42016450
Geometric permutations of higher dimensional spheres
In this paper, we prove that the maximum number of geometric permutations (induced by line transversals) of a set of n pairwise disjoint spheres with a bounded radius ratio in Rd for d ≥ 3 is at most 2⌊√2M⌋+1, where M is the ratio of the largest radius and the smallest radius of the spheres. Setting M to 1, this gives an upper bound of 4 on the maximum number of geometric permutations for congruent spheres in Rd, matching a recently independently discovered result [Y. Zhou, S. Suri, in: Proc. of 12th Annual ACM-SIAM Symp. on Discrete Algorithms, 2001, pp. 234-243] on this case. Our result settles a conjecture in combinatorial geometry.
28240465
Motion Tracking and Intensity Surface Recovery in Microscopic Nuclear Images
Current techniques for microscopic imaging do not provide necessary spatial and temporal resolutions for real time visualization of the nucleus. Images can only be acquired in time lapse mode, leading to significant loss of information between image frames. Such data, if available, can be extremely helpful in the study of nuclear organization and function. In this paper, we present a gamut of geometric-technique-based approaches for solving the problem. Our techniques, working together, can effectively recover complicated motion and deformation as well as the change of intensity surfaces from pairs of images in a microscopic image sequence, and has low time complexity, particularly desirable by many biological applications where large amount of DNA need to be processed. These techniques are also readily applicable to other types of images for reconstructing motion and intensity surfaces of deformable objects.
21098955
Topological Peeling and Implementation
We present a new approach, called topological peeling, and its implementation for traversing a portion AR of the arrangement formed by n lines within a convex region R on the plane. Topological peeling visits the cells of AR in a fashion of propagating a "wave" of a special shape (called a double-wriggle curve) starting at a single source point. This special traversal fashion enables us to solve several problems (e.g., computing shortest paths) on planar arrangements to which previously best known arrangement traversal techniques such as topological sweep and topological walkm ay not be directly applicable. Our topological peeling algorithm takes O(K + n log(n + r)) time and O(n + r) space, where K is the number of cells in AR and r is the number of boundary vertices of R. Comparing with topological walk, topological peeling uses a simpler and more efficient way to sweep different types of lines, and relies heavily on exploring small local structures, rather than a much larger global structure. Experiments show that, on average, topological peeling outperforms topological walkb y 10 - 15% in execution time.
6260724
Finding an optimal path without growing the tree
For problems on computing an optimal path as well as its length in a certain setting, the "standard" approach for finding an actual optimal path is by building (or "growing") a single-source optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of single-source optimal paths for these problems is asymptotically smaller than the space complexity of the "standard" tree-growing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a single-source optimal path tree. Our paradigm is based on the "marriage-before-conquer" strategy, the prune-and-search technique, and a new data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic solutions improve the space bounds (in certain cases, the time bounds as well) of the previously best known algorithms, and settle some open problems. Our techniques are likely to be applicable to other problems.
14297439
Geometric permutations of high dimensional spheres
We prove the maximum number of geometric permutations, induced by line transversals to a set of <i>n</i> pairwise disjoint congruent spheres in <i>R<sup>d</sup></i> with <i>d</i> ⪈ 3, is no more than 4 when <i>n</i> is sufficiently large, achieving the best known upper bound for this problem. We also prove the maximum number of geometric permutations of a set of <i>n</i> noncongruent spheres of bounded radius ratio in R<sup>d</sup>, <i>d</i> ⪈ 3, is at most 2<sup>[√2<i>M</i>]+1</sup>, where <i>M</i> is the ratio or the largest radius and the smallest radius. Our result settles a conjecture in combinatorial geometry.
29728094
Graph Bandwidth of Weighted Caterpillars
Graph bandwidth minimization (GBM) is a classical and challenging problem in graph algorithms and combinatorial optimization. Most of existing researches on this problem have focused on unweighted graphs. In this paper, we study the bandwidth minimization problem of weighted caterpillars, and propose several algorithms for solving various types of caterpillars and general graphs. More specifically, we show that the GBM problem on caterpillars with hair-length at most 2 and the GBM problem on star-shape caterpillars are NP-complete, and give a lower bound of the graph bandwidth for general weighted graphs. For caterpillars with hair-length at most 1, we present an O(n log n log(nwmax))-time algorithm to compute an optimal bandwidth layout, where n is the total number of vertices in the graph and wmax is the maximum wedge weight. For caterpillars with hair-length at most k, we give a k-approximation algorithm. For arbitrary caterpillars and general graphs, we give a heuristic algorithm and some experimental results. Experiments show that the solutions obtained by our heuristic algorithm are roughly within a factor of c log(n) of the lower bound for a small number c, which is consistent with the inapproximability results of this problem (i.e., no constant approximation for the GBM problem unless P = NP). © 2006 Published by Elsevier B.V.
