linear time algorithm to cover and hit a set of line segments optimally by two axis parallel squares abstract this paper discusses the problem of covering and hitting a set of line segments l in r 2 by a pair of axis parallel congruent squares of minimum size we also discuss the restricted version of covering where each line segment in l is to be covered completely by at least one square the proposed algorithms assume that the input segments are given in a read only array for each of these problems i e covering hitting and restricted covering problems our proposed algorithm reports the optimum result by executing only two passes of reading the input array sequentially all these algorithms need only o 1 extra space the solution of these problems also give a 2 approximation for covering and hitting those line segments l by two congruent disks of minimum radius with same computational complexity

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

on anstreicher s combined phase i phase ii projective algorithm for linear programming anstreicher has proposed a variant of karmarkar s projective algorithm that handles standard form linear programming problems nicely we suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm much of the analysis depends on a two constraint linear programming problem that is a relaxation of the scaled original problem

domains with non compact automorphism group a survey abstract we survey results arising from the study of domains in c n with non compact automorphism group beginning with a well known characterization of the unit ball we develop ideas toward a consideration of weakly pseudoconvex and even non pseudoconvex domains with particular emphasis on characterizations of i smoothly bounded domains with non compact automorphism group and ii the levi geometry of boundary orbit accumulation points particular attention will be paid to results derived in the past ten years

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

continuous location of dimensional structures abstract a natural extension of point facility location problems are those problems in which facilities are extensive i e those that cannot be represented by isolated points but as some dimensional structures such as straight lines line segments polygonal curves or circles in this paper a review of the existing work on the location of extensive facilities in continuous spaces is given gaps in the knowledge are identified and suggestions for further research are made

a randomized algorithm for fixed dimensional linear programming we give a las vegas randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is o d 3 varepsilon d d n where limd e d 0 this improves the corresponding worst case complexity o 3 d 2 n the method is based on a recent idea of clarkson two variations on the algorithm are examined briefly

low dimensional linear programming with violations two decades ago megiddo and dyer showed that linear programming lp in two and three dimensions and subsequently any constant number of dimensions can be solved in linear time in this paper we consider the lp problem with at most k violations i e finding a point inside all but at most k halfspaces given a set of n halfspaces we present a simple algorithm in two dimensions that runs in o n k2 log n expected time this is faster than earlier algorithms by everett robert and van kreveld 1993 and matousek 1994 for many values of k and is probably near optimal an extension of our algorithm in three dimensions runs in near o n k11 4n1 4 expected time interestingly the idea is based on concave chain decompositions or covers of the le k level previously used in proving combinatorial k level bounds applications in the plane include improved algorithms for finding a line that misclassifies the fewest among a set of bichromatic points and finding the smallest circle enclosing all but k points we also discuss related problems of finding local minima in levels

a two resource allocation problem solvable in linear time the allocation problem discussed here is as follows the processing time of a task is a linear function t ax by of amounts x y of resources allocated to it where t a b are positive constants given the amounts x and y of available resources we wish to allocate them among n independent tasks so as to minimize the maximal completion time we solve this problem in linear time

complete localization of domains with noncompact automorphism groups we prove a characterization of the domains in cn with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates this result generalizes the well known wong rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries

projection based polygonality measurement measuring the degree to which a shape resembles a polygon referred to as polygonality is a difficult problem due to the intrinsic diversity in the form and distortion of shapes caused by digitization and similarity transformation this paper proposes a generic approach for this problem by performing the measurement in the projection space where the radon image of some primitive shapes which compose the shape becomes apparent the obtained measures take value in 0 1 with 1 corresponding to analytical polygons they are robust to additive noise and boundary distortion and is invariant to similarity transformation the new framework generalizes existing polygonal measures such as triangularity and quadrangularity in addition the ability to estimate polygon s geometric quantities in the projection space allows approximating a shape by analytical polygons the efficiency of the proposed approach is demonstrated through a number of experiments on both synthetic and real data sets

a geometrical solution for quadratic bicriteria location models this paper deals with a single facility unweighted quadratic bicriteria location problem defined in the continuous space first we describe an analytical method for constructing the set of pareto optimal locations for this problem based on the farthest point voronoi diagram then we present an analytical method for finding the solution to the scalarized location problems based on the pareto optimal locations the analytical results enable planners to carry out some sensitivity analyses since both methods are based entirely on geometrical analysis they can be understood intuitively

