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.     

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.     

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.     

Parallel algorithms for some functions of two convex polygons

Let P andQ be two convex,n-vertex polygons. We consider the problem of computing, in parallel, some functions ofP andQ whenP andQ are disjoint. The model of parallel computation we consider is the CREW-PRAM, i.e., it is the synchronous shared-memory model where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location (even if they are trying to write the same thing). We show that a CREW-PRAM havingn ^1/k processors can compute the following functions in O(k¹⁺) time: (i) the common tangents betweenP andQ, and (ii) the distance betweenP andQ (and hence a straight line separating them). The positive constant can be made arbitrarily close to zero. Even with a linear number of processors, it was not previously known how to achieve constant time performance for computing these functions. The algorithm for problem (ii) is easily modified to detect the case of zero distance as well.This research 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.     

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.     

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.     

AE1: An extension matrix approximate method for the general covering problem

A new approach, the extension matrix approach, is introduced and used to show that some optimization problems in general covering problem areNP-hard. Approximate solutions for these problems are given. Combining these approximate solutions, this paper presents an approximately optimal covering algorithm,AE1. Implementation shows thatAE1 is efficient and gives optimal or near optimal results.This research was supported in part by the National Science Foundation under Grant DCR 84-06801, Office of Naval Research under Grant N00014-82-K-0186, Defense Advanced Research Project Agency under Grant N00014-K-85-0878, and Education Ministry of the People's Republic of China.On leave from Harbin Institute of Technology, Harbin, China.     

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.     

Designing networks with compact routing tables

Classes of network topologies are identified in which shortest-path information can be succinctly stored at the nodes, if they are assigned suitable names. The naming allows each edge at a node to be labeled with zero or more intervals of integers, representing all nodes reachable by a shortest path via that edge. Starting with the class of outerplanar networks, a natural hierarchy of networks is established, based on the number of intervals required. The outerplanar networks are shown to be precisely the networks requiring just one interval per edge. An optimal algorithm is given for determining the labels for edges in outerplanar networks.The research of this author was supported in part by the National Science Foundation under Grant DCR-8320124, and by the Office of Naval Research on contract N 00014-86-K-0689.The research of this author was supported in part by the National Science Foundation under Grant DCR-8320124.     

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.     

On the analysis of cooperation and antagonism in networks of communicating processes

We propose a new method for the analysis of cooperative and antagonistic properties of communicating finite state processes (FSPs). This algebraic technique is based on a composition operator and on the notion of possibility equivalence among FSPs. We demonstrate its utility by showing that potential blocking, termination, and lockout can be decided in polynomial time for loosely connected networks of tree FSPs. Potential blocking and termination are examples of cooperative properties, while lockout is an antagonistic one. For loosely connected networks of (the more general) acyclic FSPs, the cooperative properties become NP-complete and the antagonistic ones PSPACE-complete. For tightly coupled networks of tree FSPs, we also have NP-completeness for the cooperative properties. For the harder case of FSPs with cycles, we provide a natural extension of the method.A preliminary version of this paper appeared as an extended abstract in theProceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, August, 1985, pp. 23–38. P. C. Kanellakis was supported by ONR-DARPA Grant N00014-83-K-0146, NSF Grant DCR-8302391, and by the Office of Army Research under contract DAAG29-84-K-0058. S. A. Smolka was supported by National Science Foundation Grant DCR-8505873.     

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.     

Generalized Voronoi diagrams for a ladder: II. Efficient construction of the diagram

We present a collection of algorithms, all running in timeO(n ² logn (n) ^o((n)3)) for some fixed integers(where (n) is the inverse Ackermann's function), for constructing a skeleton representation of a suitably generalized Voronoi diagram for a ladder moving in a two-dimensional space bounded by polygonal barriers consisting ofn line segments. This diagram, which is a two-dimensional subcomplex of the dimensional configuration space of the ladder, is introduced and analyzed in a companion paper by the present authors. The construction of the diagram described in this paper yields a motion-planning algorithm for the ladder which runs within the same time bound given above.Work on this paper has been supported in part by Office of Naval Research Grant N00014-82-K-0381, and by grants from the Digital Equipment Corporation, the Sloan Foundation, the System Development Foundation, the IBM corporation, and by National Science Foundation CER Grant No. DCR-8320085. Work by the second author has also been supported in part by a grant from the US-Israeli Binational Science Foundation.     

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.The research of R. Cole was supported in part by NSF Grants CCR-8702271, CCR-8902221, and CCR-8906949, by ONR Grant N00014-85-K-0046, and by a John Simon Guggenheim Memorial Foundation fellowship. M. T. Goodrich's research was supported by the National Science Foundation under Grant CCR-8810568 and by the National Science Foundation and DARPA under Grant CCR-8908092.     

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     

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.     

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.     

On languages with a certain prefix property

A languageL has theprefix property if for allx, y L, eitherx is a prefix ofy ory is a prefix ofx. It is shown that for certain classes of languages, if a language in the class has the prefix property, then it is a regular set. The results provide new distinctions between classes as well as new profs of previously known distinctions.This research was supported in part by the National Science Foundation under Grant DCR-75-15945 and Grant MCS76-05744.     

Quasi-optimal upper bounds for simplex range searching and new zone theorems

This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ^d so that, given any query simplexq, the points inP q can be counted or reported efficiently. Ifm units of storage are available (n <m <n ^d ), then we show that it is possible to answer any query inO(n ¹⁺/m ^1/d ) query time afterO(m ¹⁺) preprocessing. This bound, which holds on a RAM or a pointer machine, is almost tight. We also show how to achieveO(logn) query time at the expense ofO(n ^d+) storage for any fixed > 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.A preliminary version of this paper has appeared in theProceedings of the Sixth Annual ACM Symposium on Computational Geometry, June 1990, pp. 23–33. Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917 and NSF Grant CCR-90-02352. Work on this paper by Micha Sharir has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grants DCR-83-20085 and CCR-8901484, 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 administered by the Israeli Academy of Sciences. Work by Emo Welzl has been supported by Deutsche Forschungsgemeinschaft Grant We 1265/1–2. Micha Sharir and Emo Welzl have also been supported by a grant from the German-Israeli Binational Science Foundation. Last but not least, all authors thank DIMACS, an NSF Science and Technology Center, for additional support under Grant STC-88-09648.     

Asymptotically optimal algorithms for approximate agreement

This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases. Alan Fekete was born in Sydney Australia in 1959. He studied Pure Mathematics and Computer Science at the University of Sydney, obtaining a B.Sc.(Hons) in 1982. He then moved to Cambridge, Massachusetts, where he obtained a distributed Ph.D. degree, awarded by Harvard University's Mathematics department for work supervised by Nancy Lynch in M.I.T.'s Laboratory for Computer Science. He spend the year 1987–1988 at M.I.T. as a postdoctoral Research Associate, and is now Lecturer in Computer Science at the University of Sydney. His research concentrates on understanding the modularity in distributed algorithms, especially those used for concurrency control in distributed databases.A preliminary version of this paper has appeared in the Proceedings of the 5th ACM Symposium on Principles of Distributed Computing (August 1986). This work was begun in the Department of Mathematics, Harvard University, and completed at the Laboratory for Computer Science at Massachusetts Institute of Technology. The work was supported in part (through Professor N. Lynch) 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 Grants MCS-8306854, DCR-83-02391, and CCR-8611442, and by the Defense Advanced Research Projects Agency (DARPA) under Contract N00014-83-K-0125