Hierarchical memory with block transfer

Abstract

In this paper we introduce a model of Hierarchical Memory with Block Transfer (BT for short). It is like a random access machine, except that access to location x takes time f(x), and a block of consecutive locations can be copied from memory to memory, taking one unit of time per element after the initial access time. We first study the model with f(x) = xα for 0 ≪ α ≪ 1. A tight bound of θ(n log log n) is shown for many simple problems: reading each input, dot product, shuffle exchange, and merging two sorted lists. The same bound holds for transposing a √n × √n matrix; we use this to compute an FFT graph in optimal θ(n log n) time. An optimal θ(n log n) sorting algorithm is also shown. Some additional issues considered are: maintaining data structures such as dictionaries, DAG simulation, and connections with PRAMs. Next we study the model f(x) = x. Using techniques similar to those developed for the previous model, we show tight bounds of θ(n log n) for the simple problems mentioned above, and provide a new technique that yields optimal lower bounds of Ω(n log2n) for sorting, computing an FFT graph, and for matrix transposition. We also obtain optimal bounds for the model f(x)= xα with α ≫ 1. Finally, we study the model f(x) = log x and obtain optimal bounds of θ(n log*n) for simple problems mentioned above and of θ(n log n) for sorting, computing an FFT graph, and for some permutations.