Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Addison-Wesley (1996)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....packets have constant size. Thus, this costs O(log P ) time, which is negligible in comparison to c n=P . Thus, we may assume, as is done in the following analysis, that r is up to date. Lemma 2 During pass l of the loop in step 3, the expected value of r is about n= P l) Proof: From [9] we use the fact that the expected number of cycles of length m equals 1=m. The possible discovery of such a cycle is an independent event, so the expected number of remaining cycles of length m equals (1 m=n) P l =m. Thus, substituting L = P l, the expected sum r l of the lengths of these ....

Sedgewick, R., Ph. Flajolet, An Introduction to the Analysis of Algorithms. Addison Wesley, Reading, Mass., 1996.

.... x : 21) Multiplying (20) by x and summing we obtain n = n This can be iterated 2 n = n 2 l to yield the nal formula : 22) Let [x ]f(x) stand for the coecient of x in the series expansion of f(x) Reading o the coecients of (22) we nd (cf. [9, 11]) 10 which nally implies (23) the formula holds for m 1. This proves (4) of Theorem 1. To derive an asymptotic expansion of t n;m we rst observe that by the combinatorial origin of the problem the uniform model ....

....the formula holds for m 1. This proves (4) of Theorem 1. To derive an asymptotic expansion of t n;m we rst observe that by the combinatorial origin of the problem the uniform model should converge as n 1 to the unbiased memoryless model that has been studied for the last thirty years. We cite [7, 9, 11] where one nds t 1;m = k 1 We shall provide an independent proof of the above. We now x m and allow n 1. Using the asymptotics (1 O(1=N) for xed k and N 1 we obtain the following chain of implications provided m 2 t n;m = # ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, AddisonWesley, 1996.

....6 (d) E(d) 7) Then, we set H 2 (d; i) H 1 (d; i)C 6 (d) and we obtain H 2 (d; i) 2H 2 (d 1; i j) i 2) H 2 (d 1; i j 1) 8) 3.2 Some generating function Equ. 8) is a perfect candidate for an exponential generating function (see Flajolet and Sedgewick [10]) We set 2 (d; v) H 2 (d; i)v (i 1) 2 2 (d 1; v) 2 (d 1; v) v] v 2 (d 1; v) v [e v] With (7) we are led to set vE(d 1) and H(d; 1) 3 (d; 0) Before establishing the corresponding equation for 3 , it is now time to nd the e ect of ....

P. Flajolet and B. Sedgewick. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....bit of the integer k is 1, thus only n 1 bits of k are involved in the processing. Therefore the expected cost is [z ]2a p (z) Alternatively, let a(z) #A then the expected cost is 1 ]2a(z 2) where the last equality follows from the scaling property of generating functions [31] A(#z) For a deeper study of the use of generating functions in the average case analysis of algorithms, we refer the reader to Sedgewick and Flajolet s book [31] and also to the series they published as INRIA research reports which form the preliminary chapters of their new book ....

.... the expected cost is 1 ]2a(z 2) where the last equality follows from the scaling property of generating functions [31] A(#z) For a deeper study of the use of generating functions in the average case analysis of algorithms, we refer the reader to Sedgewick and Flajolet s book [31] and also to the series they published as INRIA research reports which form the preliminary chapters of their new book Analytic Combinatorics. These reports are available at http: algo.inria.fr flajolet Publications books.html. 25 ....

R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. AddisonWesley, 1996.

....containing theoretical foundation. Section 3 contains experimental results demostrating applicability of the derived formulas. Derivations of theoretical results are presented in Section 4 using analytic tools of analysis of algorithms such as generating functions and complex asymptotic (cf. [15, 16]) 2 Main Results In this section we present our main theoretical results. 2.1 Formulation of the Pattern Matching Problem Given an alphabet A = fa 1 ; a 2 ; a jAj g and a pattern S = s 1 s 2 : s m of length m, we search for some occurrences of S as a subsequence within a window W of ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1995.

