Consider the following weighted digital sum (WDS) variant: write integer $n$ as $n=2^{i_1} + 2^{i_2} + \cdots + 2^{i_k}$ with $i_1 > i_2 > \cdots > i_k \geq 0$ and set $W_M(n) := \sum_{t=1}^k t^M 2^{i_t}$. This type of weighted digital sum arises (when $M=1$) in the analysis of bottom-up mergesort but is not "smooth'' enough to permit a clean analysis. We therefore analyze its average $TW_M(n) := \frac{1}{n}\sum_{j \gt n} W_M(j)$. We show that $TW_M(n)$ has a solution of the form $n G_M(\lg n) + d_M \lg ^M n + \sum\limits_{d=0}^{M-1}(\lg ^d n)G_{M,d}(\lg n)$, where $d_M$ is a constant and $G_M(u), G_{M,d}(u)$'s are periodic functions with period one (given by absolutely convergent Fourier series).

Source : oai:HAL:hal-01185583v1

Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)

Section: Proceedings

Published on: January 1, 2010

Submitted on: January 31, 2017

Keywords: Mellin Transform,Digital Sum,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]

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