Range Searching and Point Location among Fat Objects

We present a data structure that can store a set of disjoint fat objects ind-space such that point location and bounded-size range searching with arbitrarily shaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or nonconvex (fat) polytopes. The multipurpose data structure supports point location and range searching queries in timeO(log^d−1 n) and requiresO(n log^d−1 n) storage, afterO(n log^d−1 n log log n) preprocessing. The data structure and query algorithm are rather simple.     

One line and n points

We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RANDOM ‐EDGE simplex algorithm on simple polytopes with n facets in dimension n ? 2. We obtain a tight O(log² n) bound for the expected number of pivot steps. This is the first nontrivial bound for RANDOM ‐EDGE , which goes beyond bounds for specific polytopes. The process itself can be interpreted as a simple algorithm for certain 2‐variable linear programming problems, and we prove a tight Θ(n) bound for its expected runtime. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

Applications of random sampling in computational geometry,II

We use random sampling for several new geometric algorithms. The algorithms are Las Vegas, and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.     

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.     

A sublogarithmic convex hull algorithm

We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn ^1/3 divide-and-conquer and uses a simple pointer-based data structure.Part of this work was done when the last three authors were at the Department of Computer and Information Science, Linköping University. The research of the second author was supported by the Academy of Finland.     

On graphs preserving rectilinear shortest paths in the presence of obstacles

In physical VLSI design, network design (wiring) is the most time-consuming phase. For solving global wiring problems, we propose to first compute from the layout geometry a graph that preserves all shortest paths between pairs of relevant points, and then to operate on that graph for computing shortest paths, Steiner minimal tree approximations, or the like. For a set of points and a set of simple orthogonal polygons as obstacles in the plane, withn input points (polygon corner or other) altogether, we show how a shortest paths preserving graph of sizeO(n logn) can be computed in timeO(n logn) in the worst case, with spaceO(n). We illustrate the merits of this approach with a simple example: If the length of a longest edge in the graph is bounded by a polynomial inn, an assumption that is clearly fulfilled for graphs derived from VLSI layout geometries, then a shortest path can be computed in timeO(n logn log logn) in the worst case; this result improves on the known best one ofO(n(logn)^3/2).     

Fast rectangular matrix multiplication and some applications

We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary di- mensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results:(i) we decrease from O(n~2 n~(1 o)(1)logq)to O(n~(1.9998) n~(1 o(1))logq)the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic(NC)parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n×n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii)we decrease from O(m~(1.575)n)to O(m~(1.5356)n)the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.     

Pseudorandom generators for space-bounded computation

Pseudorandom generators are constructed which convertO(SlogR) truly random bits toR bits that appear random to any algorithm that runs inSPACE(S). In particular, any randomized polynomial time algorithm that runs in spaceS can be simulated using onlyO(Slogn) random bits. An application of these generators is an explicit construction of universal traversal sequences (for arbitrary graphs) of lengthn ^O(logn).The generators constructed are technically stronger than just appearing random to spacebounded machines, and have several other applications. In particular, applications are given for deterministic amplification (i.e. reducing the probability of error of randomized algorithms), as well as generalizations of it.This work was done in the Laboratory for Computer Science, MIT, supported by NSF 865727-CCR and ARO DALL03-86-K-017     

Random walks on highly symmetric graphs

We consider uniform random walks on finite graphs withn nodes. When the hitting times are symmetric, the expected covering time is at least 1/2n logn-O(n log logn) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of (n-1) logn is obtained. For hypercubes and binomial coefficient graphs, the limit law of the covering time is obtained as well.     

The weighted maximum independent set problem in permutation graphs

In this paper, sequential and parallel algorithms are presented to find a maximum independent set with largest weight in a weighted permutation graph. The sequential algorithm, which is designed based on dynamic programming, runs in timeO(nlogn) and requiresO(n) space. The parallel algorithm runs inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM.     

Random half-integral polytopes

We show that polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(logn/loglogn) with positive probability—even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank is due to certain obstructions. We determine the exact threshold number, when those cease to exist.     

Parallel algorithms for permutation graphs

In this paper, parallel algorithms are presented for solving some problems on permutation graphs. The coloring problem is solved inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM. The weighted clique problem, the weighted independent set problem, the cliques cover problem, and the maximal layers problem are all solved with the same complexities. We can also show that the longest common subsequence problem belongs to the class NC.     

