Analyzing a Weighted Digital Sum Variant

Abstract : 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).
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Communication dans un congrès
Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.93-106, 2010, DMTCS Proceedings
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Y. K. Cheung, Mordecai Golin. Analyzing a Weighted Digital Sum Variant. Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.93-106, 2010, DMTCS Proceedings. 〈hal-01185583〉

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