out of roundness problem revisited the properties and computation of the minimum radial separation mrs standard for out of roundness are discussed another standard out of roundness measurement called the minimum area difference mad center is introduced although the two centers have different characteristics the approach to finding both centers shares many commonalities an o n log n k time algorithm which is used to compute the mrs center is presented it also computes the mad center of a simple polygon g where n is the number of vertices of g and k is the number of intersection points of the medial axis and the farthest neighbor voronoi diagram of g the relationship between mrs and mad is discussed

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

an efficient algorithm for the smallest enclosing ball problem in high dimensions abstract consider the problem of computing the smallest enclosing ball of a set of m balls in r n this problem arises in many applications such as location analysis military operations and pattern recognition etc in this paper we reformulate this problem as an unconstrained convex optimization problem involving the maximum function max 0 t and then develop a simple algorithm particularly suitable for problems in high dimensions this algorithm could efficiently handle problems of dimension n up to 10 000 under a moderately large m as well as problems of dimension m up to 10 000 under a moderately large n numerical results are given to show the efficiency of algorithm

characterization of the hilbert ball by its automorphism group let be a bounded convex domain in a separable hilbert space the authors prove a version of the theorem of bun wong which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new normal families arguments in the construction of the limit biholomorphism

the discrete and mixed minimax 2 center problems abstract letting p be a set of n points in the plane the discrete minimax 2 center problem d m m 2 c p is to find two disks centered at p 1 p 2 p that minimize the maximum of two terms namely the euclidean distance between two centers and the distance of any other point to the closer center the mixed minimax 2 center problem m m m 2 c p is when one of the two centers is not in p we present algorithms solving the d m m 2 c p and m m m 2 c p the time complexities of solving the d m m 2 c p and m m m 2 c p are o n 2 log n and o n 2 log 2 n respectively furthermore we consider two steiner minimum sum dipolar spanning tree problems in which one of the two dipoles is a steiner point and the dipoles are both steiner points these two problems are shown to be solvable in o n log n and o n time respectively

up and downgrading the 1 center in a network abstract we study budget constrained network improvement and degrading problems based on the vertex 1 center problem on graphs given a graph with vertex weights and edge lengths the task is to decrease and increase the vertex weights within certain limits such that the optimal 1 center objective value with respect to the new weights is minimized and maximized respectively the upgrading improvement problem is shown to be solvable in o n 2 time provided that the distance matrix is given the downgrading 1 center problem is shown to be strongly np hard on general graphs but can be solved in o n 2 time on trees as byproduct we suggest an algorithm that solves the problem of minimizing over the upper envelope of n piecewise linear functions in o k n time where k is the total number of breakpoints

water quality modeling a review of the analysis of uncertainty this paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction more specifically four problem areas are examined in detail uncertainty about model structure uncertainty in the estimated model parameter values the propagation of prediction errors and the design of experiments in order to reduce the critical uncertainties associated with a model the review is rather lengthy and it has therefore been prepared in effect as two papers there is a shorter largely nontechnical version which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling enclosed by this shorter discussion is the main body of the review dealing in turn with 1 identifiability and experimental design 2 the generation of preliminary model hypotheses under conditions of sparse grossly uncertain field data 3 the selection and evaluation of model structure 4 parameter estimation model calibration 5 checks and balances on the identified model i e model verification and model discrimination and 6 prediction error propagation much time is spent in discussing the algorithms of system identification in particular the methods of recursive estimation and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first order error analysis there are two obvious omissions from the review it is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty although clearly some reference must be made to these topics in brief the review concludes not surprisingly that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality and much remains to be done a lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior and there is ample evidence to show that the larger more comprehensive models are easily capable of generating highly uncertain predictions of future behavior for the future of the subject it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification a need for new questions to be posed in the problem of prediction and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence

