low complexity algorithms for optimal consumer push pull partial covering in the plane this paper considers a model for locating a consumer within a bounded region in the plane with respect to a set of n existing pull push suppliers the objective is to maximize the difference of total profits and costs incurred due to the partial covering of the consumer by the suppliers pull and push influence areas we develop efficient polynomial time algorithms for the resulting problems in the rectilinear and the euclidean planes where the bounded region is either a rectangle or a constant size polygon respectively based on these solutions we develop algorithms for evaluating efficiently the objective function at any possible location of the consumer inside the bounded region we also employ the algorithms for the euclidean optimization problem and the rectilinear query computation to solve efficiently their corresponding dynamic versions where an appearance of a new supplier or an absence of an existing one occurs being easy to implement due to the extensive use of simple data structures such as the balanced and binary segment tree and the employment of standard mechanisms such as the sweep line the voronoi diagram and the circular ray shooting our solutions potentially have wide usability

a randomized algorithm for closest point queries an algorithm for closest point queries is given the problem is this given a set s of n points in d dimensional space build a data structure so that given an arbitrary query point p a closest point in s to p can be found quickly the measure of distance is the euclidean norm this is sometimes called the post office problem the new data structure will be termed an rpo tree from randomized post office the expected time required to build an rpo tree is o n lceil d 2 rceil 1 epsilon for any fixed epsilon 0 and a query can be answered in o log n worst case time an rpo tree requires o n lceil d 2 rceil 1 epsilon space in the worst case the constant factors in these bounds depend on d and epsilon the bounds are average case due to the randomization employed by the algorithm and hold for any set of input points this result approaches the omega n lceil d 2 rceil worst case time required for any algorithm that constructs the voronoi

compact location problems we consider the problem of placing a specified number p of facilities on the nodes of a network so as to minimize some measure of the distances between facilities this formulation models a number of problems arising in facility location statistical clustering pattern recognition and also a processor allocation problem in multiprocessor systems

the relative neighborhood graph with an application to minimum spanning trees le graphe de voisinage relatif rng d un ensemble v de points dans un espace euclidien est un graphe v e tel que p q appartient a e si et seulement si il n existe pas de point z de v tel que d p z d p q et d q z d p q on montre que 1 le rng de n points du plan peut etre trouve en temps o n log n lequel est optimal a une constante multiplicative pres 2 le rng ainsi que l arbre minimum des sommets d un polygone a n sommets convexe peut etre trouve en un temps o n 3 sous l hypothese que trois points d entree ne forment pas un triangle isocele le rng de n points dans un espace dimension r peut etre trouve en un temps o n 2 pour r 3

nonpositive curvature and pareto optimal coordination of robots given a collection of robots sharing a common environment assume that each possesses a graph a one dimensional complex also known as a roadmap approximating its configuration space and furthermore that each robot wishes to travel to a goal while optimizing elapsed time we consider vector valued or pareto optima for collision free coordination on the product of these roadmaps with collision type obstacles such optima are by no means unique in fact continua of pareto optimal coordinations are possible we prove a finite bound on the number of optimal coordinations in the physically relevant case where all obstacles are cylindrical i e defined by pairwise collisions the proofs rely crucially on perspectives from geometric group theory and cat 0 geometry in particular the finiteness bound depends on the fact that the associated coordination space is devoid of positive curvature we also demonstrate that the finiteness bound holds for systems with moving obstacles following known trajectories

cell edge aware precoding for downlink massive mimo cellular networks we propose a cell edge aware cea zero forcing zf precoder that exploits the excess spatial degrees of freedom provided by a large number of base station bs antennas to suppress inter cell interference at the most vulnerable user equipments ues we evaluate the downlink performance of cea zf as well as that of a conventional cell edge unaware ceu zf precoder in a network with a random bs topology our analysis and simulations show that the proposed cea zf precoder outperforms ceu zf precoding in terms of i aggregate per cell data rate ii coverage probability and iii 95 likely or edge user rate in particular when both perfect channel state information and a large number of antennas n are available at the bss we demonstrate that the outage probability under cea zf and ceu zf decay as 1 n 2 and 1 n respectively this result identifies cea zf as a more effective precoding scheme for massive multiple input multiple output mimo cellular networks our framework also reveals the importance of scheduling the optimal number of ues per bs and confirms the necessity to control the amount of pilot contamination received during the channel estimation phase

computing the k relative neighborhood graphs in euclidean plane abstract in this paper we will propose an o n 5 3 log n algorithm to construct the k relative neighborhood graph of a n points set in euclidean plane for fixed k based on this result the euclidean bottleneck matching problem can also be solved in the same time complexity the k relative neighborhood graph is a generalization of the relative neighborhood graph