Cutting a Convex Polyhedron Out of a Sphere

Given a convex polyhedron P of n vertices inside a sphere Q, we give an O(n ³)-time algorithm that cuts P out of Q by using guillotine cuts and has cutting cost O(log² n) times the optimal.     

A fast and simple randomized parallel algorithm for the maximal independent set problem

A simple parallel randomized algorithm to find a maximal independent set in a graph G = (V, E) on n vertices is presented. Its expected running time on a concurrent-read concurrent-write PRAM with O(|E|d_max) processors is O(log n), where d_max denotes the maximum degree. On an exclusive-read exclusive-write PRAM with O(|E|) processors the algorithm runs in O(log²n). Previously, an O(log⁴n) deterministic algorithm was given by Karp and Wigderson for the EREW-PRAM model. This was recently (independently of our work) improved to O(log²n) by M. Luby. In both cases randomized algorithms depending on pairwise independent choices were turned into deterministic algorithms. We comment on how randomized combinatorial algorithms whose analysis only depends on d-wise rather than fully independent random choices (for some constant d) can be converted into deterministic algorithms. We apply a technique due to A. Joffe (1974) and obtain deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments.     

Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s‐Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 ‐ o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n^{1+Ω(1/log log n)}. Recall that the Ising model with inverse temperature β defined on a graph G = (V,E) is the distribution over {±}^Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < ∞ and the Ising model defined on G (n, d/n), there exists a β_d > 0, such that for all β < β_d with probability going to 1 as n →∞, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd?s‐Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub‐graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and β < β_d, where d tanh(β_d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Convex Minimization on a Grid and Applications

This article discusses a discrete version of the convex minimization problem with applications to the efficient computation of proximity measures for pairs of convex polyhedra. Given ad-variate convex function and an isothetic grid of sizeO(n^d) in ^d, which is supposed to be finite but not necessarily regular, we want to find the grid cell containing the minimum point. With this aim, we identify a class of elementary subproblems, each resulting in the determination of a half-space in ^d, and show that the minimization problem can be solved by computingO(log n) half-spaces in the worst case foralmost uniformgrids of fixed dimensiondandO(log n) half-planes in the average for arbitrary planar grids. A major point is the potential of the approach to uniformly solve distance related problems for different configurations of a pair of convex bodies. In this respect, the case of a bivariate function is of particular interest and leads to a fast algorithm for detecting collisions between two convex polyhedra in three dimensions. The collision algorithm runs inO(log² n) average time for polyhedra withO(n) vertices whose boundaries are suitably represented; more specifically, the 1-skeletons can be embedded into layered Directed Acyclic Graphs which require justO(n) storage. The article ends with a brief discussion of a few experimental results.     

Approximation of smooth convex bodies by circumscribed polytopes with respect to the surface area

Let K be a convex body with C ² boundary in the Euclidean d-space. Following the work of L. Fejes Tóth, R. Vitale, R. Schneider, P.M. Gruber, S. Glasauer and M. Ludwig, best approximation of K by polytopes of restricted number of vertices or facets is well-understood if the approximation is with respect to the volume or the mean width. In this paper we consider the circumscribed polytope P _(n) of n facets with minimal surface area, and present an asymptotic formula in terms of n for the difference of surface areas of P _(n) and K.     

Unions of Fat Convex Polytopes Have Short Skeletons

The skeleton of a polyhedral set is the union of its edges and vertices. Let \(\mathcal {P}\) be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f _max be the maximum complexity of any face of a polytope in \(\mathcal {P}\). We prove that the total length of the skeleton of the union of the polytopes in \(\mathcal {P}\) is at most O(α(n)?log^? n?logf _max) times the sum of the skeleton lengths of the individual polytopes.     

On the construction of abstract voronoi diagrams

We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7]. This work was supported by the DFG, Me 620/6, and ESPRIT P3075 ALCOM. A preliminary version of this paper has been presented at STACS '90, Rouen, France.     

Fast broadcasting and gossiping in radio networks

We establish an O(nlog²n) upper bound on the time for deterministic distributed broadcasting in multi-hop radio networks with unknown topology. This nearly matches the known lower bound of Ω(nlogn). The fastest previously known algorithm for this problem works in time O(n^3/2). Using our broadcasting algorithm, we develop an O(n^3/2log²n) algorithm for gossiping in the same network model.