Elementary Yet Precise Worst-case Analysis of MergeSort, A short version (SV)

This paper offers two elementary yet precise derivations of an exact formula $$W(n) = \sum_{i=1} ^{n} \lceil \lg i \rceil = n \lceil \lg n \rceil - 2^{\lceil \lg n \rceil} + 1$$ for the maximum number $W(n)$ of comparisons of keys performed by ${\tt MergeSort}$ on an $n$-element array. The first of the two, due to its structural regularity, is well worth carefully studying in its own right. Close smooth bounds on $W(n)$ are derived. It seems interesting that $W(n)$ is linear between the points $n = 2^{\lfloor \lg n \rfloor}$ and it linearly interpolates its own lower bound $n \lg n - n + 1$ between these points.