efficiency of characterizing ellipses and ellipsoids by discrete moments in this paper our studies are focused on ellipses and problems related to their representation and reconstruction from the data resulting from their digitization the main result of the paper is that a finite number of discrete moments corresponded to digital ellipses is in one to one correspondence with digital ellipses which enables coding of digital ellipses with an asymptotically optimal amount of memory in addition the problem of reconstruction based on the same parameters is considered since the digitization of real shapes causes an inherent loss of information about the original objects the precision of the original shape estimation from the corresponding digital data is limited we derive a sharp upper bound for the errors in reconstruction of the center position and half axes of the ellipse in function of the applied picture resolution i e the number of pixels per unit an extension of these results to the 3d case is also given

deterministic geoleader election in disoriented anonymous systems consider a network made of n nodes scattered in the 2 dimensional space nodes are anonymous and disoriented devices being unable to communicate anonymous refers to systems made of a priori indistinguishable nodes by disoriented we mean that the nodes share no kind of coordinate system nor common sense of direction such systems are typically wireless sensor networks or swarms of robots endowed with localization capabilities for instance visibility sensors or pattern formation maps we address the geoleader election ge problem which is to ensure that solely based on their positions the nodes deterministically agree on the same position of a single node called the leader we provide a complete characterization on the node positions both for systems with common handedness chirality and for systems devoid of a common handedness the characterization is based on a particular object from combinatorics on words namely the lyndon words

a dynamic data structure for 3 d convex hulls and 2 d nearest neighbor queries we present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions where insertions take o log3n expected amortized time deletions take o log6n expected amortized time and extreme point queries take o log2n worst case time this is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions and improves the previous o ne time method by agarwal and matousek a decade ago as a consequence we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems such as levels in three dimensions and dynamic euclidean minimum spanning trees in the plane

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

finding the conditional location of a median path on a tree in this paper we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located the existing facilities may be located at any subset of vertices upper and lower bounds are proposed for both the discrete and continuous models in the discrete model a median path is not allowed to contain partial edges in the continuous model a median path may contain partial edges the proposed upper bounds for these two models are o nlogn and o nlogn a n respectively they improve the previous known bounds from o nlog 2n and o n 2 respectively the proposed lower bounds are both w nlogn

fuzzy disk for covering fuzzy points abstract in this paper we consider an important fuzzy version of the well known smallest covering circle problem which is also called the euclidean 1 center problem here we are given a set of points in the plane whose locations are crisp however the requirement for coverage of each point is fuzzy as represented by its grade of membership as such we qualify the points as fuzzy in the above framework we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points after developing a general model certain special cases are investigated in detail polynomial algorithms are presented for the special cases

measure of circularity for parts of digital boundaries and its fast computation this paper focuses on the design of an effective method that computes the measure of circularity of a part of a digital boundary an existing circularity measure of a set of discrete points which is used in computational metrology is extended to the case of parts of digital boundaries from a single digital boundary two sets of points are extracted so that the circularity measure computed from these sets is representative of the circularity of the digital boundary therefore the computation consists of two steps first the inner and outer sets of points are extracted from the input part of a digital boundary using digital geometry tools next the circularity measure of these sets is computed using classical tools of computational geometry it is proved that the algorithm is linear in time in the case of convex parts thanks to the specificity of digital data and is in o nlogn otherwise experiments done on synthetic and real images illustrate the interest of the properties of the circularity measure

the generalized trust region subproblem solution complexity and convex hull results we consider the generalized trust region subproblem gtrs of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint a lifting of this problem recasts the gtrs as minimizing a linear objective subject to two nonconvex quadratic constraints our first main contribution is structural we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil this result may be of interest in building relaxations for nonconvex quadratic programs moreover this result allows us to reformulate the gtrs as the minimization of two convex quadratic functions in the original space our next set of contributions is algorithmic we present an algorithm for solving the gtrs up to an epsilon additive error based on this reformulation we carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms notably our algorithms run in linear in the size of the input time furthermore our algorithm for computing an epsilon optimal solution has a slightly improved running time dependence on epsilon over the state of the art algorithm our analysis shows that the dominant cost in solving the gtrs lies in solving a generalized eigenvalue problem establishing a natural connection between these problems finally generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality interval and hollow constrained variants of the gtrs this gives the first linear time algorithm in the literature for these variants of the gtrs

