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Randomised Techniques in Combinatorial Algorithmics
, 1999
"... ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...
A SublinearTime Parallel Algorithm for Integer Modular Exponentiation
, 1999
"... The modular exponentiation problem is, given integers x; a; m with m ? 0, compute x a mod m. Let n denote the sum of the lengths of x, a, and m in binary. We present a parallel algorithm for this problem that takes O(n= log log n) time on the common CRCW PRAM using O(n 2+ffl ) processors. This ..."
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The modular exponentiation problem is, given integers x; a; m with m ? 0, compute x a mod m. Let n denote the sum of the lengths of x, a, and m in binary. We present a parallel algorithm for this problem that takes O(n= log log n) time on the common CRCW PRAM using O(n 2+ffl ) processors. This algorithm is based on Bernstein's Explicit Chinese Remainder Theorem combined with a fast method for parallel prefix summation. We also present a linear time algorithm for the EREW PRAM. 1 Introduction. In this paper we present a new parallel algorithm for the modular exponentiation problem. This problem is, given integers x; a and a positive integer m, compute x a mod m. Applications for this problem are quite numerous, and include primality testing, integer factoring, the discrete logarithm problem, and cryptographic protocols based on these problems such as RSA. It is not an overstatement to say that modular exponentiation is a fundamentally important problem, and fast algorithms for t...
RNC Algorithms for the Uniform Generation of Combinatorial Structures
"... We describe several RNC algorithms for generating graphs and subgraphs uniformly at random. For example, unlabelled undirected graphs are generated in O(lg 3 n) time using O i "n 2 lg 3 n j processors if their number is known in advance and in O(lg n) time using O i "n 2 lg n j ..."
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We describe several RNC algorithms for generating graphs and subgraphs uniformly at random. For example, unlabelled undirected graphs are generated in O(lg 3 n) time using O i "n 2 lg 3 n j processors if their number is known in advance and in O(lg n) time using O i "n 2 lg n j processors otherwise. In both cases the error probability is the inverse of a polynomial in ". Thus " may be chosen to tradeoff processors for error probability. Also, for an arbitrary graph, we describe RNC algorithms for the uniform generation of its subgraphs that are either nonsimple paths or spanning trees. The work measure for the subgraph algorithms is essentially determined by the transitive closure bottleneck. As for sequential algorithms, the general notion of constructing generators from counters also applies to parallel algorithms although this approach is not employed by all the algorithms of this paper.
The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It use ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
Trading Time for Space in Prime Number Sieves
 Proceedings of the Third International Algorithmic Number Theory Symposium (ANTS III
, 1998
"... . A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O( p n=(log log n) 2 ) bit ..."
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. A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O( p n=(log log n) 2 ) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l ? 1 is constant, and two superlinear time sieves that use very little space. 1 Introduction A prime number sieve is an algorithm that finds all prime numbers up to a bound n. In this paper we present four new prime number sieves, three of which accept a parameter to control their use of time versus space. The fastest known prime number sieve is the dynamic wheel sieve of Pritchard [11], which uses O(n= log log n) arithmetic operations and O(n= log log n) bits of space. Dunten, Jones, and Sorenson [6] gave an algorithm with the same asymptotic running time, while using only O(n=(log ...
Efficient and Fast RNC Algorithms for the Uniform Generation of Unlabelled Graphs
"... This paper is concerned with the problem of generating unlabelled graphs uniformly at random (u.a.r.) by means of efficient parallel algorithms. Our model of computation is the wellknown (see [6], for example) parallel random access machine (PRAM) augmented with a facility to generate random number ..."
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This paper is concerned with the problem of generating unlabelled graphs uniformly at random (u.a.r.) by means of efficient parallel algorithms. Our model of computation is the wellknown (see [6], for example) parallel random access machine (PRAM) augmented with a facility to generate random numbers in [0, 1]. By efficient algorithms we mean algorithms that run in polylogarithmic time in the problem size using a polynomial number of processors. Such problems define the class NC, or specifically RNC if the facility for generating random numbers is employed. Our algorithms generally run on the exclusiveread exclusivewrite variant of the PRAM (the EREW PRAM) with addition, subtraction, multiplication and division as basic set of operations. A PRAM algorithm, in particular, is said to be optimal (see [6]) if the product of its parallel running time t times the number of processor used p is within a constant factor from the computation time of the fastest existing sequential algorithm. The quantity pt is called work. Our domain of...
USER’S DECLARATION
, 1999
"... I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no rest ..."
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I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no restriction on photocopying will also be made available to the British Library for microfilming. The British Library may supply copies to individuals or libraries. subject to a statement from them that the copy is supplied for nonpublishing purposes. All copies supplied by the British Library will carry the following statement: “Attention is drawn to the fact that the copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the author’s written consent.” AUTHOR’S SIGNATURE:................................................
A Sublinear Time Parallel GCD Algorithm for the EREW PRAM
, 2009
"... We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRA ..."
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We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRAM for this problem.
www.elsevier.com/locate/jalgor A modular integer GCD algorithm ✩
, 1999
"... This paper describes the first algorithm to compute the greatest common divisor (GCD) of two nbit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A gener ..."
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This paper describes the first algorithm to compute the greatest common divisor (GCD) of two nbit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A gener