A 2n−2 step algorithm for routing in ann ×n array with constant-size queues

In this paper we describe a deterministic algorithm for solving any 1–1 packet-routing problem on ann ×n mesh in 2n–2 steps using constant-size queues. The time bound is optimal in the worst case. The best previous deterministic algorithm for this problem required time 2n+(n/q) using queues of size (q) for any 1qn, and the best previous randomized algorithm required time 2n+(logn) using constant-size queues.This research was supported by the Clear Center at UTD, DARPA Contracts N00014-91-J-1698 and N00014-92-J-1799, Air Force Contract F49620-92-J-0125, Army Contract DAAL-03-86-K-0171, an NSF Presidential Young Investigator Award with matching funds from AT&T and IBM, and by the Texas Advanced Research Program under Grant No. 3972. A preliminary version of this paper appeared in [5].     

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.     

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.     

Searching among intervals and compact routing tables

Shortest paths in weighted directed graphs are considered within the context of compact routing tables. Strategies are given for organizing compact routing tables so that extracting a requested shortest path will takeo(k logn) time, wherek is the number of edges in the path andn is the number of vertices in the graph. The first strategy takesO (k+logn) time to extract a requested shortest path. A second strategy takes (k) time on average, assuming alln(n–1) shortest paths are equally likely to be requested. Both strategies introduce techniques for storing collections of disjoint intervals over the integers from 1 ton, so that identifying the interval within which a given integer falls can be performed quickly.This research was supported in part by the National Science Foundation under Grants CCR-9001241 and CCR-9322501 and by the Office of Naval Research under Contract N00014-86-K-0689.     

Parallel triangulation of a polygon in two calls to the trapezoidal map

We give a parallel method for triangulating a simple polygon by two (parallel) calls to the trapezoidal map computation. The method is simpler and more elegant than previous methods. Along the way we obtain an interesting partition of one-sided monotone polygons. Using the best-known trapezoidal map algorithm, ours run in timeO(logn) usingO(n) CREW PRAM processors.This research was supported by NSF Grants No. DCR-84-01898 and No. DCR-84-01633, and ONR Contract N00014-85-K-0046.     

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].     

Checking similarity of planar figures

Two planar figures aresimilar if a scaled version of one of them can be moved so that it coincides with the second figure. The problem of checking whether two planar figures are similar is relevant to both computational geometry and pattern recognition. An efficient algorithm is known for checking whether two polygonsP andQ are similar⁽¹⁾ The purpose of this note is to give an efficient algorithm for checking whether two planar figuresP andQ are similar when the figures are no longer constrained to be polygons. We give anO(n logn) time algorithm for solving this problem when each figure consists of a collection of (possibly intersecting) straight line segments, circles, and ellipses. Our algorithm can easily be modified for figures which include other geometric patterns as well. We also prove that our algorithm is optimal.This work was partially supported by the Office of Naval Research under Contract N00014-84-K-0502.     

An efficient distributed algorithm for maximum matching in general graphs

We present a distributed algorithm for maximum cardinality matching in general graphs. On a general graph withn vertices, our algorithm requiresO(n ^5/2) messages in the worst case. On trees, our algorithm computes a maximum matching usingO(n) messages after the election of a leader.Work on this paper has been supported by the Office of Naval Research under Contract N00014-85-K-0570.     

Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires (N/d) parallel communication steps in the worst case. This is an improvement upon the (N/d ^3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in (N/logN) steps. The best previously known upper bound was (N). Our algorithm may be practical for smallN (up to about 2¹⁴ nodes).C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.     

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.     

Planar geometric location problems

We present anO(n ² log³ n) algorithm for the two-center problem, in which we are given a setS ofn points in the plane and wish to find two closed disks whose union containsS so that the larger of the two radii is as small as possible. We also give anO(n ² log⁵ n) algorithm for solving the two-line-center problem, where we want to find two strips that coverS whose maximum width is as small as possible. The best previous solutions of both problems requireO(n ³) time.Pankaj Agarwal has been supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), an NSF Science and Technology Center, under Grant STC-88-09648. Micha Sharir has been supported by the Office of Naval Research under Grants N00014-89-J-3042 and N00014-90-J-1284, by the National Science Foundation under Grant CCR-89-01484, by DIMACS, and by grants from the US-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this paper has appeared inProceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 449–458.     

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.     

An improved technique for output-sensitive hidden surface removal

We derive a new output-sensitive algorithm for hidden surface removal in a collection ofn triangles, viewed from a pointz such that they can be ordered in an acyclic fashion according to their nearness toz. Ifk is the combinatorial complexity of the outputvisibility map, then we obtain a sophisticated randomized algorithm that runs in (randomized) timeO(n^4/3 log^2.89 n +k ^3/5 n ^{4/5 +} for any > 0. The method is based on a new technique for tracing the visible contours using ray shooting.Work by the first author was partially supported by the ESPRIT II Basic Research Actions Program of the EC, under Contract No. 3075 (project ALCOM). Work by the second author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD-the Israeli National Council for Research and Development-and the Fund for Basic Research in Electronics, Computers, and Communication administered by the Israeli Academy of Sciences. A preliminary version of this paper appeared as part of the conference proceedings paper [17].     

