Analyzing a Weighted Digital Sum Variant - Archive ouverte HAL Accéder directement au contenu
Communication Dans Un Congrès Discrete Mathematics and Theoretical Computer Science Année : 2010

Analyzing a Weighted Digital Sum Variant

Résumé

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).
Fichier principal
Vignette du fichier
dmAM0107.pdf (713.14 Ko) Télécharger le fichier
Origine Fichiers éditeurs autorisés sur une archive ouverte
Loading...

Dates et versions

hal-01185583 , version 1 (20-08-2015)

Identifiants

Citer

Y. K. Cheung, Mordecai Golin. Analyzing a Weighted Digital Sum Variant. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.93-106, ⟨10.46298/dmtcs.2785⟩. ⟨hal-01185583⟩

Collections

TDS-MACS
54 Consultations
776 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More