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

Visibility-Related Problems on Parallel Computational Models

Visibility-related problems find applications in seemingly unrelated and diverse fields such as computer graphics, scene analysis, robotics and VLSI design. While there are common threads running through these problems, most existing solutions do not exploit these commonalities. With this in mind, this thesis identifies these common threads and provides a unified approach to solve these problems and develops solutions that can be viewed as template algorithms for an abstract computational model. A template algorithm provides an architecture independent solution for a problem, from which solutions can be generated for diverse computational models. In particular, the template algorithms presented in this work lead to optimal solutions to various visibility-related problems on fine-grain mesh connected computers such as meshes with multiple broadcasting and reconfigurable meshes, and also on coarse-grain multicomputers. Visibility-related problems studied in this thesis can be broadly classified into Object Visibility and Triangulation problems. To demonstrate the practical relevance of these algorithms, two of the fundamental template algorithms identified as powerful tools in almost every algorithm designed in this work were implemented on an IBM-SP2. The code was developed in the C language, using MPI, and can easily be ported to many commercially available parallel computers

Extremal properties for dissections of convex 3-polytopes

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.Comment: 19 page

Straightening out planar poly-line drawings

We show that any $y$-monotone poly-line drawing can be straightened out while maintaining $y$-coordinates and height. The width may increase much, but we also show that on some graphs exponential width is required if we do not want to increase the height. Likewise $y$-monotonicity is required: there are poly-line drawings (not $y$-monotone) that cannot be straightened out while maintaining the height. We give some applications of our result.Comment: The main result turns out to be known (Pach & Toth, J. Graph Theory 2004, http://onlinelibrary.wiley.com/doi/10.1002/jgt.10168/pdf

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure