Analysis of Pram Instruction Setsfrom a Log Cost

The log cost measure has been viewed as a more reasonable method of measuring the time complexity of an algorithm than the unit cost measure. The more widely used unit cost measure becomes unrealistic if the algorithm handles extremely large integers. Parallel machines have not been examined under the log cost measure. In this paper, we investigate the Parallel Random Access Machine under the log cost measure. Let the instruction set of a basic PRAM include addition, subtraction, and Boolean operations. We relate resource-bounded complexity classes of log cost PRAMs to complexity classes of Turing machines and circuits. We also relate log cost PRAMs with diierent instruction sets by simulations that are much more eecient than possible in the unit cost case. Let LCRCW k (CRCW k) denote the class of languages accepted by a log cost (unit cost) basic CRCW PRAM in O(log k n) time with polynomial in n number of processors. We position the log cost PRAM in the hierarchy of parallel complexity classes as: AC k = CRCW k (NC k+1 ; LCRCW k+1) AC k+1 = CRCW k+1. 1. Introduction 1.1. PRAMs and Log Cost Usually, one measures the time complexity of a Random Access Machine (RAM) algorithm as the number of steps executed. Under this unit cost measure, a RAM takes one unit of time to execute an instruction on any pair of operands stored at any pair of locations in memory, regardless of the operand and address lengths. Alternatively, one may analyze time complexity under the log cost measure, in which the time cost of an instruction execution depends on the operand and address