dynamic minimum bichromatic separating circle abstract in the minimum bichromatic separating circle problem given a set r of n red points and a set b of m blue points the goal is to find the smallest radius circle c that contains all red points and the smallest possible number of blue points in this paper we study the dynamic version of the problem which we call the dynamic minimum bichromatic separating circle problem and present data structures that allow to perform efficient reporting and updates under insertion and removal of blue points we first describe a unified linear size data structure that allows both insertions and deletions of blue points then we present a data structure that allows to report an optimal circle in o log m n time when only insertions of blue points are allowed at the expense of an increase in update time and storage finally we give a data structure that achieves o log 2 m n reporting time when only deletions of blue points are allowed with the same update time and storage as the insertions only data structure our results are the first known for the dynamic minimum bichromatic separating circle problem

digital convexity straightness and convex polygons new schemes for digitizing regions and arcs are introduced it is then shown that under these schemes sklansky s definition of digital convexity is equivalent to other definitions digital convex polygons of n vertices are defined and characterized in terms of geometric properties of digital line segments also a linear time algorithm is presented that given a digital convex region determines the smallest integer n such that the region is a digital convex n gon

algorithms for the robust 1 center problem on a tree abstract we consider the weighted 1 center problem on a network with uncertainty in node weights and edge lengths uncertainty is modelled by means of interval estimates for parameters specifically each uncertain parameter is assumed to be random with unknown distribution and can take on any value within a corresponding prespecified interval it is required to find a robust minmax regret solution that is a location which is e optimal for any possible realization of parameters with e as small as possible the problem on a general network is known to be np hard for the problem on a tree we present a polynomial algorithm

digitized circular arcs characterization and parameter estimation the digitization of a circular arc causes an inherent loss of geometrical information arcs with slightly different local curvature or position may lead to exactly the same digital pattern in this paper the authors give a characterization of all centers and radii of circular arcs yielding the same digitization pattern the radius of the arcs varies over the set however only one curvature or radius estimate can be assigned to the digital pattern the authors derive an optimal estimator and give expressions for the bound on the precision of estimation this bound due to digitization is the deterministic equivalent of the cramer rao bound known from parameter estimation theory consider the estimation of the local curvature and local radius of a smooth object typically such parameters are estimated by moving a window along the digital boundary methods in literature show a poor precision in estimating curvature values relative errors of over 40 are often found from the definition of curvature it follows that locally the curve can be considered a circular arc and hence the method presented in this paper can be applied to the pattern in the window giving estimates with optimal precision and a measure for the remaining error on the practical side the authors present examples of the residual error due to the discrete grid the estimation of the radius or curvature of a circular arc at random position with an estimation window containing 10 points coded with nine freemancodes has a relative deviation exceeding 2 for a full disk the deviation is below 1 when the radius r exceeds four grid units the presented method is particularly useful for problems where some prior knowledge on the distribution of radii is known and where there is a noise free sampling

fuzzy facility location problem with preference of candidate sites models so far considered as facility location problems treat either min max or max min criterion that is the facility is either desirable or undesirable we propose the following model considering the satisfaction degree with respect to the distance from the facility for each customer residents and preference of the site in an urban area the objective is to find the site of the facility which maximizes the minimal satisfaction degree among all demand points and maximizes the preference of the site since generally speaking there exists no site that maximizes both criteria we seek non dominated sites after defining non domination

separating point sets by circles and the recognition of digital disks given two sets a and b of points in the plane we give a polynomial time algorithm to find the set s of points which are centers of circles that contain all the members of a and none of b a consequence of the algorithm is that s is polygonal another application is an o n2 time and o n2 space algorithm for the recognition of digital disks

k center and k median problems in graded distances graded distances are generalizations of the euclidean distance on points in r1 they have been used in the study of special cases of np hard problems in this paper we study the k center and k median problems with graded distance matrices we first prove that the k center problem is polynomial time solvable when the distance matrix is graded up the rows or graded down the rows we then prove that the k median problem is np complete when the distance matrix is graded up the rows or graded down the rows an easy special case of the k median problem with graded distance matrices is also discussed

on the complexity of the elzinga hearn algorithm for the 1 center problem we show that the complexity of the elzinga hearn algorithm for the 1 center single facility minimax problem is n2 we also show that the algorithm reaches a solution within any given accuracy in on time

