Memory-constrained algorithms for simple polygons

Abstract

A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n 2 ) time and constant workspace. We also consider the problem of preprocessing a simple polygon P for shortest path queries, where P is given by the ordered sequence of its n vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n 2 /s) time.

Time-Space Tradeoffs for All-Nearest-Larger-Neighbors Problems

“…Thus, by a Θ(b)-work-space algorithm we mean an algorithm that uses work-space consisting of Θ(b) words of O(log n) bits, which can be used as counters, indices and pointers, with a range of Θ(n) values. This generalizes the constant work-space model introduced by Asano et al [1,3,2] for the study of a variety of geometric problems. It is also consistent with the framework of memory-adjustable algorithms, and the time-space tradeoffs, described in [2] and [4] respectively.…”

“…This generalizes the constant work-space model introduced by Asano et al [1,3,2] for the study of a variety of geometric problems. It is also consistent with the framework of memory-adjustable algorithms, and the time-space tradeoffs, described in [2] and [4] respectively. Of course, memory-constrained computational models and timespace tradeoffs have been the subject of study for a long time (see, for example, [7,8]); we refer the reader to [2] for a succinct overview of this background work.…”

“…In sections 5 and 6, we present algorithms for these problems that run in O(n log b n) time using Θ(b) work-space, using direct reductions to variants of the ANLN problem. This improves the earlier O(n 2 )-time constant work space algorithm of [2] for triangulating mountain polygons (a special sub-class of monotone polygons), and the algorithm of [4], for b in the range [1, log n] (including their O(n 2 )-time constant work-space and O(n log n)-time Θ(log n) work-space algorithms).…”

An Experimental Study of Algorithms for Geodesic Shortest Paths in the Constant-Workspace Model

“…Given that the general shortest path problem is unlikely to be amenable to constant-workspace algorithms (it is NL-complete [18]), it may come as a surprise that a solution for the geodesic case exists at all. By now, several algorithms are known, both for constant workspace as well as in the time-space-trade-off regime, where the number of available cells of working memory may range from constant to linear [1,12].…”

Priority Queues and Sorting for Read-Only Data

“…Analogous with the sequential-access machine, we have a read-only array for input, a write-only array for output, and a limited workspace that allows random access. Over the years, starting by a seminal paper of Munro and Paterson [20], the space-time trade-offs have been studied in this model for many problems including: sorting [4,12,24], selection [11,12], and various geometric tasks [2,3,7]. The practical motivation for some of the previous work has been the appearance of special devices, where the size of working space is limited (e.g.…”