Multidimensional Divide-and-Conquer and Weighted Digital Sums

TL;DR

This paper provides exact asymptotic solutions for functions from multidimensional divide-and-conquer algorithms and digital sums, revealing periodic fluctuations characterized by Fourier series.

Contribution

It introduces a unified Mellin transform approach to analyze three types of functions, deriving explicit asymptotic forms with periodic components.

Findings

MDC functions have solutions involving polynomial and periodic terms.

Weighted digital sums' averages are expressed with logarithmic and periodic functions.

Results include explicit formulas with Fourier series for the asymptotic behavior.

Abstract

This paper studies three types of functions arising separately in the analysis of algorithms that we analyze exactly using similar Mellin transform techniques. The first is the solution to a Multidimensional Divide-and-Conquer (MDC) recurrence that arises when solving problems on points in $d$-dimensional space. The second involves weighted digital sums. Write $n$ in its binary representation $n=(b_i b_{i-1}... b_1 b_0)_2$ and set $S_M(n) = \sum_{t=0}^i t^{\bar{M}} b_t 2^t$. We analyze the average $TS_M(n) = \frac{1}{n}\sum_{j<n} S_M(j)$. The third is a different variant of weighted digital sums. Write $n$ as $n=2^{i_1} + 2^{i_2} + ... + 2^{i_k}$ with $i_1 > i_2 > ... > i_k\geq 0$ and set $W_M(n) = \sum_{t=1}^k t^M 2^{i_t}$. We analyze the average $TW_M(n) = \frac{1}{n}\sum_{j<n} W_M(j)$. We show that both the MDC functions and $TS_M(n)$ (with $d=M+1$) have solutions of the form $\lambda_d n \lg^{d-1}n + \sum_{m=0}^{d-2}(n\lg^m n)A_{d,m}(\lg n) + c_d,$ where $\lambda_d,c_d$ are constants and $A_{d,m}(u)$'s are periodic functions with period one (given by absolutely convergent Fourier series). We also show that $TW_M(n)$ has a solution of the form $n G_M(\lg n) + d_M \lg^M n + \sum_{d=0}^{M-1}(\lg^d n)G_{M,d}(\lg n),$ where $d_M$ is a constant, $G_M(u)$ and $G_{M,d}(u)$'s are again periodic functions with period one (given by absolutely convergent Fourier series).