15266108
Optimal output-sensitive convex hull algorithms in two and three dimensions
We present simple output-sensitive algorithms that construct the convex hull of a set ofn points in two or three dimensions in worst-case optimalO (n logh) time andO (n) space, whereh denotes the number of vertices of the convex hull.
26929194
Balanced k-colorings
Abstract While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k -coloring, for k ⩾2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4 d −3, where d (the “dimension”) is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is atmost max{2 d −3,2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete.
18265362
Approximation Algorithms for Maximum Cliques in 3D Unit-Disk Graphs
10486597
A fully dynamic algorithm for planar
We show how to maintain the width of a set of $n$ planar points subjec t to insertions and deletions of points in $O(\sqrt{n}\log^3n)$ amortized time per update. Previously, no fully dynamic algorithm with a guaranteed sublinear time bound was known.
10262363
Approximate nearest neighbor queries revisited
This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in d -dimensional Euclidean space. For any fixed constant d , a data structure with O(\(\varepsilon\) (1-d)/2 n log n) preprocessing time and O(\(\varepsilon\) (1-d)/2 log n) query time achieves an approximation factor 1+\(\varepsilon\) for any given 0 < \(\varepsilon\) < 1; a variant reduces the \(\varepsilon\) -dependence by a factor of \(\varepsilon\) -1/2 . For any arbitrary d , a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves an approximation factor O(d 3/2 ) . Applications to various proximity problems are discussed.
5292943
On enumerating and selecting distances
Given an n-point set, the problems of knumerating the /z closest pairs and selecting the L-th smallest distance ace revisited. l?oc the enumeration problem, we give 8implec randomized and deterministic algorithms with O(n log n-t/~) running time in any fixed-dimensional Euclidean space. l?oc the selection problem, we describe an approach to obtaining b-sensitive time bounds. We also point out output-sensitive results for halfspace range counting that ace of use in more general distance selection problems. None of our algorithms requires pacametric search.
13840611
Dynamic cluster formation using level set methods
Density-based clustering has the advantages for: 1) allowing arbitrary shape of cluster and 2) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But, when clusters become closed to each other, CIFs still clearly reveal cluster centers, cluster boundaries, and degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation, which are often oscillatory or oversmoothed. These problems of kernel density estimation are resolved using level set methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented which illustrate the advantages of our approach.
32811638
Timing of Software Replacement
Analogous to the replacement of an old machine, such as a car, replacing an aged softwitre may contain its escalating cost of maintenance. Prior research has assumed that increasing maintenance cost is due to the deterioration of the system maintainability. However, cost of maintenance depends also on the number of incoming maintenance requests. While software maintainability is determined by its complexity and development environment, number of maintenance requests is affected by the business environment. This distinction is significant in analyzing economic tradeoffs in software maintenance because replacement improves system maintainability but will not affect the number of maintenance requests. Unlike replacement of hardware, rewriting software takes an extended length of time, Thus, the old software must still be maintained before the new software is ready. We develop an economic model that considers the number of maintenance requests and the rewriting period explicitly. The model is an extension of Gode, Barua, and Mukhopadhyay (1990), which assumes a constant number of maintenance requests and instantaneous replacement. Our model allows us to draw some additional policy implications about software inaintenatice aild replacement. For instance, we show that in certain situations delaying a system replacement can be inore cost effective when the user environment changes more rapidly, contrary to our intuition. Moreover, it is shown that rewriting should begin earlier when the instantane(,us replacement assumption is relaxed.
45840953
Dynamic Cluster Formation Using Level Set Methods
Density-based clustering has the advantages for (i) allowing arbitrary shape of cluster and (ii) not requiring the number of clusters as input. However, when clusters touch each other, both the cluster centers and cluster boundaries (as the peaks and valleys of the density distribution) become fuzzy and difficult to determine. In higher dimension, the boundaries become wiggly and over-fitting often occurs. We introduce the notion of cluster intensity function (CIF) which captures the important characteristics of clusters. When clusters are well-separated, CIFs are similar to density functions. But as clusters touch each other, CIFs still clearly reveal cluster centers, cluster boundaries, and, degree of membership of each data point to the cluster that it belongs. Clustering through bump hunting and valley seeking based on these functions are more robust than that based on kernel density functions which are often oscillatory or over-smoothed. These problems of kernel density estimation are resolved using Level Set Methods and related techniques. Comparisons with two existing density-based methods, valley seeking and DBSCAN, are presented to illustrate the advantages of our approach.
2980605
A Simplified Approach to the Order Conditions of Integration Methods
We present an approach to the numerical integration of ordinary differential equations based on the algebraic theory of Butcher (Math. Comp. 26, 79–106, 1972) and the -series theory of Hairer and Wanner (Computing 13, 1–15, 1974). We clarify the differences of these two approaches by equating the elementary weight functions and showing the differences of the composition rules. By interpreting the elementary weight function as a mapping from input values to output values and introducing some special mappings, we are able to derive the order conditions of several types of integration methods in a straight-forward way. The simplicity of the derivation is illustrated by linear multistep methods that use the second derivative as an input value, Runge-Kutta type methods that use the second as well as first derivatives, and general two-step Runge-Kutta methods. We derive new high stage-order methods in each example. In particular, we found a symmetric and stiffly-accurate method of order eight in the second example.
20538253
Parallel Complexity of Domain Decomposition Methods and Optimal Coarse Grid Size
Abstract In this paper, we analyze the parallel complexity of domain decomposition method through estimating the arithmetic time as well as the communication time. In most domain decomposition (DD) methods, a coarse grid solve is employed to provide global coupling and obtain an optimal convergence rate. Generally, a small coarse grid size H improves the convergence rate at the cost of a larger coarse grid problem and a large number P of processors results in more costly global communication, whereas a large H and a small P have the opposite effect. Therefore, there often exists an optimal H such that the execution time on the parallel machine reaches a minimum. Our main goal here is to find this optimal coarse grid size H in the parallel environment. We assume the same solver is used for the subdomains as well as the coarse problem. By expressing the parallel complexity as a function of H and h, we can derive the analytic optimal coarse grid size Hopt. When the number of processors is equal to the number of subdomains, we can prove that the choice of Hopt = √h asymptotically minimises the execution time as h tends to zero. This optimal coarse grid size is independent of the choice of subproblem solvers and the spatial dimension. However, when the number of processors is much less than the number of subdomains, the optimal coarse grid size depends on the number of processors, the subdomain solver and dimension.