On pointers versus addresses

Abstract

SUMMARY What is the cost of random access to memory? We address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it, a “pointer machine”. The model we define allows the use of a data type of our choice. A RAM program of time i and space s can be simulated in O(t log a) time using a tree. However, this is not an obvious lower bound, since a high-level data type may allow us to encode the data in a more economic way. Our major contribution is the formalization of incompressibility for general data types. The definition extends a similar property 01 strings that underlies the theory of “Kolmogorov complexity”. The main theorem states that for all incompressible data types an Q(t log s) lower bound holds. Incompressibility trivially holds for strings but is harder to prove for a powerlul data type. We prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set. This may be the richest set of operations considered in a lower bound proof. The prooi relies on the implicit functions theorem and Baire’s category theorem. We also show that the integers with arithmetic +,-,x and Lz/ZJ, any Boolean operations and left shift are incompressible. The inclusion of right shift reverses the situation, and we obtain an O(tq(s)) upper bound, where ~(s) is a functional inverse of Ackermann’s function.