a deep cut ellipsoid algorithm for convex programming theory and applications this paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems in each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided moreover for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed finally some computational results of the deep and central cut version of the algorithm applied to a min max stochastic queue location problem are reported

chebyshev center of the intersection of balls complexity relaxation and approximation we study the n dimensional problem of finding the smallest ball enclosing the intersection of p given balls the so called chebyshev center problem mathrm cc mathrm b it is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem mathrm uq when p le n mathrm uq is known to enjoy a strong duality and consequently mathrm cc mathrm b is solved via a standard convex quadratic programming mathrm sqp in this paper we first prove that mathrm cc mathrm b is np hard and the special case when n 2 is polynomially solvable with the help of a newly introduced linear programming relaxation lp the mathrm sqp relaxation is reobtained more directly and the first approximation bound for the solution obtained by mathrm sqp is established for the hard case p n finally also based on lp we show that mathrm cc mathrm b is polynomially solvable when either n or p n 0 is fixed

normal families of functions for subelliptic operators and the theorems of montel and koebe abstract a classical theorem of montel states that a family of holomorphic functions on a domain c uniformly bounded on the compact subsets of is a normal family the aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to l u 0 where l belongs to a wide class of real divergence form pdos comprising sub laplacians on carnot groups subelliptic laplacians on arbitrary lie groups as well as the laplace beltrami operator on riemannian manifolds to this end we extend another remarkable result due to koebe we characterize the solutions to l u 0 as fixed points of suitable mean value operators with non trivial kernels a suitable substitute for the cauchy integral formula is also provided finally the local boundedness assumption is relaxed by replacing it with l loc 1 boundedness

the dolbeault complex in infinite dimensions i infinite dimensional complex analysis was a popular subject in the sixties and seventies but in the last fifteen years enjoyed much less attention oddly enough this very same period began to see the emergence of examples of infinite dimensional complex manifolds in mathematical physics representation theory and geometry thus it appears to be a worthwhile undertaking to revisit infinite dimensional complex analysis and in particular to clarify fundamental properties of infinite dimensional complex manifolds the most fundamental questions at this point seem to be related to the solvability of the inhomogeneous cauchy riemann or 0 equations and more generally to the study of the dolbeault complex as an example in le3 we use a result on the a equation to describe the virasoro group as an infinite dimensional complex manifold up to now precious little has been known about the solvability of the infinite dimensional equation the available results pertain to solving the equation in domains in or over locally convex topological vector spaces and almost exclusively on the level of 0 1 forms postponing the precise definitions to sections 1 2 below we discuss those results that are of immediate relevance to this paper see d2 for more in li ligocka observes that h6rmander s proof of ehrenpreis theorem see e hor can be extended to infinite dimensions to solve the equation

reconstruction of a digital circle abstract since digitization always causes some loss of information reconstruction of the original figure from a given digitization is a challenging task reconstruction of digital circles has already been addressed in the literature however an in depth analysis of an obq image of a continuous circle as well as a solution to its domain construction problem is still lacking in this paper a detailed analysis of digital circles has been carried out a modified i r method is formulated to numerically compute the domain of each digital quarter circle for a given radius several properties of the obq image of a circle reveal that in many cases it is possible to split a digital circle into four digital quarter circles such that the domains of the individual quarter circles can be combined to obtain the domain of the full circle moreover the domain of a quarter circle is geometrically characterized

a robust method for nonlinear parameter estimation illustrated on a toxicological model the paper describes a random search method to estimate both optimal best fitting maximum likelihood values for the parameters in a nonlinear model and a measure of their uncertainty the algorithm is robust in the sense that it is relatively insensitive to local minima in addition after convergence it returns with a sample of points that directly characterizes the parameter uncertainty i e it is not necessary to make distributional assumptions about the parameters a priori the method is illustrated on a physiologically realistic toxicological model this model has a number of properties typical for such models a nonlinear likelihood function with local maxima and a non normal parameter uncertainty

the number of n point digital discs a digital disc is the set of all integer points inside some given disc let dn be the number of different digital discs consisting of n points different up to translation the upper bound d n o n2 was shown recently no corresponding lower bound is known in this paper we refine the upper bound to dn o n which seems to be the true order of magnitude and we show that the average dn d1 d2 dn n has upper and lower bounds which are of polynomial growth in n

