Ray Shooting Amidst Convex Polygons in 2D

We consider the problem of ray shooting in a two-dimensional scene consisting ofmconvex polygons with a total ofnedges. We present a data structure that requiresO(mn log m) space and preprocessing time and that answers a ray shooting query inO(log² m log² n) time. If the polygons are pairwise disjoint, the space and preprocessing time can be improved toO((m²+n)log m) andO((m²+n log n)log m), respectively. Our algorithm also works for a collection of disjoint simple polygons. We also show that if we allow onlyO(n) space, a ray shooting query among a collection of disjoint simple polygons can be answered in timeO(m/[formula]¹⁺ log² n) time, for any >0.     

Generalized delaunay triangulation for planar graphs

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graphG=(V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs inO(|V|²) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-and-conquer algorithm based on the polygon cutting theorem of Chazelle is given that runs inO(|V| log |V|) time.Supported in part by the National Science Foundation under Grants DCR 8420814 and ECS 8340031.     

Approximating Minimum-Weight Triangulations in Three Dimensions

Let S be a set of noncrossing triangular obstacles in R ³ with convex hull H . A triangulation T of H is compatible with S if every triangle of S is the union of a subset of the faces of T. The weight of T is the sum of the areas of the triangles of T. We give a polynomial-time algorithm that computes a triangulation compatible with S whose weight is at most a constant times the weight of any compatible triangulation. One motivation for studying minimum-weight triangulations is a connection with ray shooting. A particularly simple way to answer a ray-shooting query (``Report the first obstacle hit by a query ray') is to walk through a triangulation along the ray, stopping at the first obstacle. Under a reasonably natural distribution of query rays, the average cost of a ray-shooting query is proportional to triangulation weight. A similar connection exists for line-stabbing queries (``Report all obstacles hit by a query line'). Received February 3, 1997, and in revised form August 21, 1998.     

On separating two simple polygons by a single translation

LetP andQ be two disjoint simple polygons havingn sides each. We present an algorithm which determines whetherQ can be moved by a single translation to a position sufficiently far fromP, and which produces all such motions if they exist. The algorithm runs in timeO(t(n)) wheret(n) is the time needed to triangulate ann-sided polygon. Since Tarjan and Van Wyk have recently shown thatt(n)=O(n log logn) this improves the previous best result for this problem which wasO(n logn) even after triangulation.This research was supported by NSERC Grant A9293, FCAR Grant EQ-1678, and a Killam Fellowship from the Canada Council.     

Polygon triangulation inO(n log logn) time with simple data structures

We give a newO(n log logn)-time deterministic algorithm for triangulating simplen-vertex polygons, which avoids the use of complicated data structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run inO(n log*n) time. The major new techniques employed are the efficient location of horizontal visibility edges that partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. The latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.This research was partially supported by the following grants: NSERC 583584, NSERC 580485, ONR-N00014-87-0467, and by DIMACS, an NSF Science and Technology Center (NSF-STC88-09648).     

On the minimality of polygon triangulation

The problem of triangulating a polygon using the minimum number of triangles is treated. We show that the minimum number of triangles required to partition a simplen-gon is equal ton+2w –d – 2, wherew is the number of holes andd is the maximum number of independent degenerate triangles of then-gon. We also propose an algorithm for constructing the minimum triangulation of a simple hole-freen-gon. The algorithm takesO(nlog² n+DK ²) time, whereD is the maximum number of vertices lying on the same line in then-gon andK is the number of minimally degenerate triangles of then-gon.     

A fast las vegas algorithm for triangulating a simple polygon

We present a randomized algorithm that triangulates a simple polygon onn vertices inO(n log*n) expected time. The averaging in the analysis of running time is over the possible choices made by the algorithm; the bound holds for any input polygon.Research partially supported by the National Science Foundation under Grant No. DCR-8605962.     

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n ²) algorithm is presented, which is a substantial improvement over theO(n ⁷) time algorithm for the general problem.The work of the first author was supported under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692 and the work of the second author was partially supported by NSF Grants Nos. DCR-84-01898 and DCR-84-01633.     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

Isoperimetric triangular enclosures with a fixed angle

In this paper, we study four variants of the famous isoperimetric problem. Given a set S of n > 2 points in the plane (in general position), we show how to compute in O(n ²) time, a triangle with maximum (or minimum) area enclosing S among all enclosing triangles with fixed perimeter and one fixed angle. We also show how to compute in O(n ²) time, a triangle with maximum (or minimum) perimeter enclosing S among all enclosing triangles with fixed area and one fixed angle. We also provide an Ω (n log n) lower bound for these problems in the algebraic computation tree model.     

An Efficient Algorithm for a Sparse k-Edge-Connectivity Certificate

We present an efficient algorithm for finding a sparse k-edge-connectivity certificate of a multigraph G. Our algorithm runs in O((log kn)(log k)²(log n)²) time using O(k(n + m′)) processors on an ARBITRARY CRCW PRAM, where n and m′ stand for the numbers of vertices in G and edges in the simplified graph of G, respectively.     

