Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane and Related Problems

In this paper, we present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n non-intersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with very high probability and uses O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of P.T bounds since the sequential time bound for this problem is Ω(n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm which runs in O(log2 n) time using O(n) processors [13]. We obtain this result by using random sampling at two stages of our algorithm and using efficient randomized search techniques. This technique gives a direct optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use reduction from the 3-d convex hull construction)

New Results for the Minimum Weight Triangulation Problem

Given a finite set of points in a plane, a triangulation is a maximal set of non-intersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. Given a set of points in a plane, the minimum weight triangulation problem is to find a triangulation whose weight is minimal. No polynomial time algorithm is known to solve this problem, and it is unknown whether the problem is NP-hard. The current best polynomial time approximation algorithm produces a triangulation that can be 0(log n) times the weight of the optimal triangulation. We propose an algorithm that triangulates a set P, of n points in a plane in 0(n-cubed) time and that never does worse than the greedy triangulation. The algorithm produces an optimal triangulation if the points P are the vertices of a convex polygon. The algorithm has the flavor of a heuristic proposed by Lingas and analysis similar to his can be performed for our algorithm also, but experimental results indicate that our algorithm performs much better than the heuristic of Lingas. The results comparing the optimal triangulation with the performance of our algorithm, the heuristic of Lingas, and the greedy algorithm are within 0(1) of an optimal triangulation. We investigate issues of local optimality pertaining to known triangulation algorithms. We define the notion of k-optimality which suggests an interesting new approach to studying triangulation algorithms. We restate the minimum weight triangulation problem as a graph problem and show that NP-hardness of a closely related graph problem. Finally, we show that the constrained problem of computing the minimum weight of triangulation, given a set of points in a plane and enough edges to form a triangulation, is NP-hard. These results are an advance towards a proof that the minimum weight triangulation problem is NP-hard

Fast Fencing

We consider very natural "fence enclosure" problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set $S$ of $n$ points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose $n$ unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most $k$ closed curves and pay no cost per curve. For the variant with at most $k$ closed curves, we present an algorithm that is polynomial in both $n$ and $k$. For the variant with unit cost per curve, or unit disks, we present a near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most $k$ curves in $n^{O(k)}$ time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with $k$ curves is NP-hard for general $k$. Our polynomial time algorithm refutes this unless P equals NP

Approximation Schemes for Maximum Weight Independent Set of Rectangles

In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1+epsilon)-approximation algorithm for the MWISR problem with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n / log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to Arora et al. this allows us to upper bound our approximation ratio by (1+epsilon). Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1+epsilon)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem