An inverse problem for matrix processing: an improved algorithm for restoring the distance matrix for DNA chains

Abstract

We consider one of the cybernetic methods in biology related to the study of DNA chains. Namely, we are considering the problem of reconstructing the distance matrix for DNA chains. Such a matrix is formed on the basis of any of the possible algorithms for determining the distances between DNA chains. The objects of research of these algorithms (for mammals), as a rule, are one of the following 3 variants: the main histocompatibility complex, the mitochondrial DNA, and “the tail” of the Y chromosome. In the paper we give an improved algorithm for restoring the distance matrix for DNA chains. Compared to our previous publications, the following changes have been made to the algorithm. We abandoned the use of the branches and bounds method, but at the same time significantly improved the greedy auxiliary algorithm used in it. In this paper, we apply only this greedy algorithm to the general solution of the distance matrix reconstruction problem. As a result of the conducted computational experiments carried out on one of the two considered criteria for the quality of the algorithms, significant improvements were obtained compared to the results given in our previous publications. At the same time, the total running time of the algorithm remained almost the same as in the previous version.

Some more on restoring distance matrices between DNA chains: reliability coefficients

“…The connection of some algorithms for studying DNA chains with various problems of experimental physics has been demonstrated in some our previous publications; to these publications it is worth adding [Sergeenko et al, 2020], which, among other things, examines the connection with Hamiltonian cycles, and, consequently, with the traveling salesman problem (TSP), also considered in some publications of the authors of this paper, see [Melnikov, Zhang, and Chaikovskii, 2022] and some others.…”

“…In the approach considered in this paper, we consider (in contrast to the one described in [Melnikov, Zhang, and Chaikovskii, 2022] the so-called "first direction" of the development of the work), that the algorithm of the initial distance formation itself (the Needleman -Wunsch algorithm is only one of the possible examples) works in the best way. Therefore, we conditionally believe that the corresponding value of badness cannot be improved (i.e., it is impossible to get something better than it is throughout the matrix, knowing only a certain number of its elements).…”

“…In both the cases, the dimensions we can process should be of the order 7 − 8 only. In the full versions, however, we need to consider problems of dimension more than 100 for classical TSP (as a rule, the authors of this paper do not consider the geometric TSP, for which acceptable pseudo-optimal solutions for millions points have been found for a long time) and more than 500 for the restoring problem (see also [Melnikov, Zhang, and Chaikovskii, 2022]). Therefore, the main "theoretical aspect" of the presented paper is that we can do everything "step-by-step", i.e., if we consistently place the points, then in the limit we shall get the optimal solution to the problem.…”

“…Thus, the options for recovery algorithms were given in our previous papers [Melnikov, Pivneva, and Trifonov, 2017;Melnikov, Zhang, and Chaikovskii, 2022], see also the links from there. In [Melnikov, Zhang, and Chaikovskii, 2022], several directions of further research were given, which should lead to improved recovery algorithms, i.e., to the formation of such algorithms that would be performed in an acceptable time and at the same time would give a lower value of badness. In the current paper, we consider only one of these possible directions, that is, the considering so-called reliability coefficients.…”

Pseudogeometric version of the traveling salesman problem: application in quantum physics models and a heuristic variant of point placement

“…The approach to solve the pseudo-geometric variant of the traveling salesman problem, as discussed in this article aligns closely with the technique applied to the problem of reconstructing the distance matrix between DNA chains. Both problems utilize a pseudo-optimal configuration, as mentioned in [Melnikov et al, 2018;Melnikov et al, 2022]. Delving into more detail, both problems involve the placement of all absent elements (either zero or equivalent to infinity) into the depleted matrix according to certain rules.…”