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

Tomás M Coronado, Arnau Mir and Francesc Rosselló

PLOS ONE, 2022, vol. 17, issue 11, 1-37

Abstract: Divide-and-conquer dividing by a half recurrences, of the form x n = a · x ⌈ n / 2 ⌉ + a · x ⌊ n / 2 ⌋ + p ( n ) , n ⩾ 2 , 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.

Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pone00:0274448

DOI: 10.1371/journal.pone.0274448

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