Optimal parallel verification of minimum spanning trees in logarithmic time

We present the first optimal parallel algorithms for the verification and sensitivity analysis of minimum spanning trees. Our algorithms are deterministic and run inO(logn) time and require linear-work in the CREW PRAM model. These algorithms are used as a subroutine in the linear-work randomized algorithm for finding minimum spanning trees of Cole, Klein, and Tarjan. Research partially supported by a National Science Foundation Graduate Fellowship and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648. Research at Princeton University was partially supported by the National Science Foundation, Grant No. CCR-8920505, the Office of Naval Research, Contract No. N00014-91-J-1463, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648.     

Efficient PRAM simulation on a distributed memory machine

We present algorithms for the randomized simulation of a shared memory machine (PRAM) on a Distributed Memory Machine (DMM). In a PRAM, memory conflicts occur only through concurrent access to the same cell, whereas the memory of a DMM is divided into modules, one for each processor, and concurrent accesses to the same module create a conflict. Thedelay of a simulation is the time needed to simulate a parallel memory access of the PRAM. Any general simulation of anm processor PRAM on ann processor DMM will necessarily have delay at leastm/n. A randomized simulation is calledtime-processor optimal if the delay isO(m/n) with high probability. Using a novel simulation scheme based on hashing we obtain a time-processor optimal simulation with delayO(log log(n) log*(n)). The best previous simulations use a simpler scheme based on hashing and have much larger delay: (log(n)/log log(n)) for the simulation of an n processor PRAM on ann processor DMM, and (log(n)) in the case where the simulation is time-processor optimal.Our simulations use several (two or three) hash functions to distribute the shared memory among the memory modules of the PRAM. The stochastic processes modeling the behavior of our algorithms and their analyses based on powerful classes of universal hash functions may be of independent interest.Research partially supported by NSF/DARPA Grant CCR-9005448. Work was done while at the University of California at Berkeley and the International Computer Science Institute, Berkeley, CA.Research partially supported by National Science Foundation Operating Grant CCR-9016468, National Science Foundation Operating Grant CCR-9304722, United States-Israel Binational Science Foundation Grant No. 89-00312, United States-Israel Binational Science Foundation Grant No. 92-00226, and ESPRIT BR Grant EC-US 030.Part of work was done during a visit at the International Computer Science Institute at Berkeley; supported in part by DFG-Forschergruppe Effiziente Nutzung massiv paralleler Systeme, Teilprojekt 4, and by the Esprit Basic Research Action Nr. 7141 (ALCOM II).     

Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.This research was announced in preliminary form in theProceedings of the 6th ACM Symposium on Computational Geometry, 1990, pp. 73–82. The research of Michael T. Goodrich was supported by the National Science Foundation under Grants CCR-8810568 and CCR-9003299, and by the NSF and DARPA under Grant CCR-8908092.     

Parallel algorithms for arrangements

We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log^* n) time andn ²/logn processors. We generalize this to obtain anO (logn log^* n)-time algorithm usingn ^d /logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.This work was supported by the National Science Foundation, under Grants CCR-8657562 and CCR-8858799, NSF/DARPA under Grant CCR-8907960, and Digital Equipment Corporation. A preliminary version of this paper appeared at the Second Annual ACM Symposium on Parallel Algorithms and Architectures [3].     

A linear-time algorithm for finding an ambitus

We devise a linear-time algorithm for finding an ambitus ín an undirected graph. An ambitus is a cycle in a graph containing two distinguished vertices such that certain different groups of bridges (calledB ^itp-,B ^itQ-, andB ^itPQ-bridges) satisfy the property that a bridge in one group does not interlace with any bridge in the other groups. Thus, an ambitus allows the graph to be cut into pieces, where, in each piece, certain graph properties may be investigated independently and recursively, and then the pieces can be pasted together to yield information about these graph properties in the original graph. In order to achieve a good time-complexity for such an algorithm employing the divide-and-conquer paradigm, it is necessary to find an ambitus quickly. We also show that, using ambitus, linear-time algorithms can be devised for abiding-path-finding and nonseparating-induced-cycle-finding problems.The research of B. Mishra was supported in part by National Science Foundation Grants DMS-8703458 and CCR-9002819. R. E. Tarjan's research at Princeton University was partially supported by DIMACS, a National Science Foundation Science and Technology Center, Grant No. NSF-STC88-09648, and by National Science Foundation Grant CCR-8929505.     

Optimal parallel algorithms for point-set and polygon problems

In this paper we give parallel algorithms for a number of problems defined on point sets and polygons. All our algorithms have optimalT(n) * P(n) products, whereT(n) is the time complexity andP(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. Our algorithms provide parallel analogues to well-known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently than point-set problems, and that nearest-neighbor problems can be solved without explicitly constructing a Voronoi diagram.     