....digital search trees, tries, quad trees etc. that will not be discussed in this paper. Nevertheless, in all these cases the concept of generating functions can be used to rephrase the counting problem into this more analytic language (for details see Flajolet et al. 7] and Sedgewick and Flajolet [17]) Appendix 1.A: Lagrange Inversion Formula. Let a(x) be a power series with a 0 = 0 and a 1 6= 0. The Lagrange inversion formula provides an expicit representation of the coecients of inverse power series a (x) which is de ned by a(a (x) a (a(x) x: Theorem 8. Let a(x) ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison Wesley, 1996.

....in this section we will concentrate upon the case where all the functions are binary (i.e. have two arguments) and so the programs are expressed as binary trees. This avoids dealing with mixtures of arities which complicates the analysis and for which there is little existing work (for example [37] deals with two cases: binary and unrestricted arity trees) We don t expect such additional complexity to increase our understanding at this stage. There are (l 1) l 1) 2) l 1) 2) di erent programs of size l, where jT j is the number of terminals and jF j is the number of ....

....will run into the common depth limit of 17 levels in about 12 generations [17] Using the curve indicating the peak in the distribution of programs against their size and shape, a predicted depth can be converted into a predicted program size. The curve is known for programs with only two inputs [37] and can be precalculated for more complex function sets. Thus we predict that generally standard GP will run into common size limits (which can be as low as 50 or 200 nodes) within a few generations and certainly before the 50 generations commonly use, see [17] 8. Conclusions We have ....

Robert Sedgewick and Philippe Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....[5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, median of three proves the simplest and the most successful. The average number of comparisons of this variant drops to n log n O(n) cf. [15, 25, 27]) A natural question then arises: how does the standard quicksort improve from 2n log n to n log n as far as the number of comparisons is concerned Our purpose of this paper is to give a possible explanation of the underlying improvement process. For completeness, we briefly describe ....

....is in its final position, this recursive scheme sorts the input in increasing order. If we assume that each of the n permutations of n elements is equally likely, then the average number of comparisons used by quicksort when given a random permutation, denoted by C n , satisfies C 1 = 0 and (cf. [15, 24, 25, 27]) C n = n c 1 C j (n 2) # This work was done while this author was at Institute of Mathematics, Academia Sinica, Taipei. for some constant c 1 depending on implementation. The solution is easily seen to satisfy C n = 2(n 1)H n 3 n O(1) 1) where H n = 1#j#n j 1 denotes ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison Wesley, Reading, Massachusetts, 1996.

....that use randomization. In view of Theorem 3, the use of randomization is indispensable. Before continuing with the statement of the algorithms and their analysis, we will mention two formulas that will be useful in evaluating certain sums that will arise in our calculations (see, e.g. [11] for details on these, as well as, similar summation formulas) 1)la 2a (5) l 1)a 1 . 6) 5.1 A memoryless randomized algorithm The algorithm we will describe is memoryless and in order to decide at each step whether it will follow the advice of the current node or ....

.... 4n k(k 1) 11) Will we be, first, concerned with the first sum. Let and B(t) t . Then the first sum in (11) is equal to A(1) Using ordinary generation function properties (the right shift and index divide properties see [11], Section 3.1) or simply by inspection, we can see that the following holds: B(t)dt. The integral above can be computed easily and it gives [1 p(1 z) 12) from which we conclude, setting z = 1, that p(n 1) n ....

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R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, AddisonWesley, 1996. 17

....[5] Most of these variants focused on improving the average performance of quicksort or on making the quadratic worst case behavior less likely. Of these, median of three proves the simplest and the most successful. The average number of comparisons of this variant drops to n log n O(n) cf. [15, 25, 27]) A natural question then arises: how does the standard quicksort improve from 2n log n to 7 n log n as far as the number of comparisons is concerned Our purpose of this paper is to give a possible explanation of the underlying improvement process. For completeness, we brie y describe ....

