Abstract
The divide-and-conquer principle is a major 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 partitioning-merging 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.
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Flajolet, P., Golin, M. (1993). Exact asymptotics of divide-and-conquer recurrences. In: Lingas, A., Karlsson, R., Carlsson, S. (eds) Automata, Languages and Programming. ICALP 1993. Lecture Notes in Computer Science, vol 700. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56939-1_68
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DOI: https://doi.org/10.1007/3-540-56939-1_68
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