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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),
On binary representations of integers with digits 1, 0, 1
, 2000
"... Güntzer and Paul introduced a number system with base 2 and digits −1, 0, 1 which is characterized by separating nonzero digits by at least one zero. We find an explicit formula that produces the digits of the expansion of an integer n which leads us to many generalized situations. Syntactical prope ..."
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Cited by 9 (4 self)
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Güntzer and Paul introduced a number system with base 2 and digits −1, 0, 1 which is characterized by separating nonzero digits by at least one zero. We find an explicit formula that produces the digits of the expansion of an integer n which leads us to many generalized situations. Syntactical properties of such representations are also discussed.
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.
Palindrome complexity
 To appear, Theoret. Comput. Sci
, 2002
"... We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points o ..."
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Cited by 6 (2 self)
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We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to “class P ” of HofKnillSimon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block)complexity. 1
On the average growth of random Fibonacci sequences
 J. Integer Seq
"... We prove that the average value of the nth term of a sequence defined by the recurrence relation gn = gn−1 ± gn−2, where the ± sign is randomly chosen, increases exponentially, with a growth rate given by an explicit algebraic number of degree 3. The proof involves a binary tree such that the num ..."
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Cited by 5 (3 self)
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We prove that the average value of the nth term of a sequence defined by the recurrence relation gn = gn−1 ± gn−2, where the ± sign is randomly chosen, increases exponentially, with a growth rate given by an explicit algebraic number of degree 3. The proof involves a binary tree such that the number of nodes in each row is a Fibonacci number. 1
A conjecture on primes and a step towards justification, arXiv (math
 NT math CO) 0706.0786. Departments of Mathematics, BenGurion University of the Negev, BeerSheva 84105
"... Abstract. We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification. 1. Introduction and ..."
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Cited by 5 (5 self)
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Abstract. We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification. 1. Introduction and
Algebraic Aspects of Bregular Series
, 1993
"... This paper concerns power series of an arithmetic nature that arise in the analysis of divideandconquer algorithms. Two key notions are studied: that of Bregular sequence and that of Mahlerian sequence with their associated power series. Firstly we emphasize the link between rational series over ..."
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Cited by 4 (1 self)
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This paper concerns power series of an arithmetic nature that arise in the analysis of divideandconquer algorithms. Two key notions are studied: that of Bregular sequence and that of Mahlerian sequence with their associated power series. Firstly we emphasize the link between rational series over the alphabet fx0 ; x1 ; : : : ; xB\Gamma1 g and Bregular series. Secondly we extend the theorem of Christol, Kamae, Mend`es France and Rauzy about automatic sequences and algebraic series to Bregular sequences and Mahlerian series. We develop here a constructive theory of Bregular and Mahlerian series. The examples show the ubiquitous character of Bregular series in the study of arithmetic functions related to number representation systems and divideandconquer algorithms.
Summation of series defined by counting blocks of digits
 J. Number Theory
"... We discuss the summation of certain series defined by counting blocks of digits in the Bary expansion of an integer. For example, if s2(n) denotes the sum of the base2 digits of n, we show that ∑ n≥1 s2(n)/(2n(2n + 1)) = (γ + log 4 π)/2. We recover this previous result of Sondow and provide sever ..."
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Cited by 3 (0 self)
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We discuss the summation of certain series defined by counting blocks of digits in the Bary expansion of an integer. For example, if s2(n) denotes the sum of the base2 digits of n, we show that ∑ n≥1 s2(n)/(2n(2n + 1)) = (γ + log 4 π)/2. We recover this previous result of Sondow and provide several generalizations. MSC: 11A63, 11Y60. 1