....is in its nal position, this recursive scheme sorts the input in increasing order. If we assume that each of the n permutations of n elements is equally likely, then the average number of comparisons used by quicksort when given a random permutation, denoted by C n , satis es C 1 = 0 and (cf. [15, 24, 25, 27]) C n = n c 1 C j (n 2) This work was done while this author was at Institute of Mathematics, Academia Sinica, Taipei. for some constant c 1 depending on implementation. The solution is easily seen to satisfy C n = 2(n 1)H n 3 n O(1) 1) where H n = denotes the n th ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison Wesley, Reading, Massachusetts, 1996.

.... of types shows that the exponential generating function F (z) of the number of such graphs is F (z) c(t(z) where t(z) ze t(z) is the EGF (exponential generating function) of the number of labeled trees and c(z) 1 z z z is the EGF of the number of cycles of length at least 3, see [8]. The unique singularity of F (z) c(t(z) is z 0 = 1=e, since t(z 0 ) 1 i z 0 = 1=e. In [7,8] the authors show that t(z) 1 2 1=2 p 1 ez: F (z) 1 ez) which yields the number of failure calls N(K n ) on the complete graph K n : N(K n ) n [z ]F (z) n : In ....

....(z) c(t(z) where t(z) ze t(z) is the EGF (exponential generating function) of the number of labeled trees and c(z) 1 z z z is the EGF of the number of cycles of length at least 3, see [8] The unique singularity of F (z) c(t(z) is z 0 = 1=e, since t(z 0 ) 1 i z 0 = 1=e. In [7,8], the authors show that t(z) 1 2 1=2 p 1 ez: F (z) 1 ez) which yields the number of failure calls N(K n ) on the complete graph K n : N(K n ) n [z ]F (z) n : In order to get the failure probability over K n , we have to divide N(K n ) by which is the total ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison Wesley (1996).

.... implies that the asymptotic complexity to compute the first n coefficients of s(z) is O(n log n) which is much better than the previously best known bound O(n ) Many other differential difference equations arising in combinatorics and the analysis of algorithms are similar to (5) see also [2]) This is in particular so for binary splitting algorithms, such as the algorithms presented in this paper themselves Let us finally remark that for the fast expansion of power series which satisfy more complicated differential difference equations, it would be nice to extend the ideas of this ....

Flajolet, P., and Sedgewick, R. An introduction to the analysis of algorithms. Addison Wesley, Reading, Massachusetts, 1996.

....language L 1 , it applies to any weighted language L . Let us say to be complete that under a so called variability condition in [19] a Gaussian limit law holds for the coecients of F (t; u) Remark 5 If f 1, the dominant pole may not be simple ( 1) Hence, we have to use the formula ([21][Theorem 4.1] for computing the asymptotic of the coecients of the series. Remark 6 If A has a unique non trivial connected components, aperiodicy is not necessary. If A has period k, than we shall consider the words of length kn, and then achieve primitivity. Remark 7 The assumption that each ....

....the series L (t; x) t) and ;x i (t) for i 2 1; k (in which (x i ) are unknown variables) Let s the unique dominant pole of this series. 2. Put s = 1 and take as mentionned in Theorem 4. Compute the asymptotics values, say i ( of each i ( according to (17) and using ([21][Theorem 4.1] This can be done since Corollary 8 ensures that each series has got a unique pole. 3. Solve the following algebraic system in the unknown variables ( x 1 ) x k ) 1 ( v 1 . k 1 ( v k 1 D (1; 1) 0 where D (t; x) ....

R. Sedgewick and Ph. Flajolet. An introduction to the analysis of algorithms. Addison Wesley, 1996.

....circumstances. Both algorithms are based upon the divide and conquer principle and operate using similar ideas. A brief, but complete description is given in Section 2. Excellent sources for background information and further references on quicksort and quickselect and their analysis include [5, 10, 15, 16, 18, 17, 19]. Contrary to other divide and conquer algorithms, quicksort and quickselect are not guaranteed to divide the problem into subproblems of approximately the same size. Not even there is certainty that the size of the subproblems will be a fraction of the size of the original problem. Hence, their ....

