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 asymptotic estimates. The method is illustrated by a detailed average case, variance and distribution analysis of the classic top-down recursive mergesort algorithm. The approach is applicable to a large number of divide-and-conquer recurrences, and a general theorem is obtained when the partitioningmerging toll of a divide-and-conquer algorithm is a sublinear function. As another illustration the method is also used to provide an exact analysis of an efficient maxima-finding algorithm. Many algorithms are based on a recursive divide-and-conquer strategy. Accordingly, their complexity is expressed by recurrences of the usual divide-andconquer form [10]. Typical examples are heapsort, mergesort, Karatsuba's multiprecision multiplication, discrete Fourier transforms, binomial queues, sorting networks, etc. It is relatively easy to determine general orders of growth for solutions to these recurrences as explained in standard texts, see the "master theorem" of [10, p. 62]: if for example In = fin/2j + ffn/21 + en (1) and en = 0(n) then fn = 0(n log n) while if en = 0(72 1 ') for some e > 0 then en = 0(n). However, a precise asymptotic analysis is often appreciably more delicate. At a more detailed level, divide-and-conquer recurrences tend to have solutions that involve periodicities, many of which are of a fractal nature. It is our purpose here to discuss the analysis of such periodicity phenomena while focussing on the analysis of the standard top-down recursive mergesort algorithm. We will show for example that the average number of comparisons performed by mergesort satisfies U(n) = n lg n nB(1g n) 0(n1/2), while the variance is of the form nC(1g n) 0(n' 12 ): B(u) and C(u) are both periodic functions that are fractal-like and which are everywhere continuous but not differentiable at a dense set of points on the line. Our approach consists in introducing for this range of problems techniques Mellin transforms, Dirichlet series, and Perron's formula that are borrowed aks.