Flajolet, P., and Sedgewick, R. The average case analysis of algorithms: Counting and generating functions. Research Report 1888, Institut National de Recherche en Informatique et en Automatique, Apr. 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... task is to consider the number of unpaired bases, which are marked by y within F (S; z; y) We therefore compute the rst parital derivative of F (S; z; y) with respect to y taking into consideration that S is a function in z and y (for details on that part of the method the reader is refered to [8]) Afterwards we set y to 1. Denoting G : y S(z; y)j y=1 , we nd 1 SG 1 zS 1 zG G where S still is determined by F (S; z; 1) By means of resultants we can combine both polynomial equations yielding F 2 (G; z) 4z 4z (8z 8) G z = 0 for the quation which determines ....

P. Flajolet and R. Sedgewick, The average case analysis of algorithms: Counting and generating functions, INRIA rapport de recherche 1888, 1993.

.... Finally we apply a classical result (see [13] for example) if G(x) F (x) R(x) and S(x) are formal power series such that r k (F (x) R(x) S(x) F (x) S(F (x) We use this lemma together with the following version of the Lagrange inversion formula (see [9] for example) Theorem 3.1. Let f(x) k0 f k x be a formal power series such that f 0 6= 0 sand let Y (x) be the unique formal power series solution of the equation Y = x f(Y ) The coefficients of g(Y ) for an arbitrary formal power series g(x) are given by ]g(Y (x) Then we ....

....trees on [n] such that the root is smaller than its sons is Proof. We use the notations O(x) O 0 (x) and o n;0 for this family of trees. Here we have clearly r k = k and S(x) Gamma ln(1 Gamma x) If we apply Lemma 3. 1 we have that O 0 (x) ln(1 Gamma O(x) Moreover, we know (see [9]) that O(x) satisfies the functional equation 1 Gamma O(x) O 0 (x) Gamma(1 Gamma O(x) ln(1 Gamma O(x) We can apply Theorem 3.1 with the following functions 1 1 Gamma x ; g(x) Gamma(1 Gamma x) ln(1 Gamma x) x) 1 ln(1 Gamma x) Then we have o n;0 = n ....

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P. Flajolet and R. Sedgewick, The average case analysis of algorithms: counting and generating functions, Tech. Report 1888, INRIA, 1993.

....of generating functions. Over the years, this has led to several formalizations (addressing the foundational question: what is a combinatorial structure How do we specify it What is the relation of such specifications to counting ) being introduced by Goulden Jackson [GJ83] Flajolet Sedgewick [FS93, FS01], Joyal [BLL98] Stanley [Sta97, Sta99] and several others. Here we add one more formal system to the list, called the method of coinductive counting. From the enumerative point of view, it makes it possible to derive existing counting results in a new perspective. We first illustrate the method ....

....possibly in terms of some kind of grammars for tree growing. Such grammars will no doubt be closely related to an alternative approach to counting, which is based on structural properties expressed as a kind of domain equation (such as T =1 T XT for binary trees) as present in for instance [FS93] and also [BLL98] Also the work of the Florence school of Pinzani and others [BLPP99] is related to this point. ii) The issue of minimization of weighted automata has of course only been touched upon. In the presented examples, there usually was an obvious minimized candidate, but a more ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Counting and generating functions. Research Report 1888, INRIA Rocquencourt, 1993. 116 pages. 29

.... 1 = H 1 = 1; G n = 2(n Gamma 1)H n (n 1) 4) and between L n and H n L 1 = H 1 = 1; L n = n Gamma 1)H n (n 1) 5) Finally, we introduce the exponential generating function H(t) H(t) H n 1 : Using classical results on exponential generating functions and labeled structures (see [1]) we can say that: H = 1 ln ; 6) and then H 1 1 Gamma L L t : 7) Now, the rest of the proof is calculus using the previous relations. t (n Gamma 1)H n = L = tH Gamma H n By dioeerentiation and using equation (7) we obtain that: L = 1 (L ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: counting and generating functions. Technical Report 1888, INRIA, 1993.

.... (LST) O(s) and physical pointer search time through an index be IRV co with LST g2(s) Then: 1) LST of join execution time is given by the following expression: 2) and the mathematical expectation is given by: 2) IDol where (z) 1 [ VR (1 j (l z) random variable generating function (GF) [6] of j=l number of references from R to S, M (6) mathematical expectation of number of references from R to S, M(O) mathematical expectation of access time to object from via physical pointer, M (co) mathematical expectation of physical pointer search time in an index structure, VR (z) GF ....