R. Sedgewick and Ph. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....in unbalanced BSTs is (except for deletions) simple and elegant, and the cost of any of these operations is always linearly bounded by the height of the tree. For random binary search trees, the expected performance of a search, whether successful or not, and that of update operations is O(logn) [15, 17, 24], with small hidden constant factors involved (here and unless otherwise stated, n denotes the number of items in the tree or size of the tree) Random BSTs are those built using only random insertions. An insertion in a BST of size j Gamma 1 is random if there is the same probability for the ....

....of the performance of the basic algorithms is immediate, since both insertions and deletions guarantee the randomness of their results. Therefore, the large collection of results about random BSTs found in the literature may be used here. We will use three well known results (see for instance [15, 17, 24, 27]) about random BSTs of size n: the expected depth of the i th internal node, the expected depth of the i th external node (leaf) and the total expected lenght of the right and left spines of the subtree whose root is the i th node. We will denote the corresponding random variables D n , L n ....

R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....model; that means that we assume uniform random order of the items stored in the tree. Thus, the model produces trees that are isomorphic to the trees generated by the Quicksort algorithm. This model has been subject of an intensive research over the past three decades (see for instance [7, 9, 20]) and many amazing facts have been proved or conjectured on it. For example, regarding to computer simulations, the pro le of such a tree seems, with high probability, close to a gaussian one (see [20] p. 248 for more details) Among the most studied random variables of this model, we can cite in ....

....has been subject of an intensive research over the past three decades (see for instance [7, 9, 20] and many amazing facts have been proved or conjectured on it. For example, regarding to computer simulations, the pro le of such a tree seems, with high probability, close to a gaussian one (see [20], p. 248 for more details) Among the most studied random variables of this model, we can cite in particular the depth level k n of the n th insertion x n , which represents the cost of an unsuccessful search (in a query operation) the height h n which is the worst case of such an operation and ....

R. Sedgewick and P. Flajolet. \An Introduction to the Analysis of Algorithms ". Addison-Wesley. 1996. 21

....1 j d; iii) S j , 1 j d, was selected , but the resource may be allocated to another segment (satisfying certain properties) before S j completes. We say that S j was virtually taken. We use for coin tosses a probability distribution derived from the Riemann zeta function (see, e.g. [40]) for a complex number z 2 C, i(z) P n1 1 n z ; i(z) converges when jzj 1, thus we choose z = 1 ffl for some small ffl 0. Let c d = 1= d 1 ffl i(1 ffl) and p d = c d = Q d Gamma1 j=1 (1 Gamma p j ) Then c d denotes the probability that a segment in depth d will be taken; p d ....

R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....Then, both A and AGG fetch r 1 . Now, assume that A acts like AGG in the first (i Gamma 1) steps. We distinguish between three cases in the ith reference: 3 This sequence can be viewed as a special case of the q order Fibonacci sequence, in which g(n) P q j=1 g(n Gamma j) see, e.g. [17, 26]) 6 (i) A initiates a fetch of some block r l . Then since A satisfies the no harm rules, and AGG fetches in the first opportunity, clearly, AGG acts in step i like A (r l is missing in AGG s cache, and can be fetched) ii) If AGG performs no fetch, then since A satisfies the no harm ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley Publishing, 1996.

....is a definition of a function Fn in terms of the values of F at indices smaller than n; a recurrence is divide and conquer DAC, for short if the average size of these indices is a fraction of n. Our reference source here is the Master Theorem MT, for short as it can be found in [16]. Other references in this subject include the (classic) Master Theorem [1, 2, 3] several improvements [10, 17, 18, 19] as well as other related results [11] Assume that we have the recurrence Fn = t n W Delta FSn , with t n 0 and Sn = Z Delta n O(1) for some 0 Z 1. If Fn describes ....

R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

....to introduce students with little mathematical background to the math and computer science of cryptography, emphasizing some of the social and legal questions surrounding this subject. A good high school math background is sucient to understand the math parts of the course. Look at my homepage [5] http: www.math.rutgers.edu greenfie and follow the links to the two sections of Math 103. There are links to a wide variety of web pages discussing crypto and society. Even if the technical side is not attractive, I d hope that some of the issues discussed would be interesting to you. Some ....

