5208277
A Space-Efficient Algorithm for Segment Intersection
We examine the space requirement for the classic linesegment intersection problem. Using so-called implicit data structures, we show how to make the standard sweep-line algorithm run in O((n + k) log 2 n) time with only O(log 2 n) extra space, where n is the number of line segments and k is the number of intersections. If division is allowed and input can be destroyed, the algorithm can run in O((n + k) log n) time with O(1) extra space.
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.
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).
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.
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.
7475027
Metric embeddings with relaxed guarantees
We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler (2004), we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into /spl lscr//sub 1/ which exhibit gracefully degrading distortion: these is a single embedding into /spl lscr//sub 1/ that achieves distortion at most O(log 1//spl epsi/) on all but at most an /spl epsi/ fraction of distances, simultaneously for all /spl epsi/ > 0. We extend this with distortion O(log 1//spl epsi/)/sup 1/p/ to maps into general /spl lscr//sub p/, p /spl ges/ 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for /spl epsi/-slack embeddings from lower bounds for low-distortion embeddings.
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.
18182246
Total variation blind deconvolution
In this paper, we present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed. The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images as well as some blurring functions, e.g., motion blur and out-of-focus blur. An alternating minimization (AM)implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (psf). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur), and both the image and the psf can be recovered under the presence of high noise level. Finally, we remark that psf's without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.
7832316
Cosine transform based preconditioners for total variation deblurring
In PDE image restoration problems, one has to invert operators which is a sum of a blurring operator and an elliptic operator with highly varying coefficient. We present a preconditioner for such operators, which can be used with the conjugate gradient (CG) method, and compare it with Vogel and Oman's product preconditioner.