A multifacility location problem on median spaces

AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem

DisLoc: A Convex Partitioning Based Approach for Distributed 3-D Localization in Wireless Sensor Networks

Accurate localization in wireless sensor networks (WSNs) is fundamental to many applications, such as geographic routing and position-aware data processing. This, however, is challenging in large scale 3-D WSNs due to the irregular topology, such as holes in the path, of the network. The irregular topology may cause overestimated Euclidean distance between nodes as the communication path is bent and accordingly introduces severe errors in 3-D WSN localization. As an effort towards the issue, this paper develops a distributed algorithm to achieve accurate 3-D WSN localization. Our proposal is composed of two steps, segmentation and joint localization. In specific, the entire network is first divided into several subnetworks by applying the approximate convex partitioning. A spatial convex node recognition mechanism is developed to assist the network segmentation, which relies on the connectivity information only. After that, each subnetwork is accurately localized by using the multidimensional scaling-based algorithm. The proposed localization algorithm also applies a new 3-D coordinate transformation algorithm, which helps reduce the errors introduced by coordinate integration between subnetworks and improve the localization accuracy. Using extensive simulations, we show that our proposal can effectively segment a complex 3-D sensor network and significantly improve the localization rate in comparison with existing solutions

An efficient output-sensitive hidden surface removal algorithm and its parallelization

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running time is sensitive to the actual size of the visible image rather than the total number of intersections in the image plane which can be much larger than the visible image. The time complexity of this algorithm is O((k +nflognloglogn) where n and k are respectively the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time Ω(n 2) irrespective of the output size (where as the output size k is O(n 2) only in the worst case). We also present a parallel algorithm based on a similar approach which runs in time O(log4(n+k)) using O((n + k)/Iog(n+k)) processors in a CREW PRAM model. All our bounds arc obtained using ammortized analysis

Spherical distance metrics applied to protein structure classification

Observed protein structures usually represent energetically favorable conformations. While not all observed structures are necessarily functional, it is generally agreed that protein structure is closely related to protein function. Given a collection of proteins sharing a common global structure, variations in their local structures at specific, critical locations may result in different biological functions. Structural relationships among proteins are important in the study of the evolution of proteins as well as in drug design and development. Analysis of geometrical 3D protein structure has been shown to be effective with respect to classifying proteins. Prior work has shown that the Double Centroid Reduced Representation (DCRR) model is a useful geometric representation for protein structure with respect to visual models, reducing the quantity of modeled information for each amino acid, yet retaining the most important geometrical and chemical features of each: the centroids of the backbone and of the side-chain. DCRR has not yet been applied in the calculation of geometric structural similarity. Meanwhile, multi-dimensional indexing (MDI) of protein structure combines protein structural analysis with distance metrics to facilitate structural similarity queries and is also used for clustering protein structures into related groups. In this respect, the combination of geometric models with MDI has been shown to be effective. Prior work, notably Distance and Density-based Protein Indexing (DDPIn), applies MDI to protein models based on the geometry of the C-alpha backbone. DDPIn\u27s distance metrics are based on radial and density functions that incorporate spherical-based metrics, and the indices are built from metric-tree (M-tree) structures. This work combines DCRR with DDPIn for the development of new DCRR centroid-based metrics: spherical binning distance and inter-centroid spherical distance. The use of DCRR models will provide additional significant structural information via the inclusion of side-chain centroids. Additionally, the newly developed distance metric functions combined with DCRR and M-tree indexing attempt to improve upon the performance of prior work (DDPIn), given the same data set, with respect to both individual k-nearest neighbor (kNN) search queries as well as clustering all proteins in the index