P. Flajolet, B. Salvy, and P. Zimmermann. Automatic Average-Case Analysis of Algorithms. Theor. Comp. Sci., (79):37--109, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the first two sections and the generating function consequences are considered in the third section. The fourth section introduces the Maple implementation and is followed by a comparison with the LUO system for automatic average case algorithm analysis developed by the Algorithms Project of INRIA [5, 10]. 1.1. Grammars and Specifications. The structures considered here are those described in detail in [5] They are built from atoms of weight one (commonly expressed as Z) and o, ffls of weight 0. The constructors are disjoint union j, cartesian product Delta, sequence , Set( MultiSet( and ....

....section introduces the Maple implementation and is followed by a comparison with the LUO system for automatic average case algorithm analysis developed by the Algorithms Project of INRIA [5, 10] 1.1. Grammars and Specifications. The structures considered here are those described in detail in [5]. They are built from atoms of weight one (commonly expressed as Z) and o, ffls of weight 0. The constructors are disjoint union j, cartesian product Delta, sequence , Set( MultiSet( and Cycle( These latter constructors form, respectively, sequences, sets without or with duplication and ....

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, February 1991.

....size. Accommodating this requires manual or semi automatic transformation of the time or space function [17, 29] The analysis is mainly asymptotic. A challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [17, 24, 7]. But closed forms are known only for subclasses of functions. Thus, such optimization can not be done automatically for analyzing general programs. Rosendahl proposed characterizing inputs using partially known input structures [24] For example, instead of replacing an input list l with its ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37-109, Feb. 1991.

....object grammar. It is most often described with drawings. For instance, the standard decomposition of complete binary trees is an object grammar (Figure 1) The formalism given here for object grammars [7, 8] generalizes the one for context free grammars. It is akin to the work of Flajolet al... [10, 11] allowing for the specication of structures by grammars involving set, sequence and cycle constructions. One can also categorize object grammars as belonging to the domain of Universal Algebra and Magmas [9, 15, 1] Finally, our approach is related to the Theory of Species [3, 14] which gives a ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, 79(1):37109, 1991.

.... it was first developed by Gross [24] Subsequent developments are due to: Cori and Richard [9] leading to the enumeration of planar maps [7] Fliess [9] with applications to solving differential equations; Viennot [36] with applications in enumerative combinatorics; Flajolet [22], with applications to asymptotic analysis. This method is now known as the DSV methodology, following M.P. Schtzenberger s wish expressed to Viennot [35] In 1975, a theoretical computer science workshop was held in France, on the topic of formal power series. Its proceedings, edited by J. ....

....= 1 pE) x Ez ) 1 mx mx xE and we obtain the generating function for the number e n of well formed arithmetic expressions having n symbols by setting all the variables equal to x: E(x) e n x = n0 1 6x x 4 4x . This example is borrowed from the L W cookbook [22]. Recall that L U W is the software developed by the Flajolet s team at INRIA. It computes automatically asymptotic expressions for the coefficients of a series. More classical is the computation on the Dyck language which leads to the equation D = x xD 1. An elementary calculation shows ....

P. Flajolet, B. Salvy, P. Zimmerman, Automatic average-case analysis of algorithms, Theor. Comp. Sci. 79, 1 (1991), 37-109.

....e.g. 4] but it would be very di#cult to manually design appropriate models at the required fine grain level. Indeed in program analysis, the timing semantics can hardly be designed at the programming language source level (for which automatic concrete com# plexity analysis is certainly useful [47] but insu#cient since constants factors do matter [7] In practice it is indispensable to consider the program semantics at the assembler level, that is for a given compiler and for a given processor with some hypotheses on the frequency of physical interrupts [81] The model being automatically ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithm. Theoret. Comput. Sci. , 79(1):37--109, 1991.

....Performance 1. INTRODUCTION Analysis of program running time is important for reactive systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [22, 13, 14, 41], programming languages [39, 23, 32, 36, 35] and systems [37, 30, 34, 33] Being able to predict accurate time bounds automatically and efficiently is particularly important for many applications, such as reactive systems. It is also particularly desirable to be able to do so for high level ....

....independent of the input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: optimizing the time bound function to a closed form in terms of the input size [39, 6, 23, 32, 14]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. In systems, inputs are characterized indirectly using loop bounds or execution paths in programs, and ....

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, Feb. 1991.

....and formal language theory has a long story. Probabilistic tools have been used early in information theory [11] automata theory [5] and coding theory [3] 8] More recently, evidence of the usefulness of this interaction have been found in fields like asymptotic analysis of algorithms [6], allocation problems, concurrency measures [7] communication protocols specification and verification, and random generation of combinatorial structures. Most of these approaches rely on describing various subsets of A , the set of words on a finite alphabet A, defining probability measures on ....

Flajolet, P., Salvy, B., Zimmermann, P., Automatic Average-Case Analysis of Algorithms, Theoretical Computer Science 79, 1991, 37-109.

....this requires manual or semiautomatic transformation of the time or space function [18, 29] The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [18, 24, 6]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. 1 Rosendahl proposed characterizing inputs using partially known input structures [24] For ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37-109, Feb. 1991.

....this requires manual or semi automatic transformation of the time or space function [53, 32, 55] The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [53, 5, 32, 45, 12]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. Rosendahl proposed characterizing inputs using partially known input structures [45] For example, ....

....is easy to automatically generate all binary trees of height h with unknown elements or all binary trees with n nodes and unknown elements. 9 Related work There has been a large amount of work on analyzing program cost or resource complexities, but the majority of it is on time analysis, e.g. [53, 32, 11, 45, 51, 49, 12, 44, 33, 47, 34]. Some techniques for time analysis can 28 of 31 be adapted for space analysis, for example, as we did for stack space and heap allocation analysis. Analysis of live heap space has an important di erence from all these other analyses: it involves explicit analysis of the graph structure of the ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37-109, Feb. 1991.

....G omez s address: Computer Science Department, Indiana University, Bloomington, IN 47405 7104. Corresponding author: Yanhong A. Liu. Email: liu cs.sunysb.edu. Tel: 631 632 8463. Fax: 631 632 8334. URL: http: www.cs.sunysb.edu liu . 1 studied in many elds of computer science: algorithms [25, 16, 17, 53], programming languages [50, 26, 41, 44] and systems [46, 37, 43, 42] It is particularly important for many applications, such as real time systems and embedded systems, to be able to predict accurate time bounds and space bounds automatically and eciently, and it is particularly desirable to be ....

....independent of input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive cost parameters are assumed, a second problem arises, and it is theoretically 3 challenging: optimizing the cost bound function to a closed form in terms of the input size [50, 10, 26, 41, 17, 7]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. In systems, inputs are characterized indirectly using loop bounds or execution paths in programs, and ....

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37-109, Feb. 1991.

....corresponding generating functions. They are present, implicitly or explicitly, in most books that cover combinatorial enumeration, such as [81, 173, 351, 377] The most systematic approach to developing and using general rules of this type has been carried out by Flajolet and his collaborators [139]. They develop ways to see immediately (cf. 134] that if we consider mappings of a set of n labeled elements to itself, so that all n n distinct mappings are considered equally likely, then the generating function for the longest path length is given by f(z) # # k=0 # 1 1 t(z) e ....

....is precisely what computer systems do well. It is therefore possible to write programs that will derive the asymptotics behavior from the specification of the recurrence. More generally, one can analyze asymptotics of a much greater variety of generating functions. Flajolet, Salvy, and Zimmermann [124, 139] have written a powerful program for just such computations. Their system uses Maple to carry out most of the basic analytic computations. It contains a remarkable amount of automated expertise in recognizing generating functions, computing their singularities, and extracting asymptotic ....

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P. Flajolet, B. Salvy, and P. Zimmermann, Automatic average--case analysis of algorithms, Theoretical Computer Science, 79 (1991), 37--109.

....function consequences are considered in the third section, including a treatment of pathlength. The fourth section introduces the Maple implementation and is followed by a comparison with the LUO system for automatic average case algorithm analysis developed by the Algorithms Project of INRIA [5, 12]. There is a brief consideration of random generation in section seven. 1 2 MARNI MISHNA 1.1. Grammars and Specifications. The structures considered here are built from basic objects of weight 1, an Atom (commonly expressed as Z) and of weight 0, an ffl. The constructors are disjoint union j, ....

....of steps (however that is defined) an algorithm requires when a given structure is input. Generating functions summarize this information and offer a means for automatic average case complexity analysis. This idea has been explored in depth with respect to the decomposable structures defined here [4, 5, 12]. In fact, the system LUO is an implementation of this concept for a family of algorithms. The main drawback of this system is that only the final univariate generating functions are available. Attribute grammars give relations for multivariate generating functions and hence for problems it can ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, February 1991.

....have a number of useful and general tools available in the form of Mellin transforms and iteration theory of analytic functions. 6. Automatic Analysis The approach of finding general decidable asymptotic properties of combinatorial structures has been prolonged. Flajolet, Salvy and Zimmermann [1] have designed a system called Lambda Upsilon Omega ( Upsilon Omega ) that implements a number of decision procedures on combinatorial structures like the ones discussed here. The kernel specification language consists of the constructions of union, product, sequence, sets, multisets and cycles ....

Flajolet (P.), Salvy (B.), and Zimmermann (P.). -- Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, vol. 79, n 1, February 1991, pp. 37--109.

....to estimate the number of solutions of database query [25] Their method estimates the query size dynamically at run time, in contrast our method is a static analysis performed at compile time. Much of the work on automatic complexity analysis is in the context of functional programming languages [12, 15, 21, 34, 37, 41, 44]. We extend their work by being able to handle nondeterminism and 34 the generation of multiple solutions via backtracking in logic programs. Kaplan considers the analysis of the average case complexity of logic programs [18] but his approach cannot handle programs that can produce multiple ....

P. Flajolet, B. Salvy and P. Zimmermann, "Automatic average-case analysis of algorithms," Theoretical Computer Science, 79 (1991), pp. 37--109.

....just an approximation, but also a bound on the actual execution cost. Fortunately, as mentioned before, much work has been presented on (time) complexity analysis of programs (Le M etayer 1988, Wadler 1988, Rosendhal 1989, Bjerner and Holmstrom 1989, Sarkar 1989, Zimmermann and Zimmermann 1989, Flajolet et al. 1991). The most directly applicable are the methods presented by Debray and Lin (1993) and Debray et al. 1994) for statically estimating cost functions for predicates in a logic program. The two approaches have much in common but they differ in the way the approximation is done. In the first one upper ....

Flajolet, P., Salvy, B., Zimmermann, P. (1991). Automatic Average-Case Analysis of Algorithms. Theor. Comp. Sci., (79):37--109.

....lazy approach) Combinatorics and the analysis of algorithms. In combinatorics, the analysis of algorithms and for the random generation of combinatorial objects, one usually needs to expand generating functions up to a high order. Therefore, this is the ideal application for relaxed power series (Flajolet et al. 1990). In this context, multivariate power series correspond to the study of parameters in enumeration problems or the analysis of a certain algorithm (Soria, 1990) Numerical analysis. In numerical analysis, power series are mainly computed in order to be evaluated. The required number of terms ....

Flajolet, P., Salvy, B., Zimmermann, P. (1990). Automatic average-case analysis of algorithms. T.C.S., 79(1):37-109.

....this requires manual or semi automatic transformation of the time or space function [53, 32, 55] The analysis is mainly asymptotic. A theoretically challenging problem that arises in this approach is optimizing the time bound or space bound function to a closed form in terms of the input size [53, 5, 32, 45, 12]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. Rosendahl proposed characterizing inputs using partially known input structures [45] For example, ....

....time by only a constant factor. The running times of all three bound functions grow at the same rate as the corresponding space functions. 8 Related work There has been a large amount of work on analyzing program cost or resource complexities, but the majority of it is on time analysis, e.g. [53, 32, 11, 45, 51, 49, 12, 44, 33, 47, 34]. Some techniques for time analysis can be adapted for space analysis, for example, as we did for stack space and heap allocation analysis. Analysis of live heap space has an important difference from all these other analyses: it involves explicit analysis of the graph structure of the data. Most ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, Feb. 1991.

....A. Liu November 1999 1 Introduction Analysis of program running time is important for reactive systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [21, 12, 13, 40], programming languages [38, 22, 31, 35, 34] and systems [36, 29, 33, 32] Being able to predict accurate time bounds automatically and efficiently is particularly important for many applications, such as reactive systems. It is also particularly desirable to be able to do so for high level ....

....independent of the input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: optimizing the time bound function to a closed form in terms of the input size [38, 5, 22, 31, 13]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. In systems, inputs are characterized indirectly using loop bounds or execution paths in programs, and ....

[Article contains additional citation context not shown here]

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, Feb. 1991.

....investigations, mostly considered as a trivial or unimportant secondary issue, is the methodological construction of those instances that are being counted. As we will show, this very general problem is not so trivial, is interesting in its own right, and even practical. Flajolet and Zimmermann [6, 18] have defined a calculus for the description of general classes of combinatorial structures. They use this calculus both to enumerate the structures, that is to calculate the number of objects of a given size, and to uniformly generate random examples of a given size. Their method is to take a ....

....concatenation. The object z is a constant symbol (or equivalently an atom) standing for an object of size or weight one. The object 1 is a constant of weight 0 whose concatenation yields the original. The basic objects and operations on structures are in Table 2. 1, extended from Zimmermann s work [18, 6]. There are two interpretations of atoms, labeled and unlabeled, which lead to two uses of the constructors. With the labeled interpretation, all atoms in an object are marked uniquely from a set of so that they are all considered distinct and therefore no structure parts can be repeated exactly. ....

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Technical Report 1233, INRIA, Rocquencourt, August 1990.

....the calculated bounds. 1 Introduction Analysis of program running time is important for real time systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [11, 8], programming languages [25, 12, 20, 22] and systems [23, 19, 21] It is particularly important for many applications, such as real time systems, to be able to predict accurate time bounds automatically and efficiently, and it is particularly desirable to be able to do so for high level languages ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, Feb. 1991.

....object grammar. It is most often described with drawings. For instance, the standard decomposition of complete binary trees is an object grammar (Figure 1) The formalism given here for object grammars [7, 8] generalizes the one for context free grammars. It is akin to the work of Flajolet al... [10, 11] allowing for the speci cation of structures by grammars involving set, sequence and cycle constructions. One can also categorize object grammars as belonging to the domain of Universal Algebra and Magmas [9, 15, 1] Finally, our approach is related to the Theory of Species [3, 14] which gives a ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, 79(1):37109, 1991.

....series to the complex plane. This is somehow related to the infinite product Y k0 i 1 Gamma x 2 k j = X n0 ( Gamma1) n) x n : A paper on adaptive sampling [85] might also be seen in this context. 12 H. PRODINGER AND W. SZPANKOWSKI 13. Automatic analysis of algorithms The paper [96] contains the description of this very ambitious project. The system Upsilon Omega consists of two parts: the Algebraic Analyzer (Alas) and the Analytic Analyzer (Ananas) The implementation of Alas was done by Paul Zimmermann, whereas Bruno Salvy was responsible for Ananas. Historically, first ....

....certain elementary class it computes an asymptotic form of the coefficient [z n ]f(z) It uses modules Radius, Directions, Expansions, Transfer. The analyzer can also deal with entire functions, namely the class of the so called Hayman admissible functions (via saddle point analysis) The text [96] describes the theory as well as some parts of the implementation. For the purpose of illustration of the power of the system, it contains 20 automatic theorems, from classical combinatorics and the analysis of algorithms. The system Upsilon Omega still exists, but it needs probably some ....

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average--case analysis of algorithms. Theoretical Computer Science, 79:37--109, 1991.

....the calculated bounds. 1 Introduction Analysis of program running time is important for real time systems, interactive environments, compiler optimizations, performance evaluation, and many other computer applications. It has been extensively studied in many fields of computer science: algorithms [20, 12, 13, 41], programming languages [38, 21, 30, 33] and systems [35, 28, 32, 31] It is particularly important for many applications, such as real time systems, to be able to predict accurate time bounds automatically and efficiently, and it is particularly desirable to be able to do so for high level ....

....independent of input size, i.e. are constants while the computation iterates or recurses. Whatever values of the primitive parameters are assumed, a second problem arises, and it is theoretically challenging: optimizing the time bound function to a closed form in terms of the input size [38, 7, 21, 30, 13]. Although much progress has been made in this area, closed forms are known only for subclasses of functions. Thus, such optimization can not be automatically done for analyzing general programs. In systems, inputs are characterized indirectly using loop bounds or execution paths in programs, and ....

[Article contains additional citation context not shown here]

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic average-case analysis of algorithms. Theoretical Computer Science, Series A, 79(1):37--109, Feb. 1991.

....contain, at least in principle, all the necessary information needed to predict an algorithm s behaviour. This part is strongly influenced by an approach introduced earlier for the automatic analysis of some classes of algorithms over decomposable structures by Flajolet, Salvy, and Zimmermann [11]. The real dimension of generating functions lies in their complex analytic properties, especially when we contemplate them near singularities. Without this aspect, the collection of formal generating function equations would remain somewhat devoid of content. A systematic analysis of ....

....lies in their complex analytic properties, especially when we contemplate them near singularities. Without this aspect, the collection of formal generating function equations would remain somewhat devoid of content. A systematic analysis of singularities, at either a finite or infinite distance [10, 11], permits us to extract in all cases of practical interest the asymptotic costs involved. For instance, two different strategies, nicknamed little endian and big endian , when applied to Cayley trees (non plane labelled trees) lead to an average complexity of the form O(n ) and It is to ....

[Article contains additional citation context not shown here]

Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (Feb. 1991), 37--109.

.... robustness (as defined above) and the density of the graph 2 (i.e. the number of its edges) The originality of our approach consists in introducing in this range of problems methods of analytic combinatorics [14, 21] and recent research in automatic analysis based on symbolic computation [8, 13, 15, 23]. Additional threshold estimates regarding properties of multiple sourcedestination pairs are discussed in the last section of the paper. Summary of results. From earlier known results [7, 20] and this paper, a picture of robustness under the G n;p model emerges. As is usual in random graph ....

....into smaller structures either of the same type or of simpler types and then in extracting, from such a decomposition, the corresponding recurrence relations. The approach developed here is direct and symbolic , as it relies on a precise specification language for combinatorial structures [13, 15]. It is based on so called admissible constructions that have the important feature of admitting direct translations into generating functions. Let A be a class of combinatorial objects with an associated notion of size. We let A n denote 1 the subset of objects in A that have size n and write ....

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Ph. Flajolet, B. Salvy, and P. Zimmermann, "Automatic Average--case Analysis of Algorithms", Theoretical Computer Science 79(1), pp. 37--109, 1991.

....with a focus on order constraints and analysis of algorithms. Flajolet and Steyaert developed rules initially specialized to trees from which a complexity calculus could be derived for a wide class of algorithms [22, 24, 36, 37, 75] This was later extended into a much more general system [32] to be discussed in Part III. We propose now to explain the major principle of a symbolic approach to the derivation of generating functions. 5 1. Trains. 2. Formal specification. 8 : train = locomotive wagons) wagons = ....

....is compiled from the specification [x3] The generating function is then solved explicitly [x4] Currently, the analysis of this problem can be achieved automatically. A system, Lambda Upsilon Omega ( Upsilon Omega ) has been designed by B. Salvy and P. Zimmermann jointly with the author [32, 69, 84]. It does the analysis and via an implementation of complex asymptotic methods and singularity analysis, it is also able to find automatically the asymptotic form of the coefficients: The number of trains of size n satisfies the estimate trainn n 0:07097007911 Delta 1:930298068 n : 6 ....

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Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (February 1991), 37--109. 22

....contain, at least in principle, all the necessary information needed to predict an algorithm s behaviour. This part is strongly influenced by an approach introduced earlier for the automatic analysis of some classes of algorithms over decomposable structures by Flajolet, Salvy, and Zimmermann [11]. The real dimension of generating functions lies in their complex analytic properties, especially when we contemplate them near singularities. Without this aspect, the collection of formal generating function equations would remain somewhat devoid of content. A systematic analysis of ....

....lies in their complex analytic properties, especially when we contemplate them near singularities. Without this aspect, the collection of formal generating function equations would remain somewhat devoid of content. A systematic analysis of singularities, at either a finite or infinite distance [10, 11], permits us to extract in all cases of practical interest the asymptotic costs involved. For instance, two different strategies, nicknamed little endian and big endian , when applied to Cayley trees (non plane labelled trees) lead to an average complexity of the form O(n 3=2 ) and 1 It ....

[Article contains additional citation context not shown here]

Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (Feb. 1991), 37--109.

....see later. Bibliography Modern presentations of combinatorial analysis appear in the books of Comtet [4] a superb book largely example driven) Stanley [15] algebra and order structures) and Wilf [17] generating functions oriented) An encyclopedic reference is the book of Jackson Goulden [8] whose descriptive approach is very much parallel to ours. The framework presented here was used already in some surveys [5, 16] that deal more specifically with the analysis of algorithms. The sources of the modern approaches to combinatorial analysis are hard to trace since these approaches are ....

...., Udine, 1984) 6] Flajolet, P. Salvy, B. and Zimmermann, P. Automatic average case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (Feb. 1991) 37 109. 7] Flajolet, P. and Soria, M. The cycle construction. SIAM Journal on Discrete Mathematics 4, 1 (Feb. 1991) 58 60. [8] Goulden, I. P. and Jackson, D. M. Combinatorial Enumeration. John Wiley, New York, 1983. 9] Harary, F. and Palmer, E. M. Graphical Enumeration. Academic Press, 1973. 10] Joyal, A. Une th eorie combinatoire des s eries formelles. Advances in Mathematics 42, 1 (1981) 1 82. 11] Knuth, D. E. ....

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Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (Feb. 1991), 37--109.

....in this section and the next one is an informal reminder of basic properties of analytic functions intended for asymptotic analysis. For a detailed treatment, we refer the reader to one of the many excellent treatises on the subject, like the books by Dieudonn e [2] Henrici [5] Knopp [6], Titchmarsh [15] or Whittaker and Watson [16] Analytic functions. A function f(z) of the complex variable z is analytic at a point z = a if it is defined in a neighbourhood of z = a and is representable there by a convergent power series expansion f(z) X n0 f n (z Gamma a) n : 4:4) ....

....are many excellent books treating complex analysis and function theory. References most relevant to the discussion given in this chapter are the books of Titchmarsh [15] oriented towards classical analysis) Whitaker and Watson [16] with emphasis on special functions) Dieudonn e [2] and Knopp [6]. Henrici [5] presents complex analysis under the perspective of constructive methods, a valuable point of view here. References dealing specifically with asymptotic analysis are discussed at the end of the next chapter. 42 CHAPTER 4. COMPLEX ASYMPTOTIC METHODS Problems and Exercises Complex ....

