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Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

ABSTRACT: Divide-and-conquer dividing by a half recurrences, of the form [Formula: see text] appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quartet index of the maximally balanced bifurcating phylogenetic trees with n leaves, and the sum of the bitwise AND operator applied to pairs of complementary numbers up to n.

SUBMITTER: Coronado TM

PROVIDER: S-EPMC9671444 | biostudies-literature | 2022

REPOSITORIES: biostudies-literature

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Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

Coronado Tomás M TM Mir Arnau A Rosselló Francesc F

PloS one 20221117 11

Divide-and-conquer dividing by a half recurrences, of the form [Formula: see text] appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the inde ...[more]

PMID: 36395273

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Project description:Among proteins, orthologs are defined as those that are derived by vertical descent from a single progenitor in the last common ancestor of their host organisms. Our goal is to compute a complete set of protein orthologs derived from all currently available complete bacterial and archaeal genomes. Traditional approaches typically rely on all-against-all BLAST searching which is prohibitively expensive in terms of hardware requirements or computational time (requiring an estimated 18 months or more on a typical server). Here, we present xBASE-Orth, a system for ongoing ortholog annotation, which applies a "divide and conquer" approach and adopts a pragmatic scheme that trades accuracy for speed. Starting at species level, xBASE-Orth carefully constructs and uses pan-genomes as proxies for the full collections of coding sequences at each level as it progressively climbs the taxonomic tree using the previously computed data. This leads to a significant decrease in the number of alignments that need to be performed, which translates into faster computation, making ortholog computation possible on a global scale. Using xBASE-Orth, we analyzed an NCBI collection of 1,288 bacterial and 94 archaeal complete genomes with more than 4 million coding sequences in 5 weeks and predicted more than 700 million ortholog pairs, clustered in 175,531 orthologous groups. We have also identified sets of highly conserved bacterial and archaeal orthologs and in so doing have highlighted anomalies in genome annotation and in the proposed composition of the minimal bacterial genome. In summary, our approach allows for scalable and efficient computation of the bacterial and archaeal ortholog annotations. In addition, due to its hierarchical nature, it is suitable for incorporating novel complete genomes and alternative genome annotations. The computed ortholog data and a continuously evolving set of applications based on it are integrated in the xBASE database, available at http://www.xbase.ac.uk/.

Dataset Information

Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

Publications

Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

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