A Master Theorem for Discrete Divide and Conquer Recurrences

Divide-and-conquer recurrences are one of the most studied equations in computer science. Yet, discrete versions of these recurrences, namely T(n) = a(n) + Sigma(m)(j=1)b(j)T(left perpendicular p(j)n + delta(j)right perpendicular) + Sigma(m)(j=1)(b) over bar jT(inverted left perpendicularp(j)n + (delta) over bar (j)inverted right perpendicular) for some known sequence an and given b(j) bj (b) over barj and delta j (delta) over bar, present some challenges. The discrete nature of this recurrence (represented by the floor and ceiling functions) introduces certain oscillations not captured by the traditional Master Theorem, for example due to Akra and Bazzi [1998] 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 and prove the Central Limit Theorem for the phrase length. 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.