Parallel computational geometry of rectangles

Rectangles in a plane provide a very useful abstraction for a number of problems in diverse fields. In this paper we consider the problem of computing geometric properties of a set of rectangles in the plane. We give parallel algorithms for a number of problems usingn processors wheren is the number of upright rectangles. Specifically, we present algorithms for computing the area, perimeter, eccentricity, and moment of inertia of the region covered by the rectangles inO(logn) time. We also present algorithms for computing the maximum clique and connected components of the rectangles inO(logn) time. Finally, we give algorithms for finding the entire contour of the rectangles and the medial axis representation of a givenn × n binary image inO(n) time. Our results are faster than previous results and optimal (to within a constant factor).The work of Sung Kwan Kim was supported by NSF Grant CCR-87-03196 and the work of D. M. Mount was partially supported by National Science Foundation Grant CCR-89-08901.     

Expected parallel time and sequential space complexity of graph and digraph problems

This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to beO(log logn) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) andO(logn · log logn) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel timeO(log logn) for the CRCW PRAM andO(logn) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before.For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time ofO(log logn) on the CRCW PRAM, withO(n) expected number of processors only.Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential spaceO(logn · log logn) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.This research was supported by National Science Foundation Grant DCR-85-03251 and Office of Naval Research Contract N00014-80-C-0647.This research was partially supported by the National Science Foundation Grants MCS-83-00630, DCR-8503497, by the Greek Ministry of Research and Technology, and by the ESPRIT Basic Research Actions Project ALCOM.     

Randomized competitive algorithms for the list update problem

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constantd. We prove lower bounds for deterministic list update algorithms and for randomized algorithms against oblivious and adaptive on-line adversaries. In particular, we show that for this problem adaptive on-line and adaptive off-line adversaries are equally powerful.A preliminary version of these results appeared in a joint paper with S. Irani in theProceedings of the 2nd Symposium on Discrete Algorithms, 1991 [17].This research was partially supported by NSF Grants CCR-8808949 and CCR-8958528.This research was partially supported by NSF Grant CCR-9009753.This research was supported in part by the National Science Foundation under Grant CCR-8658139, by DIMACS, a National Science Foundation Science and Technology center, Grant No. NSF-STC88-09648.     

Constructing the Voronoi diagram of a set of line segments in parallel

In this paper we give a parallel algorithm for constructing the Voronoi diagram of a polygonal scene, i.e., a set of line segments in the plane such that no two segments intersect except possibly at their endpoints. Our algorithm runs inO(log² n) time usingO(n) processors in the CREW PRAM model.The research of M. T. Goodrich was supported by NSF under Grants CCR-8810568 and CCR-9003299 and by NSF/DARPA under Grant CCR-8908092. C. K. Yap's research was supported in part by NSF Grants DCR-8401898 and CCR-9002819.     

Algorithms for bichromatic line-segment problems and polyhedral terrains

We consider a variety of problems on the interaction between two sets of line segments in two and three dimensions. These problems range from counting the number of intersecting pairs between m blue segments andn red segments in the plane (assuming that two line segments are disjoint if they have the same color) to finding the smallest vertical distance between two nonintersecting polyhedral terrains in three-dimensional space. We solve these problems efficiently by using a variant of the segment tree. For the three-dimensional problems we also apply a variety of recent combinatorial and algorithmic techniques involving arrangements of lines in three-dimensional space, as developed in a companion paper.Work on this paper by the first author has been supported in part by the National Science Foundation under Grant CCR-9002352. Work by the second author was supported in part by the National Science Foundation under Grant CCR-8714565. The fourth author has been supported in part by the Office of Naval Research under Grant N0014-87-K-0129, by the National Science Foundation under Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a grant from the US-Israeli Binational Science Foundation.     

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.     

A path ordering for proving termination of AC rewrite systems

We describe a method that extends the lexicographic recursive path ordering of Dershowitz and Kamin and Levy for proving termination of associative-commutative (AC) rewrite systems. Instead of comparing the arguments of an AC-operator using the multiset extension, wepartition them into disjoint subsets and use each subset only once for comparison. To preserve transitivity, we introduce two techniques —pseudocopying andelevating of arguments of an AC operator. This method imposesno restrictions at all on the underlying precedence relation on function symbols. It can therefore prove termination of a much more extensive class of AC rewrite systems than can previous methods, such as associative path ordering, that restrict AC operators to be minimal or subminimal in precedence. A number of examples illustrating the power of the approach are discussed. The method has been implemented inRRL, Rewrite Rule Laboratory, a theorem-proving environment based on rewrite techniques and completion.A preliminary version appears in Proc. of10th Conference on Foundations of Software Technology and Theoretical Computer Science, Bangalore, India (1990).Partially supported by the National Science Foundation Grant no. CCR-8906678.Partially supported by the National Science Foundation Grant no. CCR-9009755Partially supported by the National Science Foundation under Grants CCR-9202838 and CCR-9357851.     