Robert Sedgewick, An Introduction to the Analysis of Algorithms, Addison-Wesley, 1996 (about 500 pages, $45).

....of the number of key exchanges of Quicksort, linear combinations of key exchanges and comparison. Several random trees, such as the random m ary search tree, the random median of (2k 1) search tree, and the random quadtree, see for the de nitions Mahmoud [27] Sedgewick and Flajolet [39], Knuth [25] and Flajolet, Labelle, Laforest, and Salvy [14] for probabilistic analysis of quadtrees, have an important parameter, the total internal path length I n (the sum of the distances from the nodes to the root) which satis es (I n E I n ) n X for a di erent limit law X and some ....

Sedgewick, R. and P. Flajolet (1996). An introduction to the analysis of algorithms.. AddisonWesley, Amsterdam.

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Robert Sedgewick and Philippe Flajolet, An introduction to the analysis of algorithms, Addison-Wesley Publishing Company, 1996.

....of occurrences can be described by means of regular expressions extended with disjoint The notation [z ]f(z) represents the coecient of z in the series f(z) unions, and Cartesian products. Thus a minimal set of rules must rst be given in order to translate such basic constructions; see [16, 36, 38] for a general framework. Take A; B; C to be weighted sets with respective weights ; Here is a brief summary of translation rules from weighted sets to generating functions: Disjoint unions. Assume that C = A[B where the union is disjoint (A B = and that the weight on C is ....

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1995.

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Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Addison-Wesley (1996)

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Robert Sedgewick and Philippe Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

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Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Addison-Wesley (1996)

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Robert Sedgewick and Philippe Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

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R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

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Robert Sedgewick and Philippe Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wiley, 1996.

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Sedgewick, R., and Flajolet, P. An Introduction to the Analysis of Algorithms. Addison-Wiley, 1996.

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R. Sedgewick, P. Flajolet, An Introduction to the Analysis of Algorithms, AddisonWesley, Reading, Massachusetts, 1996.

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R. Sedgewick, P. Flajolet, An Introduction to the Analysis of Algorithms, AddisonWesley, Reading, Massachusetts, 1996.

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R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

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Robert Sedgewick and Philippe Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley Publishing Company, 1996.

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R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading, MA, USA, 1996.

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Sedgewick, R., and Flajolet, P. An Introduction to the Analysis of Algorithms. Addison-Wesley Publishing Company, 1996.

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Sedgewick, R. and P. Flajolet (1996). An introduction to the analysis of algorithms.. AddisonWesley, Amsterdam.

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R. Sedgewick and P. Flajolet (1995), An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA.

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R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. Addison-Wesley, 1996.

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R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, 1996.

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Sedgewick, R., and Flajolet, P. An Introduction to Analysis of Algorithm. Addison-Wesley, 1996.

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Robert Sedgewick and Philippe Flajolet, An Introduction to the Analysis of Algorithms (Reading, Massachusetts: Addison-Wesley, 1996).

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Sedgewick (Robert) and Flajolet (Philippe). { An introduction to the analysis of algorithms. { Addison-Wesley Publishing Co., Reading, MA, 1996.

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Robert Sedgewick and Philippe Flajolet, An Introduction to the Analysis of Algorithms (Reading, Massachusetts: Addison-Wesley, 1996).

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R. Sedgewick, and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1995. 19

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R. Sedgewick and P. Flajolet. An introduction to the analysis of algorithms. Addison Wesley, 1996.

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R. Sedgewick and P. Flajolet. An Introduction to the Analysis of Algorithms. AddisonWesley Publishing Company, USA, 1996.

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Sedgewick, R., and Flajolet, P. An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading MA, 1996.

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R. Sedgewick et P. Flajolet, (1996) An Introduction to the Analysis of Algorithms, Addison-Wesley. 13

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R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, Massachusetts, 1996.

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