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

We consider space-bounded computations on a random-access machine (RAM) where the input is given on a read-only random-access medium, the output is to be produced to a writeonly 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. We present a state-of-the-art priority queue—called an adjustable navigation pile—for this restricted RAM model. Under some reasonable assumptions, our priority queue supports minimum and insert in O(1) worst-case time and extract in O(N/S + lgS) worst-case time for any S ≥ lgN . (We use lg x as a shorthand for log2(max {2, x}).) We show how to use this data structure to sort N elements and to compute the convex hull of N points in the two-dimensional Euclidean space in O(N2/S+N lgS) worst-case time for any S ≥ lgN . 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 spaceefficient algorithms, and we expect that it will find its way to yet other applications.