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 recurrences that relate the cost at size n to the cost at fractions of n. For instance, the complexity of Karatsuba’s multiplication algorithm for polynomials of degree n is governed by c(n) = n+ 3c (dn/2e) . (1) The first values taken by this sequence with c(1) = 1 are displayed in Figure 1. The linear term n is of course dependent on the complexity model. However, the analysis is quite robust and any function growing linearly would lend itself to this analysis, leading to similar results with minor technical adjustments. Notation In this presentation, general consideration are interlaced with this particular example. In order to help distinguish between the general and the specific, we use blue characters to display the particulars of the example. 1 Divide-and-conquer Recurrences A more complicated example is provided by the analysis of a recent “dichopile” algorithm due to J. Oudinet. This algorithm performs random generation from a regular language with uniform Figure 1: The sequence c(n) (black) and its upper bound from Section 3 (red)