The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph can be realized in a given space (generally Euclidean) so that a given set of distance constraints (associated to the edges of the graph) is satisfied. The Discretizable DGP (DDGP) represents a subclass of instances where the search space can be reduced to a discrete domain having the structure of a tree. In the ideal case where all distances are precise, the tree is binary and one singleton, representing one possible position for a vertex of the graph, is associated to every tree node. When the distance information is however not precise, the uncertainty on the distance values implies that a three-dimensional region of the search space needs to be assigned to some nodes of the tree. By using a recently proposed coarse-grained representation for DDGP solutions, we extend in this work the branch-and-prune (BP) algorithm so that it can efficiently perform an exhaustive search of the search domain, even when the uncertainty on the distances is important. Instead of associating singletons to nodes, we consider a pair consisting of a box and of a most-likely position for the vertex in this box. Initial estimations of the vertex positions in every box can be subsequently refined by using local optimization. The aim of this paper is twofold: (i) we propose a new simple method for the computation of the three-dimensional boxes to be associated to the nodes of the search tree; (ii) we introduce the resolution parameter ρ, with the aim of controling the similarity between pairs of solutions in the solution set. Some initial computational experiments show that our algorithm extension, differently from previously proposed variants of the BP algorithm, is actually able to terminate the enumeration of the solution set by providing solutions that differ from one another accordingly to the given resolution parameter.
With the most recent releases of MD-JEEP, new relevant features have been included to our software tool. MD-JEEP solves instances of the class of Discretizable Distance Geometry Problems (DDGPs), which ask to find possible realizations, in a Euclidean space, of a simple weighted undirected graph for which distance constraints between vertices are given, and for which a discretization of the search space can be supplied. Since its version 0.3.0, MD-JEEP is able to deal with instances containing interval data. We focus in this short paper on the most recent release MD-JEEP 0.3.2: among the new implemented features, we will focus our attention on three features: (i) an improved procedure for the generation and update of the boxes used in the coarse-grained representation (necessary to deal with instances containing interval data); (ii) a new procedure for the selection of the so-called discretization vertices (necessary to perform the discretization of the search space); (iii) the implementation of a general parser which allows the user to easily load DDGP instances in a given specified format. The source code of MD-JEEP 0.3.2 is available on GitHub, where the reader can find all additional details about the implementation of such new features, as well as verify the effectiveness of such features by comparing MD-JEEP 0.3.2 with its previous releases.
Abstract-Given a set of points in a Euclidean space having dimension K > 0, we are interested in the problem of finding a realization of the same set in a Euclidean space having a lower dimension. In most situations, it is not possible to preserve all available interpoint distances in the new space, so that the best possible realization, which gives the minimal error on the distances, needs to be searched. This problem is known in the scientific literature as the Multidimensional Scaling (MDS). We propose a new methodology to discretize the search space of MDS instances, with the aim of performing an efficient enumeration of their solution sets. Some preliminary computational experiments on a set of artificially generated instances are presented. We conclude our paper with some future research directions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.