and weighted versions, and the relations between the two, and develop a very general method to prove lower bounds on the complexity as a tradeoff between number of messages and “distributedness.” The resulting lower bounds are tight in all cases we have examined. We present a success-stop version of distributed match-making that is analysed in terms of a weight distribution that in all cases results in approximately halving the (expected) number of messages required in the corresponding strategy that does not use these weights. The second author did part of this work at the Laboratory for Computer Science, M.I.T., Cambridge, MA. He was supported in part by the Office of Naval Research under Contract N00014-85-K-0168, by the Office of Army Research under Contract DAAG29-84-K-0058, by the National Science Foundation under Grant DCR-83-02391, and by the Defence Advanced Research Projects Agency (DARPA) under Contract N00014-83-K-0125. A preliminary version of this paper appeared inProc. VLSI Algorithms and Architectures, 3rd Aegean Workshop on Computing (AWOC 88), Lecture Notes in Computer Science, vol. 319, Springer-Verlag, Berlin, 1988, pp. 361–368.     

Fast algorithms for bit-serial routing on a hypercube

In this paper we describe anO(logN)-bit-step randomized algorithm for bit-serial message routing on a hypercube. The result is asymptotically optimal, and improves upon the best previously known algorithms by a logarithmic factor. The result also solves the problem of on-line circuit switching in anO(1)-dilated hypercube (i.e., the problem of establishing edge-disjoint paths between the nodes of the dilated hypercube for any one-to-one mapping).Our algorithm is adaptive and we show that this is necessary to achieve the logarithmic speedup. We generalize the Borodin-Hopcroft lower bound on oblivious routing by proving that any randomized oblivious algorithm on a polylogarithmic degree network requires at least (log² N/log logN) bit steps with high probability for almost all permutations.This research was supported by the Defense Advanced Research Projects Agency under Contracts N00014-87-K-825 and N00014-89-J-1988, the Air Force under Contract AFOSR-89-0271, and the Army under Contract DAAL-03-86-K-0171. This work was completed while the third and fourth authors were at the Laboratory for Computer Science, Massachusetts Institute of Technology.     

An assessment of multilisp: Lessons from experience

Multilisp is a parallel programming language derived from the Scheme dialect of Lisp by addition of thefuture construct. It has been implemented on Concert, a 32-processor shared-memory multiprocessor. A statistics-gathering feature of Concert Multilisp producesparallelism profiles showing the number of processors busy with computing or overhead, as a function of time. Experience gained using parallelism profiles and other measurement tools on several application programs has revealed three basic ways in whichfuture generates concurrency. These ways are illustrated on two example programs: the Lisp mapping functionmapcar and the partitioning routine from Quicksort. Experience with Multilisp programming exposes issues relating to side effects, error and exception handling, low-level operations for explicit manipulation of futures and tasks, and speculative computing, which are also discussed. The basic outlines of Multilisp are now fairly clear and have stood the test of being used for several applications, but further language design work is especially needed in the areas of speculative computing and exception handling.This research was supported in part by the Defense Advanced Research Projects Agency and was monitored by the Office of Naval Research under contract numbers N00014-83-K-0125 and N00014-84-K-0099.