9771502
Geometric Optimization Problems Over Sliding Windows
We study the problem of maintaining a (1+e)-factor approximation of the diameter of a stream of points under the sliding window model In one dimension, we give a simple algorithm that only needs to store $O(\frac{1}{\epsilon}{\rm log}R)$ points at any time, where the parameter R denotes the “spread” of the point set This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space Related optimization problems, such as the width, are also considered in the two-dimensional case.
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.
15785799
Semi-online maintenance of geometric optima and measures
We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.
18265362
Approximation Algorithms for Maximum Cliques in 3D Unit-Disk Graphs
15997426
On levels in arrangements of curves
Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk/sup 1-2/3/*)) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s=1 and s=2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk/sup 7/9/log/sup 2/3/ k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.
15654193
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 <i>n</i> points in three dimensions, where insertions take <i>O</i>(log<sup>3</sup> <i>n</i>) expected amortized time, deletions take <i>O</i>(log<sup>6</sup> <i>n</i>) expected amortized time, and extreme-point queries take <i>O</i>(log<sup>2</sup> <i>n</i>) 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 <i>O</i>(<i>n</i><sup>ε</sup>)-time method by Agarwal and Matoušek 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).
6323270
Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs
A conditionally Gaussian channel is a vector channel in which the channel output, given the channel input, has a Gaussian distribution with (well-behaved) input-dependent mean and covariance. We study the capacity-achieving probability measure for conditionally Gaussian channels subject to bounded-input constraints and average cost constraints. Many practical communication systems, including additive Gaussian noise channels, certain optical channels, fading channels, and interference channels fall within this framework. Subject to bounded-input constraint (and average cost constraints), we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving. Under certain conditions, the capacity-achieving measure is proved to be discrete.
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.
20186109
Balanced information inequalities
In this correspondence, we are interested in linear information inequalities, both discrete and continuous ones. We show that every discrete information inequality is associated with a "balanced" information inequality and a set of "residual weights." To prove the inequality, it is necessary and sufficient to prove that its "balanced" version is valid and all its residual weights are nonnegative. For a continuous information inequality, we prove that it is valid if and only if its discrete counterpart is balanced and valid.
15943063
Enterprise Systems Success: A Measurement Model
This paper presents a validated measurement model and instrument for assessing enterprise systems success from multiple perspectives. The final validated study model employs 27 measures of the four dimensions: information quality, system quality, individual impact, and organizational impact. The model is empirically tested with survey data gathered from 27 public sector organizations that implemented SAP R/3 in the late 1990s. The study consists of an exploratory inventory survey (model building) to identify the salient success dimensions and measures, followed by a confirmatory weights survey, for testing model validity (model testing). Test results demonstrate the discriminant validity of the four dimensions, as well as their convergence on a single higher-order phenomenon: enterprise systems success (ESS). Criterion validity testing further demonstrates the additivity of the four dimensions of success, and the completeness of the resultant overarching, second-order measure of ESS.
16750516
Spanners with Slack
Given a metric (V,d), a spanner is a sparse graph whose shortest-path metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distances, but are then required to incur only a constant (independent of n) distortion on the remaining distances? We answer this question in the affirmative, thus complementing similar recent results on embeddings with slack into l p spaces. For instance, we show that if we ignore an e fraction of the distances, we can get spanners with O(n) edges and O(log 1 ∈) distortion for the remaining distances. We also show how to obtain sparse and low-weight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces.