Abstract

Abstract. Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divide-and-conquer algorithms. This note illustrates the techniques by providing a precise analysis of the standard topdown recursive mergesort algorithm, in the average case, as well as in the worst and best cases. It also derives the variance and shows that the cost of mergesort has a Gaussian limiting distribution. The approach is applicable to a number of divide-and-conquer recurrences. Many algorithms are based on a recursive divide-and-conquer strategy of splitting a problem into two subproblems of equal or almost equal size, separately solving the subproblems, and then knitting their solutions together to find the solution to the original problem. Accordingly, their complexity is expressed by recurrences of the usual divide-and-conquer form where the initial condition,f, , and the ‘‘knitting costs”, e,, depend on the problem being studied. Typical examples are mergesort, heapsort, 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 [6, p. 621. 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. For example, as we shall soon see, the average cost of running mergesort on n keys satisfies u (n) = n lg n + nB (lg n) + 0 (n),

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