Best-case Analysis of MergeSort with an Application to the Sum of Digits Problem, A manuscript (MS) v2

TL;DR

This paper derives a simple, efficient formula for the minimal number of comparisons in MergeSort and connects it to the sum of binary digits problem, providing insights into algorithm analysis and number theory.

Contribution

It introduces a new, less complex formula for MergeSort comparisons and links it to the sum of binary digits, showing no closed-form exists for the formula.

Findings

The formula can be evaluated in logarithmic time.

There is no closed-form expression for the minimal comparisons.

Results extend to the sum of binary digits problem.

Abstract

An exact formula \[ B(n) = \frac{n}{2}(\lfloor \lg n \rfloor + 1) - \sum _{k=0} ^{\lfloor \lg n \rfloor} 2^k Zigzag(\frac{n}{2^{k+1}}), \] where \[ Zigzag (x) = \min (x - \lfloor x \rfloor, \lceil x \rceil - x), \] for the minimal number $ B(n) $ of comparisons of keys performed by $ {\tt MergeSort} $ on an $ n $-element array is derived and analyzed. The said formula is less complex than any other known formula for the same and can be evaluated in $ O(\log ^{c}) $ time, where $ c $ is a constant. It is shown that there is no closed-form formula for the above. Since the recurrence relation for the minimal number of comparisons of keys for $ {\tt MergeSort} $ is identical with a recurrence relation for the number of 1s in binary expansions of all integers between $ 0 $ and $ n $ (exclusively), the above results extend to the sum of binary digits problem.