an optimal algorithm for constructing the weighted voronoi diagram in the plane let s denote a set of n points in the plane such that each point p has assigned a positive weight w p which expresses its capability to influence its neighbourhood in this sense the weighted distance of an arbitrary point x from p is given by de x p w p where de denotes the euclidean distance function the weighted voronoi diagram for s is a subdivision of the plane such that each point p in s is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of s an algorithm which constructs the weighted voronoi diagram for s in o n2 time is outlined in this paper the method is optimal as the diagram can consist of n2 faces edges and vertices

heavy tails of ols suppose the tails of the noise distribution in a regression exhibit power law behavior then the distribution of the ols regression estimator inherits this tail behavior this is relevant for regressions involving financial data we derive explicit finite sample expressions for the tail probabilities of the distribution of the ols estimator these are useful for inference simulations for medium sized samples reveal considerable deviations of the coefficient estimates from their true values in line with our theoretical formulas the formulas provide a benchmark for judging the observed highly variable cross country estimates of the expectations coefficient in yield curve regressions

an improved algorithm for constructing kth order voronoi diagrams the kth order voronoi diagram of a finite set of sites in the euclidean plane e2 subdivides e2 into maximal regions such that all points within a given region have the same k nearest sites two versions of an algorithm are developed for constructing the kth order voronoi diagram of a set of n sites in o n2 log n k n k log2 n time o k n k storage and in o n2 k n k log2 n time o n2 storage respectively

linear and nonlinear regression with stable errors abstract in this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution the asymptotic fisher information matrix for both the regression coefficients and the error distribution parameters are derived giving large sample confidence intervals for all parameters simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable as well as a real example using financial data the results are then extended to nonlinear models and to non homogeneous error terms

shortest paths in the plane with polygonal obstacles we present a practical algorithm for finding minimum length paths between points in the euclidean plane with not necessarily convex polygonal obstacles prior to this work the best known algorithm for finding the shortest path between two points in the plane required o n 2 log n time and o n 2 space where n denotes the number of obstacle edges assuming that a triangulation or a voronoi diagram for the obstacle space is provided with the input if is not either one can be precomputed in o n log n time we present an o kn time algorithm where k denotes the number of islands connected components in the obstacle space the algorithm uses only o n space and given a source point s produces an o n size data structure such that the distance between s and any other point x in the plane x is not necessarily an obstacle vertex or a point on an obstacle edge can be computed in o 1 time the algorithm can also be used to compute shortest paths for the movement of a disk so that optimal movement for arbitrary objects can be computed to the accuracy of enclosing them with the smallest possible disk

fully dynamic point location in a monotone subdivision in this paper a dynamic technique for locating a point in a monotone planar subdivision whose current number of vertices is n is presented the complete set of update operations are insertion o

optimal cover of points by disks in a simple polygon let p be a simple polygon and let q be a set of points in p we present an almost linear time algorithm for computing a minimum cover of q by disks that are contained in p we then generalize the algorithm so that it can compute a minimum cover of q by homothets of any fixed compact convex set cal o of constant description complexity that are contained in p this improves previous results of katz and morgenstern lecture notes in comput sci 5664 2009 pp 447 458 we also consider the minimum disk cover problem when q is contained in a sufficiently narrow annulus and present a nearly linear algorithm for this case too

r estimation in semiparametric dynamic location scale models we propose rank based estimation r estimators as an alternative to gaussian quasi likelihood and standard semiparametric estimation in time series models where conditional location and or scale depend on a euclidean parameter of interest while the unspecified innovation density is a nuisance we show how to construct r estimators achieving semiparametric efficiency at some predetermined reference density while preserving root n consistency and asymptotic normality irrespective of the actual density contrary to the standard semiparametric estimators our r estimators neither require tangent space calculations nor innovation density estimation numerical examples illustrate their good performances on simulated and real data

