Space-Efficient Plane-Sweep Algorithms

TL;DR

This paper develops space-efficient plane-sweep algorithms for geometric problems, optimizing time and space complexity for tasks like closest pair, segment intersection enumeration, and Klee's measure, using novel techniques.

Contribution

It introduces three general techniques for space-efficient algorithms and provides new algorithms with improved time bounds for geometric problems under limited workspace.

Findings

Closest pair algorithm runs in O(n^2/s + n log s) time.

Segment intersection enumeration runs in O((n^2/s^{2/3}) log s + k) time.

Klee's measure algorithm runs in O((n^2/s + n log s) sqrt((n/s) log n)) time.

Abstract

We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of $n$ items and that the available workspace is $\Theta(s)$ bits, where $\lg n \leq s \leq n \cdot \lg n$. Three techniques that can be used as general tools in different space-efficient algorithms are introduced and employed within our algorithms. In particular, we give an almost-optimal algorithm for finding the closest pair among a set of $n$ points that runs in $O(n^2/s + n \cdot \lg s)$ time. We also give a simple algorithm to enumerate the intersections of $n$ line segments that runs in $O((n^2/s^{2/3}) \cdot \lg s + k)$ time, where $k$ is the number of intersections. The counting version can be solved in $O((n^2/s^{2/3}) \cdot \lg s)$~time. When the segments are axis-parallel, we give an $O((n^2/s) \cdot \lg^{4/3} s + n^{4/3} \cdot \lg^{1/3} n)$-time algorithm for counting the intersections, and an algorithm for enumerating the intersections that runs in $O((n^2/s) \cdot \lg s \cdot \lg \lg s + n \cdot \lg s + k)$ time, where $k$ is the number of intersections. We finally present an algorithm that runs in $O((n^2/s + n \cdot \lg s) \cdot \sqrt{(n/s) \cdot \lg n})$ time to calculate Klee's measure of axis-parallel rectangles.