Introduction to algorithms
Mellin transforms and asymptotics
Acta Informatica
Mellin transforms and asymptotics: harmonic sums
Theoretical Computer Science - Special volume on mathematical analysis of algorithms (dedicated to D. E. Knuth)
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
On the Solution of Linear Recurrence Equations
Computational Optimization and Applications
Limiting distributions for the costs of partial match retrievals in multidimensional tries
Proceedings of the ninth international conference on on Random structures and algorithms
Improved master theorems for divide-and-conquer recurrences
Journal of the ACM (JACM)
Average Case Analysis of Algorithms on Sequences
Average Case Analysis of Algorithms on Sequences
Analytic Combinatorics
Tunstall code, Khodak variations, and random walks
IEEE Transactions on Information Theory
Generalized Tunstall codes for sources with memory
IEEE Transactions on Information Theory
Block arithmetic coding for source compression
IEEE Transactions on Information Theory
| Hi-index | 0.00 |
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 an and given bj, bj, pj and δj, δj, 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.