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

a space optimal solution of general region location abstract in 1979 kirpatrick obtained a practically feasible algorithm for planar region location working in linear space and logarithmic time provided the regions are bounded by straight line segments no algorithm requiring only linear space and log polynomial time was known so far for general planar region location i e for the case where regions are bounded by curves more complicated than straight line segments as main result of this paper such an algorithm is presented

cooperative base station coloring for pair wise multi cell coordination this paper proposes a method for designing base station bs clusters and cluster patterns for pair wise bs coordination the key idea is that each bs cluster is formed by using the second order voronoi region and the bs clusters are assigned to a specific cluster pattern by using edge coloring for a graph drawn by delaunay triangulation the main advantage of the proposed method is that the bs selection conflict problem is prevented while users are guaranteed to communicate with their two closest bss in any irregular bs topology with the proposed coordination method analytical expressions for the rate distribution and the ergodic spectral efficiency are derived as a function of relevant system parameters in a fixed irregular network model in a random network model with a homogeneous poisson point process a lower bound on the ergodic spectral efficiency is characterized through system level simulations the performance of the proposed method is compared with that of conventional coordination methods dynamic clustering and static clustering our major finding is that when users are dense enough in a network the proposed method provides the same level of coordination benefit with dynamic clustering to edge users

geometric retrieval problems a large class of geometric retrieval problems has the following form given a set x of geometric objects preprocess to obtain a data structure d x now use d x to rapidly answer queries on x we say an algorithm for such a problem has worst case space time complexity o f n g n if the space requirement for d x is o f and the locate run time required for each retrieval is o g we show three techniques which can consistently be exploited in solving such problems for instance using our techniques we obtain an o n2 e log n log n log 1 e space time algorithm for the polygon retrieval problem for arbitrarily small e improving on the previous solution having complexity o n7 log n

query point visibility constrained shortest paths in simple polygons in this paper we study the problem of finding the shortest path between two points inside a simple polygon such that there is at least one point on the path from which a query point is visible we provide an algorithm which preprocesses the input in o n 2 nk time and space and provides logarithmic query time the input polygon has n vertices and k is a parameter dependent on the input polygon which is o n 2 in the worst case but is much smaller for most polygons the preprocessing algorithm sweeps an angular interval around every reflex vertex of the polygon to store the optimal contact points between the shortest paths and the windows separating the visibility polygons of the query points from the source and the destination

discrete voronoi diagrams and the skiz operator a dynamic algorithm the voronoi diagram vd is a popular tool for partitioning the support of an image an algorithm is presented for constructing vd when the seed set which determines the voronoi regions can be modified by adding and removing seeds the number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized a result on cocircular seeds is presented the adjacency or dual graph of the vd is readily obtained the use of the algorithm for constructing skeletons by influence zones skiz is demonstrated

a unified approach to dynamic point location ray shooting and shortest paths in planar maps we describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map cal m with n vertices and apply it to the development of a unified dynamic data structure that supports point location ray shooting and shortest path queries in cal m the space requirement is o n log n point location queries take time o log n ray shooting and shortest path queries take time o log 3 n plus o k time if the k edges of the shortest path are reported in addition to its length updates consist of insertions and deletions of vertices and edges and take o log 3 n time amortized for vertex updates this is the first polylog time dynamic data structure for shortest path and ray shooting queries it is also the first dynamic point location data structure for connected planar maps that achieves optimal query time

semi lagrangian methods for level set equations a new numerical method for solving geometric moving interface problems is presented the method combines a level set approach and a semi lagrangian time stepping scheme which is explicit yet unconditionally stable the combination decouples each mesh point from the others and the time step from the cfl stability condition permitting the construction of methods which are efficient adaptive and modular analysis of a linear one dimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of two dimensional numerical results the new method computes correct viscosity solutions to problems involving geometry anisotropy curvature and complex topological events

a unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids abstract given a set of n points in r 2 bounded within a rectangular floor f and a rectangular plate p of specified size we consider the following two problems find an isothetic position of p such that it encloses i maximum and ii minimum number of points keeping p totally contained within f for both of these problems a new algorithm based on interval tree data structure is presented which runs in o nlogn time and consumes o n space if polygonal objects of arbitrary size and shape are distributed in r 2 the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity finally the algorithm is extended for identifying a cuboid i e a rectangular parallelepiped that encloses maximum number of polyhedral objects in r 3 thus the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition

rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles we study the rectilinear shortest paths and minimum spanning tree mst problems for a set of points in the plane in the presence of rectilinear obstacles we use the track graph a suitably defined grid like structure to obtain efficient solutions for both problems the track graph consists of rectilinear tracks defined by the obstacles and the points for which shortest paths and a minimum spanning tree are sought we use a growth process like dijkstra s on the track graph to find shortest paths from any point in the set to all other points the one to all shortest paths problem for the one to all shortest paths problem for n points we derive an o n min log n log e e k log t time algorithm where e is the total number of edges of all obstacles t is the number of extreme edges of all obstacles and k is the number of intersections among obstacle tracks all bounds are for the worst case the mst for the points is constructed also in time o n log n e k log t by a hybrid method of searching for shortest paths while simultaneously constructing an mst an interesting application of the mst algorithm is the approximation of steiner trees in graphs

