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

TL;DR

This paper derives an explicit solution for a class of divide-and-conquer recurrences with polynomial independent terms, expanding beyond the typical asymptotic analysis to provide exact formulas based on binary decomposition.

Contribution

It introduces a method to explicitly solve divide-and-conquer recurrences with polynomial terms, which were previously only understood through asymptotic bounds.

Findings

Provides explicit formulas for solutions based on binary decomposition

Extends analysis of divide-and-conquer recurrences beyond big-Theta bounds

Applicable to various areas including algorithms and phylogenetics

Abstract

Divide-and-conquer dividing by a half recurrences, of the form $x_n =a\cdot x_{\left\lceil{n}/{2}\right\rceil}+a\cdot x_{\left\lfloor{n}/{2}\right\rfloor}+p(n)$, $n\geq 2$, appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. The Master Theorems that solve these equations do not provide the solution's explicit expression, only its big-$\Theta$ order of growth. In this paper we give an explicit expression (in terms of the binary decomposition of $n$) for the solution $x_n$ of a recurrence of this form, with given initial condition $x_1$, when the independent term $p(n)$ is a polynomial in $\lceil{n}/{2}\rceil$ and $\lfloor{n}/{2}\rfloor$.