Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Chapter 15 – Geometric Shortest Paths and Network Optimization

Akinetic data structure(KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a concentual framework for kinetic data structures, we propose a number of criteria for the quality of such structures, and we describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behave well according to the proposed quality criteria for KDSs.

Data structures for mobile data

The UBI-QEP method: A practical theoretical approach to understanding chemistry on transition metal surfaces

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.

Applications of a semi-dynamic convex hull algorithm

Finding tailored partitions

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for three problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. We then apply these results to solve the following problems: (1) [matching] given n red points and n blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, (2) [scheduling] given n jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty, and (3) [covering] given n points in the plane and two real numbers r1 and r2, determine if one can cover all the points with two disks of radii r1 and r2. Our time bounds are O(n log n) for problems (1) and (2) and O(n2 log n) for problem (3), an improvement by a factor of log n over the previous bounds.

We consider the following problem: given a planar set of points S, a measure m acting on S, and a pair of values m1 and m2, does there exist a bipartition S = S1 U S2 satisfying m(Si) ≤ mi for i = 1,2? We present algorithms of complexity O(n log n) for several natural measures, including the diameter (set measure), the area, perimeter or diagonal of the smallest enclosing axes-parallel rectangle (rectangular measure), and the side length of the smallest enclosing axes-parallel square (square measure). The problem of partitioning S into k subsets, where k ≥ 3, is known to be NP-complete for many of these measures.

We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = {e 1 ,e 2 ,..., e p } between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e i . We show that the replacement paths problem requires Ω(mn) time in the worst case whenever m = O(nn). Our construction also implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest paths) can be solved in near linear time for undirected graphs.

John Hershberger

Papers

Applications of a semi-dynamic convex hull algorithm

Finding tailored partitions

Applications of a semi-dynamic convex hull algorithm

Finding tailored partitions

On the difficulty of some shortest path problems