second order boundary schemes for the lattice boltzmann method with general propagation abstract this paper is concerned with the boundary treatment of the lattice boltzmann method lbm with general propagation for the incompressible navier stokes equations since the propagation step involves three grid points constructing effective boundary schemes for such method is highly nontrivial to address this problem we first construct an auxiliary boundary scheme under the diffusive scaling by using the maxwell iteration based on this scheme and the interpolations of both distribution functions and fluid velocities we propose a parameterized boundary scheme for the lbm with general propagation the scheme thus obtained not only has second order accuracy under the diffusive scaling for curved boundaries but also possesses good stability several specific schemes are also provided and their accuracy and stability are verified via four flow problems with both straight and curved boundaries it should be noted that the proposed boundary scheme can also be used for the standard lbm

discrete unified gas kinetic scheme for all knudsen number flows low speed isothermal case based on the boltzmann bgk bhatnagar gross krook equation in this paper a discrete unified gas kinetic scheme dugks is developed for low speed isothermal flows the dugks is a finite volume scheme with the discretization of particle velocity space after the introduction of two auxiliary distribution functions with the inclusion of collision effect the dugks becomes a fully explicit scheme for the update of distribution function furthermore the scheme is an asymptotic preserving method where the time step is only determined by the courant friedricks lewy condition in the continuum limit numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current dugks the comparison between the dugks and the well defined lattice boltzmann equation method d2q9 is presented as well

lattice boltzmann method for incompressible flows with large pressure gradients conventional lattice boltzmann bhatnagar gross krook lbgk models can simulate incompressible flows correctly only if the mach number m and the density variation are negligibly small however the equation of state p rt resulting from the conventional models limits their application to incompressible flows with a rather small pressure gradient in this paper based on the enskog equation we propose a finite difference lattice bgk model for isothermal incompressible flows with the resulting equation of state and transport properties suitable for nonideal fluids we validated this model by simulating the plane poiseuille flow the two dimensional womersley flow and the backward facing step flow and found that the numerical results obtained by the proposed model are more accurate than those by the conventional lbgk models when the pressure gradient imposed on the flows increases

the method of simulated quantiles we introduce an inference method based on quantiles matching which is useful for situations where the density function does not have a closed form but it is simple to simulate and or moments do not exist functions of theoretical quantiles which depend on the parameters of the assumed probability law are matched with sample quantiles which depend on observations since the theoretical quantiles may not be available analytically the optimization is based on simulations we illustrate the method with the estimation of stable distributions a thorough monte carlo study and an illustration to 22 nancial indexes show

convergence of lattice boltzmann methods for stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method m junk z yang asymptotic analysis of lattice boltzmann boundary conditions j stat phys 121 2005 3 35 with the stability estimate presented in m junk w a yong weighted l 2 stability of the lattice boltzmann equation preprint we are able to prove some strict convergence results the proof applies to the lattice boltzmann method with linear collision operator both in the case of periodic domains and bounded domains if the dirichlet boundary condition is realized with the bounce back rule

novel transformation techniques using q heaps with applications to computational geometry using the notions of q heaps and fusion trees developed by fredman and willard we develop general transformation techniques to reduce a number of computational geometry problems to their special versions in partially ranked spaces in particular we develop a fast fractional cascading technique which uses linear space and enables sublogarithmic iterative search on catalog trees in the case when the degree of each node is bounded by o log epsilon n for some constant epsilon 0 where n is the total size of all the lists stored in the tree we apply the fast fractional cascading technique in combination with the other techniques to derive the first linear space sublogarithmic time algorithms for two fundamental geometric retrieval problems orthogonal segment intersection and rectangular point enclosure

new upper bounds for neighbor searching this paper investigates the circular retrieval problem and the k nearest neighbor problem for sets of n points in the euclidean plane two similar data structures each solve both problems a deterministic structure uses space o n log n log log n 2 and a probabilistic structure uses space o n log2 n for both problems these two structures answer a query that returns k points in o k log n time

how to search history this paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times we present a new scheme for recording a history of h updates over an ordered set s of n objects which allows fast neighbor computation at any time in the history the novelty of the method is to allow the set s to be only partially ordered with respect to queries and the time measure to be multi dimensional the generality of the method makes it useful for a number of problems in 3 dimensional geometry for example we are able to give fast algorithms for locating a point in a 3 dimensional complex using linear space or for finding which of n given points is closest to a query plane using a simpler yet conceptually similar technique we show that with o n 2 preprocessing it is possible to determine in o log 2 n time which of n given points in e 3 is closest to an arbitrary query point

