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Models of Computation  Exploring the Power of Computing
"... Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and oper ..."
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Cited by 57 (7 self)
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Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although
Greed is Good: Approximating Independent Sets in Sparse and . . .
, 1994
"... ... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/3 for approximating independent sets in graphs with degree bounded by A. The analysis directs us tow ..."
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Cited by 56 (7 self)
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... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/3 for approximating independent sets in graphs with degree bounded by A. The analysis directs us towards a simple parallel and distributed algorithm with identical performance, which on constantdegree graphs runs in O(log ” n) time using linear number of processors. We also analyze the Greedy algorithm when run in combination with a fractional relaxation technique of Nemhauser and Trotter, and obtain an improved (2Z + 3)/5 performance ratio on graphs with average degree ~. Finally, we introduce a generally applicable technique for improving the approximation ratios of independent set algorithms, and illustrate it by improving the performance ratio of Greedy for large ∆.
Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 56 (11 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
Undirected connectivity in O(log 1.5 (n)) space
 In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science
, 1992
"... We present a deterministic algorithm for the connectivity problem on undirected graphs that runs in O(log 1.5 n) space. Thus, the recursive doubling technique of Savich which requires Θ(log 2 n) space is not optimal for this problem. 1 ..."
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Cited by 52 (5 self)
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We present a deterministic algorithm for the connectivity problem on undirected graphs that runs in O(log 1.5 n) space. Thus, the recursive doubling technique of Savich which requires Θ(log 2 n) space is not optimal for this problem. 1
Planar Separators and Parallel Polygon Triangulation
, 1992
"... We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree ..."
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Cited by 51 (7 self)
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We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+ffl )separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n= log n) processors on a CRCW PRAM. Keywords: Computational geometry, algorithmic graph theory, planar graphs, planar separators, polygon triangulation, parallel algorithms, PRAM model. 1 Introduction Let G = (V; E) be an nnode graph. An f(n)separator is an f(n)sized subset of V whose removal disconnects G into two subgraphs G 1 and G 2 each...
Topological Queries in Spatial Databases
 Journal of Computer and System Sciences
, 1996
"... We study topological queries over twodimensional spatial databases. First, we show that the topological properties of semialgebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invar ..."
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Cited by 45 (2 self)
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We study topological queries over twodimensional spatial databases. First, we show that the topological properties of semialgebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invariant characterizing semialgebraic regions up to homeomorphism. All topological queries on semialgebraic regions can be answered by queries on the invariant whose complexity is polynomially related to the original. Also, we show that for the purpose of answering topological queries, semialgebraic regions can always be represented simply as polygonal regions. We then study query languages for topological properties of twodimensional spatial databases, starting from the topological relationships between pairs of planar regions introduced by Egenhofer. We show that the closure of these relationships under appropriate logical operators yields languages which are complete for topological prope...
Can a SharedMemory Model Serve as a Bridging Model for Parallel Computation?
, 1999
"... There has been a great deal of interest recently in the development of generalpurpose bridging models for parallel computation. Models such as the BSP and LogP have been proposed as more realistic alternatives to the widely used PRAM model. The BSP and LogP models imply a rather different style fo ..."
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Cited by 42 (11 self)
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There has been a great deal of interest recently in the development of generalpurpose bridging models for parallel computation. Models such as the BSP and LogP have been proposed as more realistic alternatives to the widely used PRAM model. The BSP and LogP models imply a rather different style for designing algorithms when compared with the PRAM model. Indeed, while many consider data parallelism as a convenient style, and the sharedmemory abstraction as an easytouse platform, the bandwidth limitations of current machines have diverted much attention to messagepassing and distributedmemory models (such as the BSP and LogP) that account more properly for these limitations. In this paper we consider the question of whether a sharedmemory model can serve as an effective bridging model for parallel computation. In particular, can a sharedmemory model be as effective as, say, the BSP? As a candidate for a bridging model, we introduce the Queuing SharedMemory (QSM) model, which accounts for limited communication bandwidth while still providing a simple sharedmemory abstraction. We substantiate the ability of the QSM to serve as a bridging model by providing a simple workpreserving emulation of the QSM on both the BSP, and on a related model, the (d, x)BSP. We present evidence that the features of the QSM are essential to its effectiveness as a bridging model. In addition, we describe scenarios
Synthesizers and Their Application to the Parallel Construction of PseudoRandom Functions
, 1995
"... A pseudorandom function is a fundamental cryptographic primitive that is essential for encryption, identification and authentication. We present a new cryptographic primitive called pseudorandom synthesizer and show how to use it in order to get a parallel construction of a pseudorandom function. ..."
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Cited by 42 (10 self)
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A pseudorandom function is a fundamental cryptographic primitive that is essential for encryption, identification and authentication. We present a new cryptographic primitive called pseudorandom synthesizer and show how to use it in order to get a parallel construction of a pseudorandom function. We show several NC¹ implementations of synthesizers based on concrete intractability assumptions as factoring and the DiffieHellman assumption. This yields the first parallel pseudorandom functions (based on standard intractability assumptions) and the only alternative to the original construction of Goldreich, Goldwasser and Micali. In addition, we show parallel constructions of synthesizers based on other primitives such as weak pseudorandom functions or trapdoor oneway permutations. The security of all our constructions is similar to the security of the underlying assumptions. The connection with problems in Computational Learning Theory is discussed.
Parallel Algorithms with Processor Failures and Delays
, 1995
"... We study efficient deterministic parallel algorithms on two models: restartable failstop CRCW PRAMs and asynchronous PRAMs. In the first model, synchronous processors are subject to arbitrary stop failures and restarts determined by an online adversary and involving loss of private but not shared ..."
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Cited by 41 (7 self)
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We study efficient deterministic parallel algorithms on two models: restartable failstop CRCW PRAMs and asynchronous PRAMs. In the first model, synchronous processors are subject to arbitrary stop failures and restarts determined by an online adversary and involving loss of private but not shared memory; the complexity measures are completed work (where processors are charged for completed fixedsize update cycles) and overhead ratio (completed work amortized over necessary work and failures). In the second model, the result of the computation is a serializaton of the actions of the processors determined by an online adversary; the complexity measure is total work (number of steps taken by all processors). Despite their differences the two models share key algorithmic techniques. We present new algorithms for the WriteAll problem (in which P processors write ones into an array of size N ) for the two models. These algorithms can be used to implement a simulation strategy for any N ...
Horizons of Parallel Computation
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1993
"... This paper considers the ultimate impact of fundamental physical limitationsnotably, speed of light and device sizeon parallel computing machines. Although we fully expect an innovative and very gradual evolution to the limiting situation, we take here the provocative view of exploring the ..."
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Cited by 39 (3 self)
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This paper considers the ultimate impact of fundamental physical limitationsnotably, speed of light and device sizeon parallel computing machines. Although we fully expect an innovative and very gradual evolution to the limiting situation, we take here the provocative view of exploring the consequences of the accomplished attainment of the physical bounds. The main result is that scalability holds only for neighborly interconnections, such as the square mesh, of boundedsize synchronous modules, presumably of the areauniversal type. We also discuss the ultimate infeasibility of latencyhiding, the violation of intuitive maximal speedups, and the emerging novel processortime tradeoffs.