Piecewise linear wavelets on sierpinski gasket type fractals

Let SG denote the Sierpinski gasket with Hausdorff measure μ of dimensionlog 3/log 2, let PL_k denote the continuous piecewise linear functions with respect to the usual triangulation of SG into 3^k triangles, and let W_k denote the orthogonal complement of PL_k−1 in PL_k. We construct a basis for each W_k, so that the entire collection is a frame for L²(dμ). This wavelet basis is obtained from three wavelet generators by scaling, translation and rotation, and the wavelets are supported either by one corner triangle or a pair of adjacent triangles in the triangulation of level k − 1. Analogous bases are constructed in the von Koch curve, the hexagasket, and the n-dimensional analog of SG.     

The furthest-site geodesic voronoi diagram

We present anO((n+k) log(n+k))-time,O(n+k)-space algorithm for computing the furthest-site Voronoi diagram ofk point sites with respect to the geodesic metric within a simplen-sided polygon.A preliminary version of this paper appeared in theProceedings of the Fourth ACM Symposium on Computational Geometry, 1988. The work of Boris Aronov was supported by an AT&T Bell Laboratories Ph.D. Scholarship. Part of the work was performed while he was at AT&T Bell Laboratories, Murray Hill, NJ, USA. His current address is Computer Science Department, Polytechnic University, Brooklyn, NY 11201, USA.     

A Parallel Reduction of Hamiltonian Cycle to Hamiltonian Path in Tournaments

We propose a parallel algorithm which reduces the problem of computing Hamiltonian cycles in tournaments to the problem of computing Hamiltonian paths. The running time of our algorithm is O(log n) using O(n²/log n) processors on a CRCW PRAM, and O(log n log log n) on an EREW PRAM using O(n²/log n log log n) processors. As a corollary, we obtain a new parallel algorithm for computing Hamiltonian cycles in tournaments. This algorithm can be implemented in time O(log n) using O(n²/log n) processors in the CRCW model and in time O(log²n) with O(n²/log n log log n) processors in the EREW model.     

Stirling Number Asymptotics from Recursion Equations Using the Ray Method

A technique for computing asymptotic expansions of combinatorial quantities from their recursion relations is presented. It is applied to the Stirling numbers of the first and second kinds, s(n, k) and S(n, k), for n ? 1 and three ranges of k: (i) k = O(1), (ii) n ? k = O(1), (iii) k ? 1, 0 < k / n < 1, The technique uses asymptotic methods of applied mathematics such as the ray method and the method of matched asymptotic expansions.     

Improved dynamic programming algorithms for bandwidth minimization and the MinCut Linear Arrangement problem

The dynamic programming algorithm of [12.] for the bandwidth minimization problem is improved. It is shown that, for all k > 1, BANDWIDTH(k) can be solved in O(n^k) steps and simultaneous O(n^k) space, where n is the number of vertices in the graph, and that each such problem is in NSPACE(log n). The same improved dynamic programming algorithm approach works to show that the MINCUT LINEAR ARRANGEMENT problem restricted to the fixed value k, denoted by MINCUT(k), is solvable in O(n^k) steps and simultaneous O(n^k) space and is in the class NSPACE(log n).     

Approximating the minimum weight steiner triangulation

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. InO(n logn) time we can compute a triangulation withO(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher-dimensional triangulation problems. No previous polynomial-time triangulation algorithm was known to approximate the MWST within a factor better thanO(logn).     

Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random k ‐coloring of an n ‐vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k₀(ε) and either: (i) k/Δ > α* + ε where α*≈? 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈? 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n²) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle‐free “neighborhood‐amenable” graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Optimally Computing the Shortest Weakly Visible Subedge of a Simple Polygon

Given ann-vertex simple polygonP, the problem of computing the shortest weakly visible subedge ofPis that of finding a shortest line segmentson the boundary ofPsuch thatPis weakly visible froms(ifsexists). In this paper, we present new geometric observations that are useful for solving this problem. Based on these geometric observations, we obtain optimal sequential and parallel algorithms for solving this problem. Our sequential algorithm runs inO(n) time, and our parallel algorithm runs inO(log n) time usingO(n/log n) processors in the CREW PRAM computational model. Using the previously best known sequential algorithms to solve this problem would takeO(n²) time. We also give geometric observations that lead to extremely simple and optimal algorithms for solving, both sequentially and in parallel, the case of this problem where the polygons are rectilinear.     

Finding intersection of rectangles by range search

Three related rectangle intersection problems in k-dimensional space are considered: (1) find the intersections of a rectangle with a given set of rectangles, (2) find the intersecting pairs of rectangles as they are inserted into or deleted from an existing set of rectangles, and (3) find the intersecting pairs of a given set of rectangles. By transforming these problems into range search problems, one need not divide the intersection problem into two subproblems, namely, the edge-intersecting problem and the containment problem, as done by many previous studies. Furthermore, this approach can also solve these subproblems separately, if required. For the first problem the running time is O((log n)^2k−1 + s), where s is the number of intersecting pairs of rectangles. For the second problem the time needed to generate and maintain n rectangles is O(n(log n)^2k) and the time for each query is O((log n)^2k−1 + s). For the third problem the total time is O(n log n + n(log n)^2(k−1) + s) for k ≥ 1.