computing relative neighbourhood graphs in the plane abstract the relative neighbourhood graph rng of a set of n points connects the relative neighbours i e a pair of points is connected by an edge if those points are at least as close to each other as to any other point the paper presents two new algorithms for constructing rng in two dimensional euclidean space the method is to determine a supergraph for rng which can then be thinned efficiently from the extra edges the first algorithm is simple and works in o n 2 time the worst case running time of the second algorithm is also o n 2 but its average case running time is o n for points from a homogeneous planar poisson point process experimental tests have shown the usefulness of the approach

computing closest and farthest points for a query segment in this paper we present an improved algorithm for finding k closest farthest points for a given arbitrary query segment we show how to preprocess a planar set p of n given points in o n 2logn expected time or alternatively in o n 2log 2n deterministic time and a subquadratic space in order to report k closest points to an arbitrary given query line segment in o k log 2nloglogn time here for the first time the data structure that provides polylogarithmic query time and uses subquadratic space is presented we also show an algorithm for reporting the k farthest points from an arbitrary given query line segment

finite difference based lattice boltzmann model for dense binary mixtures we propose a finite difference based lattice boltzmann model for dense binary mixtures based on the enskog theory the model is applicable to a mixture composed of two dense fluids with different shear viscosities the macroscopic hydrodynamic and diffusion equations are derived from the model through the chapmann enskog procedure the model is also validated numerically

an approximate algorithm for computing multidimensional convex hulls to find a convex hull for n points in d dimensional space the optimal algorithm has time complexity o n d 2 when n and d are large the execution time is very long in this paper we propose an approximate algorithm for computing multidimensional convex hulls this algorithm finds quasi two side approximation to the hull to reduce the time for computing the exact hull boundary to yield an e approximate convex hull it has time complexity o e 1 d 1 n and storage complexity o e 1 d 1 the approximate algorithm has several advantages 1 it can easily be implemented 2 it is suitable for parallel implementation 3 it is much faster than the exact algorithm 4 the user can choose to get more accurate results using longer computation time and 5 it can be applied to solve many problems related to convex hull computation

maximum likelihood estimation for α stable autoregressive processes we consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non gaussian stable noise a nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution the estimators for the autoregressive parameters are n 1 consistent and converge in distribution to the maximizer of a random function the form of this limiting distribution is intractable but the shape of the distribution for these estimators can be examined using the bootstrap procedure the bootstrap is asymptotically valid under general conditions the estimators for the parameters of the stable noise distribution have the traditional n 1 2 rate of convergence and are asymptotically normal the behavior of the estimators for finite samples is studied via simulation and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of volumes of wal mart stock traded daily on the new york stock exchange

instance optimal geometric algorithms we prove the existence of an algorithm a for computing 2d or 3d convex hulls that is optimal for every point set in the following sense for every sequence of n points and for every algorithm a in a certain class a the running time of a on input is at most a constant factor times the running time of a on the worst possible permutation of for a in fact we can establish a stronger property for every sequence of points and every algorithm a the running time of a on is at most a constant factor times the average running time of a over all permutations of we call algorithms satisfying these properties instance optimal in the order oblivious and random order setting such instance optimal algorithms simultaneously subsume output sensitive algorithms and distribution dependent average case algorithms and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order the class a under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments to establish an instance specific lower bound we deviate from traditional ben or style proofs and adopt a new adversary argument for 2d convex hulls we prove that a version of the well known algorithm by kirkpatrick and seidel 1986 or chan snoeyink and yap 1995 already attains this lower bound for 3d convex hulls we propose a new algorithm we further obtain instance optimal results for a few other standard problems in computational geometry such as maxima in 2d and 3d orthogonal line segment intersection in 2d finding bichromatic l close pairs in 2d offline orthogonal range searching in 2d offline dominance reporting in 2d and 3d offline half space range reporting in 2d and 3d and offline point location in 2d our framework also reveals a connection to distribution sensitive data structures and yields new results as a byproduct for example on online orthogonal range searching in 2d and online half space range reporting in 2d and 3d

transdichotomous results in computational geometry i point location in sublogarithmic time given a planar subdivision whose coordinates are integers bounded by u leq2 w we present a linear space data structure that can answer point location queries in o min lg n lg lg n sqrt lg u lg lg u time on the unit cost random access machine ram with word size w this is the first result to beat the standard theta lg n bound for infinite precision models as a consequence we obtain the first o n lg n randomized algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word ram including constructing the convex hull of a three dimensional 3d point set computing the voronoi diagram or the euclidean minimum spanning tree of a planar point set triangulating a polygon with holes and finding intersections among a set of line segments higher dimensional extensions and applications are also discussed though computational geometry with bounded precision input has been investigated for a long time improvements have been limited largely to problems of an orthogonal flavor our results surpass this long standing limitation answering for example a question of willard soda 92

