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31
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 82 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Mellin Transforms and Asymptotics: The Mergesort Recurrence, Acta Informatica
, 1994
"... Abstract. Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divideandconquer algorithms. This note illustrates the techniques by providing a precise analysis of the standard topdown recursive mergesort algorithm, in the average case, as wel ..."
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Cited by 27 (5 self)
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Abstract. Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divideandconquer 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 divideandconquer recurrences. Many algorithms are based on a recursive divideandconquer 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 divideandconquer 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, divideandconquer 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 topdown 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),
The Average Case Analysis of Algorithms: Mellin Transform Asymptotics
, 1996
"... This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: divideandconquer ..."
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Cited by 12 (0 self)
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This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: divideandconquer recurrences, maxima finding, mergesort, digital trees and plane trees.
The Cost Distribution of QueueMergesort, Optimal Mergesorts, and PowerofTwo Rules
"... Queuemergesort is recently introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queuemergesort, including the best, average and variance cases. The asymptotic normality of its cost is also ..."
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Cited by 9 (5 self)
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Queuemergesort is recently introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queuemergesort, including the best, average and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (topdown mergesort). On the other hand, the variance of queuemergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of topdown mergesort, bottomup mergesort and queuemergesort. We derive an "invariance principle" for asymptotic linearity of divideandconquer recurrences based on general "poweroftwo" rules of which the underlying dividing rule of queuemergesort is a special case. These analyses reveal an interesting algorithmic feature for general "poweroftwo" rules.
On the number of optimal base 2 representations of integers, Des
 Codes Cryptogr
"... Abstract. We study representations of integers n in binary expansions using the digits 0, ±1. We analyze the average number of such representations of minimal “weight ” ( = number of nonzero digits). The asymptotic main term of this average involves a periodically oscillating function, which is ana ..."
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Cited by 9 (5 self)
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Abstract. We study representations of integers n in binary expansions using the digits 0, ±1. We analyze the average number of such representations of minimal “weight ” ( = number of nonzero digits). The asymptotic main term of this average involves a periodically oscillating function, which is analyzed in some detail. The main tool is the construction of a measure on [−1,1], which encodes the number of representations. 1.
Analysis of the expected number of bit comparisons required by Quickselect
 Algorithmica
"... When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard ..."
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Cited by 8 (4 self)
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When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank. AMS 2000 subject classifications. Primary 68W40; secondary 68P10, 60C05. Key words and phrases. Quickselect,Find, searching algorithms, asymptotics, averagecase analysis, key comparisons, bit comparisons.
Exact Asymptotics of DivideandConquer Recurrences
"... The divideandconquer 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 performanc ..."
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Cited by 7 (1 self)
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The divideandconquer 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 topdown recursive mergesort algorithm. The approach is applicable to a large number of divideandconquer recurrences, and a general theorem is obtained when the partitioningmerging toll of a divideandconquer algorithm is a sublinear function. As another illustration the method is also used to provide an exact analysis of an efficient maximafinding algorithm.
Digital Sums And DivideAndConquer Recurrences: Fourier Expansions And Absolute Convergence
, 2004
"... We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing ..."
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Cited by 7 (2 self)
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We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing numerically the coefficients involved to high precision.
An Asymptotic Theory for Recurrence Relations Based on Minimization and Maximization
 Journal of Algorithms
, 2001
"... We derive asymptotic approximations for the sequence f(n) defined recursively by f(n) = min 1#j
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Cited by 6 (3 self)
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We derive asymptotic approximations for the sequence f(n) defined recursively by f(n) = min 1#j<n {f(j) j)} + g(n), when the asymptotic behavior of g(n) is known. Our tools are general enough and applicable to another sequence F (n) = max 1#j<n {F min{g(j), g(n j)}}, also frequently encountered in divideandconquer problems. Applications of our results to algorithms, group testing, dichotomous search, etc. are discussed.