The I/O complexity of sorting and related problems

Abstract

We provide tight upper and lower bounds, up to a constant factor, for the number of accesses (I/Os) to secondary storage required for four sorting-related problems: sorting, the fast Fourier transform (FFT), permutation networks, and permuting records. The bounds hold both in the worst case and in the average case. Secondary storage is modeled as a magnetic disk capable of transfering P blocks each containing B records in a single time unit; the records in each block must be input from or output to B contiguous locations on the disk. We give two optimal algorithms for the problems, which are variants of merge sorting and distribution sorting. In particular we show for P=1 that the standard merge sorting algorithm is an optimal external sorting method. Our sorting algorithms use the same number of I/Os as does the permutation phase of key sorting, except when the internal memory size is extremely small, thus substantiating the popular adage that key sorting is not better. We also give a simpler and more direct derivation of a result due to Hong and Kung, which gives the FFT lower bound for the special case B=P=O(1).

Support was provided in part by NSF research grant DCR-84-03613, by an NSF Presidential Young Investigator Award with matching funds from an IBM Faculty Development Award and an AT&T research grant, and by a Guggenheim Fellowship.

Research was also done while the author was at the Mathematical Sciences Research Institute in Berkeley, Brown University in Providence, and Ecole Normale Supérieure in Paris. Correspondence should be addressed to the author at Brown University, Dept. of Computer Science, Box 1910, Providence, R. I. 02912, USA.