A Master Theorem for Discrete Divide and Conquer Recurrences

• Published in: Journal of the ACM Vol. 60; no. 3; pp. 1 - 49
• Main Authors: Drmota, Michael, Szpankowski, Wojciech
• Format: Journal Article
• Language: English
• Published: New York, NY Association for Computing Machinery 01.06.2013
• Subjects: Algebra; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Coding, codes; Computer science; Computer science; control theory; systems; Dirichlet problem; Divide & conquer algorithms; Exact sciences and technology; Information, signal and communications theory; Mathematical analysis; Mathematical functions; Mathematics; Number theory; Sciences and techniques of general use; Sequences, series, summability; Signal and communications theory; Studies; Telecommunications and information theory; Theorems; Theoretical computing; Recurrence; Mellin-Perron formula; Perron formula; Redundancy; Data compression; Matrix product; Modeling; Karatsuba algorithm; Master theorem; Sorting; Tauberian theorem; Central limit theorem; Algorithms; Divide-and-conquer recurrence; Arithmetic code; Dirichlet series; Asymptotic approximation; Boncelet's data compression algorithm; Fast algorithm; Strassen algorithm; Variable length code; Divide and conquer method; mergesort; Mellin transformation
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/2487241.2487242

Divide-and-conquer recurrences are one of the most studied equations in computer science. Yet, discrete versions of these recurrences, namely for some known sequence a n and given b j , b j , p j and δ j , δ j , 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.