Multiple-size divide-and-conquer recurrences

Asymptotics of linear divide-and-conquer recurrences

Asymptotics of divide-and-conquer recurrences is usually dealt either with elementary inequalities or with sophisticated methods coming from analytic number theory. Philippe Dumas proposes a new approach based on linear algebra. The example of the complexity of Karatsuba’s algorithm is used as a guide in this summary. The complexity analysis of divide-and-conquer algorithms gives rise to recurr...

Exact Asymptotics of Divide-and-Conquer Recurrences

The divide-and-conquer principle is a majoi paradigm of algorithms design. Corresponding cost functions satisfy recurrences that directly reflect the decomposition mechanism used in the algorithm. This work shows that periodicity phenomena, often of a fractal nature, are ubiquitous in the performances of these algorithms. Mellin transforms and Dirichlet series are used to attain precise asympto...

Divide - and - Conquer Multiple Sequence

Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems

fn = (k −mod(n, k))fbn/kc +mod(n, k)fdn/ke + an, n ≥ k, there is a unique continuous periodic function f∗ : R→ R with period 1 such that fn = nf(logk n)+o(n). If (an) is periodic with period k, ak = 0, and the initial conditions (fi : 1 ≤ i ≤ k − 1) are all zero, we obtain a specific iterated function system S, consisting of k continuous functions from [0, 1] × R into itself, such that the attr...

An Improved Master Theorem for Divide-and-Conquer Recurrences

We present a new master theorem for the study of divide-and-conquer recursive deenitions, which improves the old one in several aspects. In particular, it provides more information, frees us from technicalities like oors and ceilings, and covers a wider set of toll functions and weight distributions, stochastic recurrences included.