On approximating the ideal random access machine by physical machines

Abstract

The capability of the Random Access Machine (RAM) to execute any instruction in constant time is not realizable, due to fundamental physical constraints on the minimum size of devices and on the maximum speed of signals. This work explores how well the ideal RAM performance can be approximated, for significant classes of computations, by machines whose building blocks have constant size and are connected at a constant distance. A novel memory structure is proposed, which is pipelined (can accept a new request at each cycle) and hierarchical, exhibiting optimal latency a(x) = O(x1/d) to address x, in d-dimensional realizations. In spite of block-transfer or other memory-pipeline capabilities, a number of previous machine models do not achieve a full overlap of memory accesses. These are examples of machines with explicit data movement. It is shown that there are direct-flow computations (without branches and indirect accesses) that require time superlinear in the number of instructions, on all such machines. To circumvent the explicit-data-movement constraints, the Speculative Prefetcher (SP) and the Speculative Prefetcher and Evaluator (SPE) processors are developed. Both processors can execute any direct-flow program in linear time. The SPE also executes in linear time a class of loop programs that includes many significant algorithms. Even quicksort, a somewhat irregular, recursive algorithm admits a linear-time SPE implementation. A relation between instructions called address dependence is introduced, which limits memory-access overlap and can lead to superlinear time, as illustrated with the classical merging algorithm.