Optimal randomized parallel algorithms for computational geometry

We present parallel algorithms for some fundamental problems in computational geometry which have a running time ofO(logn) usingn processors, with very high probability (approaching 1 asn ). These include planar-point location, triangulation, and trapezoidal decomposition. We also present optimal algorithms for three-dimensional maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on a CREW PRAM model and have optimal processor-time product which improve on the previously best-known algorithms of Atallah and Goodrich [5] for these problems. The crux of these algorithms is a useful data structure which emulates the plane-sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk [26] and Reif and Valiant [25] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.This is a substantially revised version of the paper that appeared as Optimal Randomized Parallel Algorithms for Computational Geometry in theProceedings of the 16th International Conference on Parallel Processing, St. Charles, Illinois, August 1987.This research was supported by DARPA/ARO Contract DAAL03-88-K-0195, Air Force Contract AFOSR-87-0386, DARPA/ISTO Contracts N00014-88-K-0458 and N00014-91-J-1985, and by NASA Subcontract 550-63 of Primecontract NAS5-30428.     

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.     

The general maximum matching algorithm of micali and vazirani

We give a clear exposition of the algorithm of Micali and Vazirani for computing a maximum matching in a general graph. This is the most efficient algorithm known for general matching. On a graph withn vertices andm edges this algorithm runs inO(n ^1/2 m) time.Work on this paper has been supported by the Office of Naval Research under Contract N00014-85-K-0570 and by the Eastman Kodak Company.     

Shortest paths in euclidean graphs

We analyze a simple method for finding shortest paths inEuclidean graphs (where vertices are points in a Euclidean space and edge weights are Euclidean distances between points). For many graph models, the average running time of the algorithm to find the shortest path between a specified pair of vertices in a graph withV vertices andE edges is shown to beO(V) as compared withO(E +V logV) required by the classical algorithm due to Dijkstra.Support for the first author was provided in part by NSF Grant MCS-83-08806. Support for the second author was provided in part by NSF Grants MCS-81-05324 and DCR-84-03613, an NSF Presidential Young Investigator Award, an IBM research contract, and an IBM Faculty Development Award. Support for this research was also provided in part by an ONR and DARPA under Contract N00014-83-K-0146 and ARPA Order No. 4786. Equipment support was provided by NSF Grant MCS-81-218106.     

Optimal routing algorithms for mesh-connected processor arrays

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than (logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.Preliminary reports of portions of the results contained in this paper have appeared in theProceedings of the 1988 Aegean Workshop on Computing [5], and in theProceedings of the 1987 Conference on Foundations of Software Technology and Theoretical Computer Science [18]. The work of the first author was supported in part by NSF Grant NSF-DCR-85-03251 and ONR contract N00014-80-C-0647. The work of the second author was supported in part by NSF Grant NSF-DCR-86-00379.     

Communication-efficient parallel algorithms for distributed random-access machines

This paper introduces a model for parallel computation, called thedistributed randomaccess machine (DRAM), in which the communication requirements of parallel algorithms can be evaluated. A DRAM is an abstraction of a parallel computer in which memory accesses are implemented by routing messages through a communication network. A DRAM explicitly models the congestion of messages across cuts of the network.We introduce the notion of aconservative algorithm as one whose communication requirements at each step can be bounded by the congestion of pointers of the input data structure across cuts of a DRAM. We give a simple lemma that shows how to shortcut pointers in a data structure so that remote processors can communicate without causing undue congestion. We giveO(lgn)-step, linear-processor, linear-space, conservative algorithms for a variety of problems onn-node trees, such as computing treewalk numberings, finding the separator of a tree, and evaluating all subexpressions in an expression tree. We giveO(lg² n)-step, linear-processor, linear-space, conservative algorithms for problems on graphs of sizen, including finding a minimum-cost spanning forest, computing biconnected components, and constructing an Eulerian cycle. Most of these algorithms use as a subroutine a generalization of the prefix computation to trees. We show that any suchtreefix computation can be performed inO(lgn) steps using a conservative variant of Miller and Reif's tree-contraction technique.This research was supported in part by the Defense Advanced Research Projects Agency under Contract N00014-80-C-0622 and by the Office of Naval Research under Contract N00014-86-K-0593. Charles Leiserson is supported in part by an NSF Presidential Young Investigator Award with matching funds provided by AT&T Bell Laboratories and Xerox Corporation. Bruce Maggs is supported in part by an NSF Fellowship.     

On parallel integer sorting

We present an optimal algorithm for sortingn integers in the range [1,n ^c ] (for any constantc) for the EREW PRAM model where the word length isn , for any >0. Using this algorithm, the best known upper bound for integer sorting on the (O(logn) word length) EREW PRAM model is improved. In addition, a novel parallel range reduction algorithm which results in a near optimal randomized integer sorting algorthm is presented. For the case when the keys are uniformly distributed integers in an arbitrary range, we give an algorithm whose expected running time is optimal.Supported by NSF-DCR-85-03251 and ONR contract N00014-87-K-0310