17229097
Distance irredundance and connected domination numbers of a graph
Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number @c"k(G), the connected k-domination number @c"k^c(G); the k-independent domination number @c"k^i(G) and the k-irredundance number ir"k(G). The authors prove that if an ir"k-set X is a k-independent set of G, then ir"k(G)[email protected]"k(G)[email protected]"k^i(G), and that for k>=2, @c"k^c(G)=1 if ir"k(G)=1, @c"k^c(G)=
15417765
The proof of a conjecture of Bouabdallah and Sotteau
Let G be a connected graph of order n. A routing in G is a set of n(n 1) fixed paths for all ordered pairs of vertices of G. The edge-forwarding index of G, (G), is the minimum of the maximum number of paths specified by a routing passing through any edge of G taken over all routings in G, and ,n is the minimum of (G) taken over all graphs of order n with maximum degree at most . To determine n 2p 1,n for 4p 2p/3 1 ≤ n ≤ 6p, A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), 249–255]. The set 3 {1, 2, . . . , (4p)/3} can be partitioned into 2p pairs plus singletons such that the set of differences of the pairs is the set 2 {1, 2, . . . , p}. This article gives a proof of this conjecture and determines that n 2p 1,n is equal to 5 if 4p 2p/3 1 ≤ n ≤ 6p and to 8 if 3p p/3 1 ≤ n ≤ 3p (3p)/5 for any p ≥ 2. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 292–296 2004
240982
Data streaming algorithms for estimating entropy of network traffic
Using entropy of traffic distributions has been shown to aid a wide variety of network monitoring applications such as anomaly detection, clustering to reveal interesting patterns, and traffic classification. However, realizing this potential benefit in practice requires accurate algorithms that can operate on high-speed links, with low CPU and memory requirements. In this paper, we investigate the problem of estimating the entropy in a streaming computation model. We give lower bounds for this problem, showing that neither approximation nor randomization alone will let us compute the entropy efficiently. We present two algorithms for randomly approximating the entropy in a time and space efficient manner, applicable for use on very high speed (greater than OC-48) links. The first algorithm for entropy estimation is inspired by the structural similarity with the seminal work of Alon et al. for estimating frequency moments, and we provide strong theoretical guarantees on the error and resource usage. Our second algorithm utilizes the observation that the performance of the streaming algorithm can be enhanced by separating the high-frequency items (or elephants) from the low-frequency items (or mice). We evaluate our algorithms on traffic traces from different deployment scenarios.
12535990
Fast integer-pel and fractional-pel motion estimation for H.264/AVC
Abstract This paper gives out a fast motion estimation algorithm for H.264/AVC, which has almost the same quality performance as that of the Full Search scheme and also provides a strategy to balance the quality performance and the search speed. Integer-pel search is the most time consuming module for motion estimation. In this paper a hybrid Unsymmetrical-cross Multi-hexagon-grid Search (UMHexagonS) algorithm is introduced, which well solves the false motion vector estimation problem because of the local-minimum. As results, it saves more than 90% on search time whereas the averaging PSNR loss is less than 0.056 dB for all tested sequences with different motion extent compared with the Fast Full Search scheme. Fractional-pel fast search is not negligible when the integer motion estimation has been extensively speeded up. By utilizing the property of uni-modal error surface inside the fractional-pel search window, a novel Center Biased Fractional-pel Search (CBFPS) algorithm is proposed in this paper, which can save 30–50% computation compared with the Full Fractional-pel Search scheme. Early termination is another problem discussed in this paper, to construct a complete motion estimation method, giving a uniform method which can cover a wide range of applications, terminating the program at early and right stage is certainly required. A model based on the Normative SAD Difference (NSD) is given to assist the termination decision-making, and a tradeoff between the search speed and the reconstructed quality can be achieved by changing a modulation factor.
10735574
Forwarding indices of folded n-cubes
For a given connected graph G of order n, a routing R is a set of n(n - 1) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of a graph is the maximum number of paths of R passing through any vertex (resp. edge) in the graph. In this paper, the authors determine the vertex and the edge forwarding indices of a folded n-cube as (n-1)2n-1 +1-((n+l)/2) (n ⌊n+1/2⌋) and 2n-(n,⌊n+1/2⌋),respectively.