Memory-Adjustable Navigation Piles with Applications to Sorting and Convex Hulls

Abstract

We consider space-bounded computations on a random-access machine, where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the input is N elements, the length of the output is limited by the computation, and the capacity of the workspace is O ( S ) bits for some predetermined parameter S ≥ lg N . We present a state-of-the-art priority queue—called an adjustable navigation pile —for this restricted model. This priority queue supports M inimum in O (1) time, C onstruct in O ( N ) time, and E xtract - min in O ( N / S + lg S ) time for any S ≥ lg N . The priority queue can be further augmented in O ( N ) time to deal with a batch of at most S elements in a specified range of values at a time, and allow to I nsert (activate) or E xtract (deactivate) an element among these elements, such that I nsert and E xtract take O ( N / S + lg S ) time for any S ≥ lg N . We show how to use our data structure to sort N elements and to compute the convex hull of N points in the Euclidean plane in O ( N 2 / S + N lg S ) time for any S ≥ lg N . Following a known lower bound for the space-time product of any branching program for finding unique elements, both our sorting and convex-hull algorithms are optimal. The adjustable navigation pile has turned out to be useful when designing other space-efficient algorithms, and we expect that it will find its way to yet other applications.