9903009
A note on maximum independent sets in rectangle intersection graphs
Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209–218] gave a (1+1/k)-factor algorithm with an O(nlogn+n2k−1) time bound for any integer constant k⩾1; we describe a similar algorithm running in only O(nlogn+nΔk−1) time, where Δ⩽n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443–470] gave a ⌈logkn⌉-factor algorithm with an O(nk+1) time bound for any integer constant k⩾2; we describe similar algorithms running in O(nlogn+nΔk−2) and nO(k/logk) time.
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.
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.
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
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).
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.
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.
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.
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.
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.