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...

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...

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 trade-off processors for error probability. Also, for an arbitrary graph, we describe RNC algorithms for the uniform generation of its subgraphs that are either non-simple 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. 1 Introduction. This paper is concerned with problems of generating combinatori...

. 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 super-linear 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 ...

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, space-efficient implementation of the wheel datastructure. 1