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

TL;DR

This paper introduces a space-efficient priority queue called an adjustable navigation pile, enabling optimal sorting and convex hull computations in space-bounded RAM models, with significant improvements in worst-case time complexity.

Contribution

The paper presents a novel space-bounded priority queue supporting constant-time insert and minimum operations, and near-linear time convex hull and sorting algorithms, achieving optimal space-time trade-offs.

Findings

Supports minimum and insert in O(1) worst-case time

Computes convex hull in O(N^2/S + N log S) worst-case time

Algorithms are proven optimal under known lower bounds

Abstract

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 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 $S$. 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 $\mathit{minimum}$ and $\mathit{insert}$ in $O(1)$ worst-case time and $\mathit{extract}$ in $O(N/S + \lg{} S)$ worst-case time for any $S \geq \lg{} N$. 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 $O(N^2/S + N \lg{} S)$ worst-case time for any $S \geq \lg{} N$. 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.