....S. The cardinality of this set is given by the following expression: k (4) Mathematical expectation of the number of selected objects may be found from (3) through its first derivative: Mo) 5; 5; z) 5) JD . Jk ) Qk J=JD. Jk In (3) and (5) zi]V(z) designates coefficient of zifrom GF V(z) [6]. For the case of equal probabilities j of pointers from R to S (see Fig. 1) formula (5) will be transformed and reduced to the following form: o) xs (6) here NR and K s number of selected objects from extents R and S, satisfying conditions F R and F s respectively; K denotes the number of ....

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Flajolet, P., Sedgewick, R. The average case analysis of algorithms: counting and generating functions. Technical Report No. RR-1888, INRIA, Rocquencourt, 1993, 113 p.

....and hence an extension is necessary. The notation here is different, but the final q series functional equations are similar. From the point of view of decomposable structures, certainly values have been given to structures before under the names additive attributes and recursive parameters [1, 6]. However, this work proposes a framework and makes explicit all the general enumerative consequences. In this document, the suitable attributes are defined and further specified in the first two sections and the generating function consequences are considered in the third section, including a ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Counting and generating functions. Research Report 1888, Institut National de Recherche en Informatique et en Automatique, April 1993. 115 pages.

.... S(x) are formal power series such that G(x) X k0 1 (k 1) r k (F (x) k ; R(x) X k0 r k x k k ; S(x) Z x 0 R(t)dt = X k1 r k Gamma1 x k k ; then G(x) x F (x) S(F (x) We use this lemma together with the following version of the Lagrange inversion formula (see [9] for example) 8 CEDRIC CHAUVE, SERGE DULUCQ AND OLIVIER GUIBERT Theorem 3.1. Let f(x) P k0 f k x k be a formal power series such that f 0 6= 0 sand let Y (x) be the unique formal power series solution of the equation Y = x f(Y ) The coefficients of g(Y ) for an arbitrary formal power ....

....k=0 (n k Gamma 1) n Gamma k Gamma 1)k : Proof. We use the notations O(x) O 0 (x) and o n;0 for this family of trees. Here we have clearly r k = k and S(x) Gamma ln(1 Gamma x) If we apply Lemma 3. 1 we have that O 0 (x) Gammax O(x) ln(1 Gamma O(x) Moreover, we know (see [9]) that O(x) satisfies the functional equation O(x) x 1 Gamma O(x) So O 0 (x) Gamma(1 Gamma O(x) ln(1 Gamma O(x) We can apply Theorem 3.1 with the following functions f(x) 1 1 Gamma x ; g(x) Gamma(1 Gamma x) ln(1 Gamma x) g x (x) 1 ln(1 Gamma x) Then we have o ....

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P. Flajolet and R. Sedgewick, The average case analysis of algorithms: counting and generating functions, Tech. Report 1888, INRIA, 1993.

.... = 1; G n = 2(n Gamma 1)H n (n 1) 4) and between L n and H n L 1 = H 1 = 1; L n = n Gamma 1)H n (n 1) 5) Finally, we introduce the exponential generating function H(t) H(t) X n0 H n 1 t n n : Using classical results on exponential generating functions and labeled structures (see [1]) we can say that: H = 1 ln 1 1 Gamma L ; 6) and then H t = 1 1 Gamma L L t : 7) Now, the rest of the proof is calculus using the previous relations. X n1 L n t n n = t X n2 (n Gamma 1)H n t n n = L = tH Gamma X n2 H n t n n By dioeerentiation and ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: counting and generating functions. Technical Report 1888, INRIA, 1993.

....to compute a function encoding the counting sequence of a class. The second section is dedicated to complex analysis. The aim is Lecture notes for a course given during the workshop AL EA 01 in Luminy (France) This summary is inspired by the book in preparation of Flajolet and Sedgewick [2, 3]. to give a method to extract the asymptotic behavior of a counting sequence encoded by a complex function. The nal section illustrates these methods throughout two examples: clouds and d 4997 1. A Symbolic Method for Enumerative Combinatorics A counting sequence fA n g n 0 can be encoded by ....

Flajolet (Philippe) and Sedgewick (Robert). { The Average Case Analysis of Algorithms: Counting and Generating Functions. { Research Report n 1888, Institut National de Recherche en Informatique et en Automatique, 1993. 116 pages. A generalisation of transfer theorems exists for the case of multiple of singularities; see [2, p. 85].

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Flajolet, P., and Sedgewick, R. The average case analysis of algorithms: Counting and generating functions. Research Report 1888, Institut National de Recherche en Informatique et en Automatique, Apr. 1993.

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P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Counting and generating functions. Research Report 1888, INRIA, 1993. 116 pages.

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