Abstract
Divide-and-conquer recurrences are one of the most studied equations in computer science. Yet, discrete versions of these recurrences, namely[EQUATION]for some known sequence an and given bj, pj and δj, present some challenges. The discrete nature of this recurrence (represented by the floor function) introduces certain oscillations not captured by the traditional Master Theorem, for example due to Akra and Bazzi who primary studied the continuous version of the recurrence. We apply powerful techniques such as Dirichlet series, Mellin-Perron formula, and (extended) Tauberian theorems of Wiener-Ikehara to provide a complete and precise solution to this basic computer science recurrence. We illustrate applicability of our results on several examples including a popular and fast arithmetic coding algorithm due to Boncelet for which we estimate its average redundancy. To the best of our knowledge, discrete divide and conquer recurrences were not studied in this generality and such detail; in particular, this allows us to compare the redundancy of Boncelet's algorithm to the (asymptotically) optimal Tunstall scheme.
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