A Census Algorithm for Chinese Remainder Pseudorank with Experimental Results

This paper describes a polynomial time census algorithm for chinese remainder pseudorank. That is, we can rapidly estimate the density of integers with either good or bad pseudorank. The pseudorank of an integer is an important approximation technique that makes possible efficient parallel computation of the standard order relation on integers in chinese remainder (also known as multiresidue) representation. In this paper we also describe an implementation of the algorithm in C and report on its performance, and on the census information for chinese remainder representation systems up to 989-bit integers. Introduction The chinese remainder representation (CRR) pseudorank was introduced and studied by Davida and Litow in [2]. Pseudorank is an approximate computation of what is known in multiresidue literature as rank. Pseudorank was used by Davida and Litow to provide for compact, logdepth Boolean circuits to compute comparison of integers in CRR. Each integer in the CRR has a pseudorank, which is either good, i.e., equals that integer’s rank, or is bad, i.e. is one less than that integer’s rank. The critical range is the lower quarter of the integers in the CRR so that any direct count of good and bad integers is out of the question. Recently, pseudorank has been used to study the circuit size of NP-complete problems. See [6]. The crucial fact in this circuit size study is that the densities of integers of both good and bad pseudorank in the critical range are both bounded from below by Ω(1/n), where the largest integer representable by the CRR is roughly 2n. These terms will be precisely defined in Section 1. We prove this Ω(1/n) lower bound for good and bad pseudorank densities in the critical range, and describe and analyze a polytime algorithm that usefully estimates these densities. This algorithm (the census algorithm) has been implemented in C. We describe this implementation of the census algorithm, and present and interpret some experimental results. These results may be of use in investigations into other applications of CRR. Section 1 defines pseudorank and discusses pseudorank density. Section 2 describes the census algorithm, and establishes its polynomial running time. Section 3 describes an implementation of the census algorithm in C, and experimental findings are reported and analyzed in Section 4. Integer, unless stated otherwise, will mean nonnegative integer, and log is the base 2 logarithm. §School of Information Technology, James Cook University, Townsville, Qld. 4811, Australia [email protected] ¶School of Information Technology, James Cook University, Townsville, Qld. 4811, Australia [email protected]