Efficient parallel algorithms forr-dominating set andp-center problems on trees

We develop efficient parallel algorithms for ther-dominating set and thep-center problems on trees. On a concurrent-read exclusive-write PRAM, our algorithm for ther-dominating set problem runs inO(logn log logn) time withn processors. The algorithm for thep-center problem runs inO(log² n log logn) time withn processors.Xin He was supported in part by an Ohio State University Presidential Fellowship, and by the Office of Research and Graduate Studies of Ohio State University. Yaacov Yesha was supported in part by the National Science Foundation under Grant No. DCR-8606366.     

On deterministic approximation of DNF

We develop several quasi-polynomial-time deterministic algorithms for approximating the fraction of truth assignments that satisfy a disjunctive normal form formula. The most efficient algorithm computes for a given DNF formulaF onn variables withm clauses and > 0 an estimateY such that ¦Pr[F] –Y¦ in time which is , for any constant. Although the algorithms themselves are deterministic, their analysis is probabilistic and uses the notion of limited independence between random variables.Research supported in part by National Science Foundation Operating Grant CCR-9016468, National Science Foundation Operating Grant CCR-9304722, United States-Israel Binational Science Foundation Grant No. 89-00312, United States-Israel Binational Science Foundation Grant No. 92-00226, and ESPRIT Basic Research Grant EC-US 030.Research partially done while visiting the International Computer Science Institute and while at Carnegie Mellon University.     

Optimal cooperative search in fractional cascaded data structures

Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total sizen. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel. The preprocessing takesO(logn) time withn/logn processors on an EREW PRAM. For a balanced binary tree, cooperative search along root-to-leaf paths can be done inO((logn)/logp) time usingp processors on a CREW PRAM. Both of these time/processor constraints are optimal. The searching in the fractional cascaded data structure can be either explicit, in which the search path is specified before the search starts, or implicit, in which the branching is determined at each node. We apply this technique to a variety of geometric problems, including point location, range search, and segment intersection search.An earlier version of this work appears inProceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, July 1990, pp. 307–316. The first author's support was provided in part by National Science Foundation Grant CCR-9007851, by the U.S. Army Research Office under Grants DAAL03-91-G-0035 and DAAH04-93-0134, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052, ARPA Order 8225. This research was performed while the second author was at Brown University. Support was provided in part by an NSF Presidential Young Investigator Award CCR-9047466, with matching funds from IBM, by National Science Foundation Grant CCR-9007851, by the U.S. Army Research Office under 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.     

Selecting distances in the plane

We present a randomized algorithm for computing the kth smallest distance in a set ofn points in the plane, based on the parametric search technique of Megiddo [Mel]. The expected running time of our algorithm is O(n^4/3 log^8/3 n). The algorithm can also be made deterministic, using a more complicated technique, with only a slight increase in its running time. A much simpler deterministic version of our procedure runs in time O(n^3/2 log^5/2 n). All versions improve the previously best-known upper bound ofO(@#@ n^9/5 log^4/5 n) by Chazelle [Ch]. A simpleO(n logn)-time algorithm for computing an approximation of the median distance is also presented.Part of this work was done while the first two authors were visting DIMACS, Rutgers University, New Brunswick, NJ. Work by the first three authors has been partly supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648. Work by the second author has also been supported by National Security Agency Grant MDA 904-89-H-2030. Work by the third author has also been supported by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Incremental topological flipping works for regular triangulations

A set ofn weighted points in general position in ^d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n ^[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.The research of both authors was supported by the National Science Foundation under Grant CCR-8921421 and the research by the first author was also supported under the Alan T. Waterman award, Grant CCR-9118874. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation.     

Searching for empty convex polygons

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear-time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r> 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.The first author is pleased to acknowledge support by the National Science Foundation under Grant CCR-8700917. The research of the second author was supported by Amoco Foundation Faculty Development Grant CS 1-6-44862 and by the National Science Foundation under Grant CCR-8714565.     

An optimal algorithm for shortest paths on weighted interval and circular-arc graphs,with applications

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcsand the sorted list of the interval endpoints. Our algorithm solves this problem optimally inO(n) time, wheren is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is anO(qn + n logn)-time algorithm for the minimum-weight circle-cover problem, whereq is the minimum number of arcs crossing any point on the circle; then logn term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were0(n logn) for this shortest paths problem, andO(qn logn) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (logn)-factor improvements in other problems on such graphs.The research of M. J. Atallah was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the Air Force Office of Scientific Research under Contract AFOSR-90-0107, and by the National Science Foundation under Grant CCR-9202807. D. Z. Chen's research was supported in part by the Leonardo Fibonacci Institute, Trento, Italy. The research of D. T. Lee was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the National Science Foundation, and the Office of Naval Research under Grants CCR-8901815, CCR-9309743, and N00014-93-1-0272.