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

semiparametrically efficient rank based inference for shape ii optimal r estimation of shape a class of r estimators based on the concepts of multivariate signed ranks and the optimal rank based tests developed in hallin and paindaveine 2006a is proposed for the estimation of the shape matrix of an elliptical distribution these r estimators are root n consistent under any radial density g without any moment assumptions and semiparametrically efficient at some prespecified density f when based on normal scores they are uniformly more efficient thanthe traditional normal theory estimator based on empirical covariance matrices the asymptotic normality of which moreover requires finite moments of order four irrespective of the actual underlying elliptical density they rely on an original rank based version of le cam s one step methodology which avoids the unpleasant nonparametric estimation of cross information quantities that is generally required in the context of r estimation although they are not strictly equivariant they are shown to be equivariant in a weak asymptotic sense simulations confirm their feasability and excellent finite sample performances

one step r estimation in linear models with stable errors classical estimation techniques for linear models either are inconsistent or perform rather poorly under stable error densities most of them are not even rate optimal in this paper we propose an original one step r estimation method and investigate its asymptotic performances under stable densities contrary to traditional least squares the proposed r estimators remain root n consistent the optimal rate under the whole family of stable distributions irrespective of their asymmetry and tail index while parametric stable likelihood estimation due to the absence of a closed form for stable densities is quite cumbersome our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values 0 b0 of the tail index and skewness parameter b while preserving root n consistency under any b as well as under usual light tailed densities the method furthermore avoids all forms of multidimensional argmin computation simulations confirm its excellent finite sample performances

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

convex partitions of polyhedra a lower bound and worst case optimal algorithm the problem of partitioning a polyhedron into a minimum number of convex pieces is known to be np hard we establish here a quadratic lower bound on the complexity of this problem and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case the algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles since in most applications areas the former quantity greatly exceeds the latter the algorithm is viable in practice

an optimal algorithm for euclidean shortest paths in the plane we propose an optimal time algorithm for a classical problem in plane computational geometry computing a shortest path between two points in the presence of polygonal obstacles our algorithm runs in worst case time o n log n and requires o n log n space where n is the total number of vertices in the obstacle polygons the algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane the map can be used to answer single source shortest path queries in o log n time the time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem finally we also discuss extensions to more general shortest path problems involving nonpoint and multiple sources

on the convex layers of a planar set let s be a set of n points in the euclidean plane the convex layers of s are the convex polygons obtained by iterating on the following procedure compute the convex hull of s and remove its vertices from s this process of peeling a planar point set is central in the study of robust estimators in statistics it also provides valuable information on the morphology of a set of sites and has proven to be an efficient preconditioning for range search problems an optimal algorithm is described for computing the convex layers of s the algorithm runs in 0 n log n time and requires o n space also addressed is the problem of determining the depth of a query point within the convex layers of s i e the number of layers that enclose the query point this is essentially a planar point location problem for which optimal solutions are therefore known taking advantage of structural properties of the problem however a much simpler optimal solution is derived

solving related two and three dimensional linear programming problems in logarithmic time abstract given n linear inequalities in three variables we show how to construct a corresponding spherical subdivision using great circle arcs in time o n log n and space o n this subdivision in turn allows us to compute the point in space satisfying all inequalities and maximizing any desired linear objective function in time o log n

optimal search in planar subdivisions a planar subdivision is any partition of the plane into possibly unbounded polygonal regions the subdivision search problem is the following given a subdivision s with n line segments and a query point p determine which region of s contains p we present a practical algorithm for subdivision search that achieves the same optimal worst case complexity bounds as the significantly more complex algorithm of lipton and tarjan namely o log n search time with o n storage our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications

relative neighborhood graphs in the li metric abstract the relative neighborhood graph of a set of n points in the plane under the l 1 metric is considered an algorithm that runs in o n log n time for constructing the relative neighborhood graph based on the delaunay triangulation is presented improving a previously known algorithm that runs in o n 2 log n time