structural properties of voronoi diagrams in facility location problems with continuous demand we consider facility location problems where the demand is continuously and uniformly distributed over a convex polygon with m vertices in the rectilinear plane n facilities are already present and the goal is to find an optimal location for an additional facility based on an analysis of structural properties of incremental voronoi diagrams we develop polynomial exact algorithms for five conditional location problems the developed methodology is applicable to a variety of other facility location problems with continuous demand moreover we briefly discuss the euclidean case

deferred data structuring we consider the problem of answering a series of on line queries on a static data set the conventional approach to such problems involves a preprocessing phase which constructs a data structure with good search behavior the data structure representing the data set then remains fixed throughout the processing of the queries our approach involves dynamic or query driven structuring of the data set our algorithm processes the data set only when doing so is required for answering a query a data structure constructed progressively in this fashion is called a deferred data structure we develop the notion of deferred data structures by solving the problem of answering membership queries on an ordered set we obtain a randomized algorithm which achieves asymptotically optimal performance with high probability we then present optimal deferred data structures for the following problems in the plane testing convex hull membership half plane intersection queries and fixed constraint multi objective linear programming we also apply the deferred data structuring technique to multidimensional dominance query problems

efficient algorithms for center problems in cactus networks efficient algorithms for solving the center problems in weighted cactus networks are presented in particular we have proposed the following algorithms for the weighted cactus networks of size n an o nlogn time algorithm to solve the 1 center problem and an o nlog 3n time algorithm to solve the weighted continuous 2 center problem we have also provided improved solutions to the general p center problems in cactus networks the developed ideas are then applied to solve the obnoxious 1 center problem in weighted cactus networks

minimum polygonal separation abstract in this paper we study the problem of polygonal separation in the plane i e finding a convex polygon with minimum number k of sides separating two given finite point sets k separator if it exists we show that for k n n log n is a lower bound to the running time of any algorithm for this problem and exhibit two algorithms of distinctly different flavors the first relies on an o n log n time preprocessing task which constructs the convex hull of the internal set and a nested star shaped polygon determined by the external set the k separator is contained in the annulus between the boundaries of these two polygons and is constructed in additional linear time the second algorithm adapts the prune and search approach and constructs in each iteration one side of the separator its running time is o kn but the separator may have one more side than the minimum

minmax regret 1 facility location on uncertain path networks abstract let p be an undirected path graph of n vertices each edge of p has a positive length and a constant capacity every vertex has a nonnegative supply which is an unknown value but is known to be in a given interval the goal is to find a point on p to build a facility and move all vertex supplies to the facility such that the maximum regret is minimized the previous best algorithm solves the problem in o n log 2 n time and o n log n space in this paper we present an o n log n time and o n space algorithm and our approach is based on new observations and algorithmic techniques

an o n algorithm for the multiple choice knapsack linear program an algorithm for solving the linear program associated with the multiple choice knapsack problem is described the algorithm is shown to work in time linear in the number of variables this improves the previously known best bound for this problem and is optimal to within a constant factor

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

algorithms for a minimum volume enclosing simplex in three dimensions we develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in cal r 3 if the convex hull of the points has n vertices then our algorithm takes theta n 4 time combining our exact but slow algorithm with a simple but crude approximation technique we also develop an varepsilon approximation algorithm the algorithm computes in o n 1 varepsilon 6 time a simplex whose volume is within 1 varepsilon factor of the optimal for any varepsilon 0

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc

center problems with pos neg weights on trees abstract in a network with positive and negative vertex weights the pos neg 1 center problem asks to minimize a linear combination of the maximum weighted distances of the center to the vertices with positive weights and to the vertices with negative weights respectively we show that in a network with n vertices and m edges the pos neg 1 center problem can be solved in o mn log n time in trees a better complexity can be achieved in the case of a path or of a star graph this problem can be solved in linear time further this problem is studied for a cactus with vertex weights 1 and 1 moreover an algorithm for the discrete anti p center problem on a tree with the improved time complexity o n log 2 n is given finally the pos neg discrete p center on a tree is treated and solved by an algorithm of time complexity o n 2 log n