Space–Time Trade-offs for Stack-Based Algorithms

Abstract

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memoryconstrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Θ(n), we can modify it so that it runs in O(n 2 /2 s ) time using a workspace of O(s) variables (for any s ∈ o(log n)) or O(n log n/ log p) time using O(p log n/ log p) variables (for any 2 ≤ p ≤ n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling

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

“…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).…”

“…It would be very interesting to strengthen our lower bound results, when the number of pointers grows as a function of the input size. It would be desirable to characterize, in the spirit of time-space tradeoff results for stack-based algorithms in [4], those applications for which time-space tradeoffs can be achieved by reduction to variants of the ANLN problem.…”

Priority Queues and Sorting for Read-Only Data

“…In our experience, very few data structures can be made memory adjustable as elegantly as priority queues. A stack is another candidate [3]. For example, a dictionary must maintain a permutation of a set of size N ; this means that it is difficult to manage with much less than N lg N bits.…”

“…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.…”