detecting spatial homogeneity in the world trade web with detrended fluctuation analysis in a spatially embedded network that is a network where nodes can be uniquely determined in a system of coordinates links weights might be affected by metric distances coupling every pair of nodes dyads in order to assess to what extent metric distances affect relationships link s weights in a spatially embedded network we propose a methodology based on dfa detrended fluctuation analysis dfa is a well developed methodology to evaluate autocorrelations and estimate long range behaviour in time series we argue it can be further extended to spatially ordered series in order to assess autocorrelations in values a scaling exponent of 0 5 uncorrelated data would thereby signal a perfect homogeneous space embedding the network we apply the proposed methodology to the world trade web wtw during the years 1949 2000 and we find in some contrast with predictions of gravity models a declining influence of distances on trading relationships

weighted mathbb l 2 stability of the lattice boltzmann method this article is concerned with the linearized stability of the lattice boltzmann method both on periodic domains and on bounded domains with the bounce back rule used at the boundaries under a structural hypothesis we prove that a weighted mathbb l 2 norm of the solutions to the linearized lattice boltzmann method is decreasing with time moreover we show that the structural hypothesis holds true for many lattice boltzmann models

on the randomized construction of the delaunay tree the delaunay tree is a hierarchical data structure which is defined from the delaunay triangulation and roughly speaking represents a triangulation as a hierarchy of balls it allows a smidynamic construction of the delaunay triangulation of a finite set of n points in any dimension in this paper we prove that a randomized construction of the delaunay tree and thus of the delaunay triangulation can be done in o n log n expected time in the plane and in o n d2 expected time in d dimensional space these results are optimal for fixed d the algorithm is extremely simple and experimental results are given

l1 shortest path queries in simple polygons abstract let p be a simple polygon of n vertices we consider two point l 1 shortest path queries in p we build a data structure of o n size in o n time such that given any two query points s and t the length of an l 1 shortest path from s to t in p can be computed in o log n time or in o 1 time if both s and t are vertices of p and an actual shortest path can be output in additional linear time in the number of edges of the path to achieve the result we propose a mountain decomposition of simple polygons which may be interesting in its own right most importantly our approach is much simpler than the previous work on this problem

a stability notion for lattice boltzmann equations lattice boltzmann equations are usually constructed to satisfy some physical requirements such as galilean invariance and isotropy to possess a velocity independent pressure and no compressible effects and so on in this paper a stability requirement is introduced as a new criterion for the constructions with this requirement we derive some relations of parameters for several lattice boltzmann models interestingly these relations are satisfied by many choices of parameters used in the literature

parallel transitive closure and point location in planar structures parallel algorithms for several graph and geometric problems are presented including transitive closure and topological sorting in planar st graphs preprocessing planar subdivisions for point location queries and construction of visibility representations and drawings of planar graphs most of these algorithms achieve optimal o log n running time using n log n processors in the erew pram model n being the number of vertices

on k nearest neighbor voronoi diagrams in the plane the notion of voronoi diagram for a set of n points in the euclidean plane is generalized to the voronoi diagram of order k and an iterative algorithm to construct the generalized diagram in 0 k2n log n time using 0 k2 n k space is presented it is shown that the k nearest neighbor problem and other seemingly unrelated problems can be solved efficiently with the diagram

dynamic voronoi diagrams a new dynamizing technique is introduced whereby n point voronoi diagrams both closest and farthest point can be updated in o n time per insertion or deletion in the worst case general properties of these dynamic voronoi diagrams are explored including a storage deletion time trade off in addition their application to such problems as nearest neighbor search and the 2 minimum spanning circle problem is discussed

range queries on uncertain data given a set p of n uncertain points on the real line each represented by its one dimensional probability density function we consider the problem of building data structures on p to answer range queries of the following three types for any query interval i 1 top 1 query find a point in p that lies in i with the highest probability 2 top k query given any integer k n as part of the query return the k points in p that lie in i with the highest probabilities and 3 threshold query given any threshold as part of the query return all points of p that lie in i with probabilities at least we present data structures for these range queries with linear or nearly linear space and efficient query time

delaunay configurations and multivariate splines a generalization of a result of b n delaunay in the 1920s b n delaunay proved that the dual graph of the voronoi diagram of a discrete set of points in a euclidean space gives rise to a collection of simplices whose circumspheres contain no points from this set in their interior such delaunay simplices tessellate the convex hull of these points an equivalent formulation of this property is that the characteristic functions of the delaunay simplices form a partition of unity in the paper this result is generalized to the so called delaunay configurations these are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set in contrast to the delaunay simplices whose circumspheres are empty it is proved that every family of delaunay configurations generates a partition of unity formed by the so called simplex splines these are compactly supported piecewise polynomial functions which are multivariate analogs of the well known univariate b splines it is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines

