Penalty-Based Aggregation of Strings

International Summer School on Aggregation Operators (2019. Olomouc, Czech Republic

Pivot Selection for Median String Problem

The Median String Problem is W[1]-Hard under the Levenshtein distance, thus, approximation heuristics are used. Perturbation-based heuristics have been proved to be very competitive as regards the ratio approximation accuracy/convergence speed. However, the computational burden increase with the size of the set. In this paper, we explore the idea of reducing the size of the problem by selecting a subset of representative elements, i.e. pivots, that are used to compute the approximate median instead of the whole set. We aim to reduce the computation time through a reduction of the problem size while achieving similar approximation accuracy. We explain how we find those pivots and how to compute the median string from them. Results on commonly used test data suggest that our approach can reduce the computational requirements (measured in computed edit distances) by $8$\% with approximation accuracy as good as the state of the art heuristic. This work has been supported in part by CONICYT-PCHA/Doctorado Nacional/$2014-63140074$ through a Ph.D. Scholarship; Universidad Cat\'{o}lica de la Sant\'{i}sima Concepci\'{o}n through the research project DIN-01/2016; European Union's Horizon 2020 under the Marie Sk\l odowska-Curie grant agreement $690941$; Millennium Institute for Foundational Research on Data (IMFD); FONDECYT-CONICYT grant number $1170497$; and for O. Pedreira, Xunta de Galicia/FEDER-UE refs. CSI ED431G/01 and GRC: ED431C 2017/58

A new iterative algorithm for computing a quality approximate median of strings based on edit operations

This paper presents a new algorithm that can be used to compute an approximation to the median of a set of strings. The approximate median is obtained through the successive improvements of a partial solution. The edit distance from the partial solution to all the strings in the set is computed in each iteration, thus accounting for the frequency of each of the edit operations in all the positions of the approximate median. A goodness index for edit operations is later computed by multiplying their frequency by the cost. Each operation is tested, starting from that with the highest index, in order to verify whether applying it to the partial solution leads to an improvement. If successful, a new iteration begins from the new approximate median. The algorithm finishes when all the operations have been examined without a better solution being found. Comparative experiments involving Freeman chain codes encoding 2D shapes and the Copenhagen chromosome database show that the quality of the approximate median string is similar to benchmark approaches but achieves a much faster convergence.This work is partially supported by the Spanish CICYT under project DPI2006-15542-C04-01, the Spanish MICINN through project TIN2009-14205-CO4-01 and by the Spanish research program Consolider Ingenio 2010: MIPRCV (CSD2007-00018)

Boosting Perturbation-Based Iterative Algorithms to Compute the Median String

[Abstract] The most competitive heuristics for calculating the median string are those that use perturbation-based iterative algorithms. Given the complexity of this problem, which under many formulations is NP-hard, the computational cost involved in the exact solution is not affordable. In this work, the heuristic algorithms that solve this problem are addressed, emphasizing its initialization and the policy to order possible editing operations. Both factors have a significant weight in the solution of this problem. Initial string selection influences the algorithm’s speed of convergence, as does the criterion chosen to select the modification to be made in each iteration of the algorithm. To obtain the initial string, we use the median of a subset of the original dataset; to obtain this subset, we employ the Half Space Proximal (HSP) test to the median of the dataset. This test provides sufficient diversity within the members of the subset while at the same time fulfilling the centrality criterion. Similarly, we provide an analysis of the stop condition of the algorithm, improving its performance without substantially damaging the quality of the solution. To analyze the results of our experiments, we computed the execution time of each proposed modification of the algorithms, the number of computed editing distances, and the quality of the solution obtained. With these experiments, we empirically validated our proposal.This work was supported in part by the Comisión Nacional de Investigación Científica y Tecnológica - Programa de Formación de Capital Humano Avanzado (CONICYT-PCHA)/Doctorado Nacional/2014-63140074 through the Ph.D. Scholarship, in part by the European Union's Horizon 2020 under the Marie Sklodowska-Curie under Grant 690941, in part by the Millennium Institute for Foundational Research on Data (IMFD), and in part by the FONDECYT-CONICYT under Grant 1170497. The work of ÓSCAR PEDREIRA was supported in part by the Xunta de Galicia/FEDER-UE refs under Grant CSI ED431G/01 and Grant GRC: ED431C 2017/58, in part by the Office of the Vice President for Research and Postgraduate Studies of the Universidad Católica de Temuco, VIPUCT Project 2020EM-PS-08, and in part by the FEQUIP 2019-INRN-03 of the Universidad Católica de TemucoXunta de Galicia; ED431G/01Xunta de Galicia; ED431C 2017/58Chile. Comisión Nacional de Investigación Científica y Tecnológica; 2014-63140074Chile. Comisión Nacional de Investigación Científica y Tecnológica; 1170497Universidad Católica de Temuco (Chile); 2020EM-PS-08Universidad Católica de Temuco (Chile); 2019-INRN-0

The Fine-Grained Complexity of Median and Center String Problems Under Edit Distance

We present the first fine-grained complexity results on two classic problems on strings. The first one is the k-Median-Edit-Distance problem, where the input is a collection of k strings, each of length at most n, and the task is to find a new string that minimizes the sum of the edit distances from itself to all other strings in the input. Arising frequently in computational biology, this problem provides an important generalization of edit distance to multiple strings and is similar to the multiple sequence alignment problem in bioinformatics. We demonstrate that for any ? > 0 and k ? 2, an O(n^{k-?}) time solution for the k-Median-Edit-Distance problem over an alphabet of size O(k) refutes the Strong Exponential Time Hypothesis (SETH). This provides the first matching conditional lower bound for the O(n^k) time algorithm established in 1975 by Sankoff. The second problem we study is the k-Center-Edit-Distance problem. Here also, the input is a collection of k strings, each of length at most n. The task is to find a new string that minimizes the maximum edit distance from itself to any other string in the input. We prove that the same conditional lower bound as before holds. Our results also imply new conditional lower bounds for the k-Tree-Alignment and the k-Bottleneck-Tree-Alignment problems studied in phylogenetics