On the Separability of Stochastic Geometric Objects, with Applications

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min{|S_R|, |S_B|} and N = max{|S_R|, |S_B|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d-1}) time for d >= 3 and O(min{nN log N, N^2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^{d-1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems

Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design

AbstractA parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria

New Bounds for Range Closest-Pair Problems

Given a dataset S of points in R^2, the range closest-pair (RCP) problem aims to preprocess S into a data structure such that when a query range X is specified, the closest-pair in S cap X can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new

Searching for the Closest-Pair in a Query Translate

We consider a range-search variant of the closest-pair problem. Let Gamma be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Gamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Gamma is a polygon (possibly with holes) and when Gamma is a general convex body whose boundary is smooth. When Gamma is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Gamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log^2 n) query time. Our results settle some open questions posed by Xue et al. at SoCG 2018

Multiple structure alignment and consensus identification for proteins

<p>Abstract</p> <p>Background</p> <p>An algorithm is presented to compute a multiple structure alignment for a set of proteins and to generate a consensus (pseudo) protein which captures common substructures present in the given proteins. The algorithm represents each protein as a sequence of triples of coordinates of the alpha-carbon atoms along the backbone. It then computes iteratively a sequence of transformation matrices (i.e., translations and rotations) to align the proteins in space and generate the consensus. The algorithm is a heuristic in that it computes an approximation to the optimal alignment that minimizes the sum of the pairwise distances between the consensus and the transformed proteins.</p> <p>Results</p> <p>Experimental results show that the algorithm converges quite rapidly and generates consensus structures that are visually similar to the input proteins. A comparison with other coordinate-based alignment algorithms (MAMMOTH and MATT) shows that the proposed algorithm is competitive in terms of speed and the sizes of the conserved regions discovered in an extensive benchmark dataset derived from the HOMSTRAD and SABmark databases.</p> <p>The algorithm has been implemented in C++ and can be downloaded from the project's web page. Alternatively, the algorithm can be used via a web server which makes it possible to align protein structures by uploading files from local disk or by downloading protein data from the RCSB Protein Data Bank.</p> <p>Conclusions</p> <p>An algorithm is presented to compute a multiple structure alignment for a set of proteins, together with their consensus structure. Experimental results show its effectiveness in terms of the quality of the alignment and computational cost.</p

Space-efficient schemes for message routing in distributed networks

A primary function in a distributed network is to route messages between pairs of nodes. Often, a cost is associated with each edge, making it desirable to route along shortest paths. Although this can be accomplished easily by storing a complete routing table at each node, this approach is expensive, using $\Theta$(n\sp2) items of routing information for an n-node network. The focus of this research is to identify classes of networks for which considerably less routing information can be maintained, given the freedom to name nodes suitably. Node naming and message routing schemes are given for a number of network classes, including outerplanar networks, c-decomposable networks, for c a constant, and planar networks. These schemes use considerably less space than complete routing tables, keep node names to O(log n) bits, and still route along either shortest or near-shortest paths. The scheme for outerplanar networks uses $\Theta$(n) items and routes along shortest paths. The key idea here is a naming of the nodes with integers between 1 and n in a way that allows shortest paths information at every node to be encoded succinctly as a subinterval of (1,n) labeling each incident edge. Outerplanar networks are shown to be precisely the networks possessing this interval property. For c-decomposable networks, O(cn log n) items are stored, and, in worst case, any routing is at most 2/$\alpha$ + 1 times longer than a shortest routing, where $\alpha$ $\u3e$ 1 is a certain function of c. A novel approach is used where information about relative magnitudes of distances is compactly encoded within the node names. The scheme for planar networks uses O(n\sp{1+\epsilon}) items and O((1/$\epsilon$) log n)-bit names, for any constant $\epsilon$, 0 $\u3c$ $\epsilon$ $\u3c$ 1/3. A worst-case bound of 7 on the routings is achieved. To obtain this result, the network is decomposed using structured cycle-separators and a powerful multi-interval graph labeling technique is applied to succinctly encode routing information. The issue of fault-tolerance is also addressed. For outerplanar networks, a scheme is given which, in the presence of t node and edge faults, uses O(t$\alpha$n) items to route with a worst case bound of (($\alpha$ + 1)/($\alpha$-1))\sp{\rm t}, where $\alpha$ $\u3e$ 1 is an odd-valued integer parameter