optimal expected case planar point location point location is the problem of preprocessing a planar polygonal subdivision s of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point we consider this problem from the perspective of expected query time we are given the probabilities p z that the query point lies within each cell z in s the entropy h of the resulting discrete probability distribution is the dominant term in the lower bound on the expected case query time we show that it is possible to achieve query time h o sqrt h 1 with space o n which is optimal up to lower order terms in the query time we extend this result to subdivisions with convex cells assuming a uniform query distribution within each cell in order to achieve space efficiency we introduce the concept of entropy preserving cuttings

convergence of lattice boltzmann methods for navier stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method with a stability estimate we are able to prove some convergence results which establish a strict relation to the incompressible navier stokes equation the proof applies to the lattice boltzmann method in the case of periodic domains and for specific bounded domains if the dirichlet boundary condition is realized with the bounce back rule

a nearly optimal parallel algorithm for constructing maximal independent set in planar graphs abstract this paper presents an efficient parallel algorithm for constructing maximal independent set in planar graphs the algorithm runs in o log 2 n time with o n processors on a pram and is within an o log 2 n factor of optimal the best previously known algorithm for this problem takes o log 2 n time with o n 3 processors the key subroutine of the algorithm is an o log n time o n processor parallel algorithm for constructing a maximal independent set in a one page planar graph this subroutine may be of independent interest

weighted l2 stability of a discrete kinetic approximation for the incompressible navier stokes equations on bounded domains abstract this paper is concerned with the stability of a discrete kinetic approximation with a boundary scheme introduced by the authors in a previous work we prove the weighted l 2 stability of the approximation by using an identity on three point difference schemes for convection equations with the weighted l 2 stability the convergence of the discrete kinetic approximation can be directly established moreover the present stability analysis particularly the identity on three point difference schemes may be adapted to other kinetic methods numerical experiments are conducted to verify our stability results

a new approach to planar point location given a planar straight line graph g with n vertices and a point p 0 locating p 0 means to find the region of the planar subdivision induced by g which contains p 0 recently lipton and tarjan presented a brilliant but extremely complex point location algorithm which runs in time o log n on a data structure using o n storage this paper presents a practical algorithm which runs in less than 6 lceil log 2 n rceil comparisons on a data structure which uses o n log n storage in the worst case the method rests crucially on a simple partition of each edge of g into o log n segments

power diagrams properties algorithms and applications the power pow x s of a point x with respect to a sphere s in euclidean d space e d is given by d 2 x z r 2 where d denotes the euclidean distance function and z and r are the center and the radius of s the power diagram of a finite set s of spheres in e d is a cell complex that associates each s in s with the convex domain x in e d operatorname pow x s operatorname pow x t text for all t in s s the close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited efficient algorithms that compute the power diagram and its order k modifications are obtained among the applications of these results are algorithms for detecting k sets for union and intersection problems for cones and paraboloids and for constructing weighted voronoi diagrams and voronoi diagrams for spheres upper space bounds for these geometric problems are derived

a fast planar partition algorithm ii randomized optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree and a set of linear chains of a bounded degree are given this paper also provides a new technique for clipping called virtual clipping whose overhead per window w depends logarithmically on the number if intersections between the borders of w and the input segments in contrast the overhead of the conventional clipping technique depends linearly on this number of intersections as an application of virtual clipping a new simple and efficient algorithm for plannar point location is given

the discrete geodesic problem we present an algorithm for determining the shortest path between a source and a destination on an arbitrary possibly nonconvex polyhedral surface the path is constrained to lie on the surface and distances are measured according to the euclidean metric our algorithm runs in time o n log n and requires o n2 space where n is the number ofedges ofthe surface afterwe run our algorithm the distance from the source to any other destination may be determined using standard techniques in time o log n by locating the destination in the subdivision created by the algorithm the actual shortest path from the source to a destination can be reported in time o k log n where k is the number of faces crossed by the path the algorithm generalizes to the case of multiple source points to build the voronoi diagram on the surface where n is now the maximum of the number of vertices and the number of sources

generalization of voronoi diagrams in the plane in this paper we study the voronoi diagram for a set of n line segments and circles in the euclidean plane the diagram is a generalization of the voronoi diagram for a set of points in the plane and has applications in wire layout facility location clustering and contouring problems we present an o n log n 2 algorithm for constructing the diagram it is an improvement of a previous known result which takes o nc sqrt log n time the algorithm described in this paper is also shown to be applicable under a more general metric if certain conditions are satisfied

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

multilevel solvers for unstructured surface meshes parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing such parameterization approaches give rise to large and exceedingly ill conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies since the underlying meshes are very fine to begin with such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy in this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature we introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner