Space-Time Trade-offs for Stack-Based Algorithms

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 memory-constrained algorithms. Given an algorithm \alg\ that runs in O(n) time using $\Theta(n)$ variables, we can modify it so that it runs in $O(n^2/s)$ time using a workspace of O(s) variables (for any $s\in o(\log n)$) or $O(n\log n/\log p)$ time using $O(p\log n/\log p)$ variables (for any $2\leq p\leq n$). 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 the first 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

Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Simple and Robust Boolean Operations for Triangulated Surfaces

Boolean operations of geometric models is an essential issue in computational geometry. In this paper, we develop a simple and robust approach to perform Boolean operations on closed and open triangulated surfaces. Our method mainly has two stages: (1) We firstly find out candidate intersected-triangles pairs based on Octree and then compute the inter-section lines for all pairs of triangles with parallel algorithm; (2) We form closed or open intersection-loops, sub-surfaces and sub-blocks quite robustly only according to the cleared and updated topology of meshes while without coordinate computations for geometric enti-ties. A novel technique instead of inside/outside classification is also proposed to distinguish the resulting union, subtraction and intersection. Several examples have been given to illus-trate the effectiveness of our approach.Comment: Novel method for determining Union, Subtraction and Intersectio