28th Annual Symposium on Foundations of Computer Science (sfcs 1987) (1987)
Oct. 12, 1987 to Oct. 14, 1987
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/SFCS.1987.31
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.
A. K. Chandra, M. Snir and A. Aggarwal, "Hierarchical memory with block transfer," 28th Annual Symposium on Foundations of Computer Science (sfcs 1987)(FOCS), vol. 00, no. , pp. 204-216, 1987.