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Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science 79, 1 (Feb. 1991), 37--109.

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic Average--case Analysis of Algorithms. Theoretical Computer Science, 79(1):37--109, February 1991.

....We feel that symbolic asymptotics is a young field of symbolic computation which has plenty of potential applications. Functional inversion is one of them and others should follow, such as asymptotic expansions of integrals or of coefficients of generating functions (see [14] for ideas on that and [3] for applications to the automatic analysis of algorithms) Acknowledgements. The second author would like to thank INRIA Rocquencourt, and in particular Philippe Flajolet and Bruno Salvy, for their hospitality during the week 24 30 November, when the groundwork for this paper was laid. This ....

Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science, Series A 79, 1 (Feb. 1991), 37--109.

....analysis, generating functions. 1 Introduction Since Wegbreit s paper [11] different authors tried to formalize the analysis of algorithms: Ramshaw in his PhD Thesis [9] Cohen and Hickey [2, 7] Recent works are more concerned in the design of systems that really perform an automatic analysis [8, 13, 5]. In the particular field of average case analysis, generating functions have proven to be a very useful tool [6] Briefly, programs translate into equations for generating functions (Algebraic Analysis [12] that contain all the required informations for average case analysis; in particular, some ....

....for example in the Lambda Upsilon Omega system. Before we reveal the aim of this paper, let us just give a brief description of this system. Lambda Upsilon Omega (or simply Upsilon Omega ) is an assistant system, that analyzes in the average case some well defined classes of algorithms [4, 5]. To describe an algorithm, one writes a program y Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex. This research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM) in a special purpose language called Adl (Algorithm ....

[Article contains additional citation context not shown here]

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic Average--case Analysis of Algorithms. Theoretical Computer Science, 79(1):37--109, February 1991.

....of Section 1 is the nth Taylor coefficient at the origin of exp X j 0 2z j j(1 Gamma z j ) 1 2z 5z 2 10z 3 20z 4 36z 5 65z 6 110z 7 185z 8 O(z 9 ) Proof. The proof is based on generating function techniques and we select to present it using the framework of [4]. The generating function of a class of combinatorial objects C is C(z) X n0 c n z n ; where c n denotes the number of objects of size n in C (it is assumed here that all the c n s are finite) Combinatorial constructs on classes of objects have direct translations into generating ....

.... are either elements of B or elements of C, B and C having no common element is B(z) C(z) Similar reasoning shows that the generating function of the cartesian product B Theta C is B(z)C(z) A classical lemma gives the generating function of multisets (sets where repetition is allowed) see [2, 4]) Lemma 2 Let A be the class of multisets of elements of B, B having no element of size 0. Then the generating functions A(z) and B(z) of A and B are related by A(z) exp 2 4 X j 0 B(z j ) j 3 5 : From the combinatorial viewpoint, the classification of Section 1 translates directly into ....

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P. Flajolet, B. Salvy, and P. Zimmermann, Automatic average-case analysis of algorithms, Theoretical Computer Science Series A, 79 1, (1991) 37--109.

....tree decomposes recursively as a root node to which is attached a sequence of trees. Let A n be the number of trees with n nodes; the ordinary generating function of the sequence fA n g is defined by A(z) 1 X n=1 A n z n : Then, by the classical laws of combinatorial analysis (see, e.g. [33, 38, 78, 87]) the decomposition translates into a functional equation that admits an explicit solution A(z) z 1 Gamma A(z) and A(z) 1 Gamma p 1 Gamma 4z 2 : We have A n 1 = C n with C n the Catalan number of (37) Let similarly A h (z) be the generating function of trees of height at most h. As ....

Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average--case analysis of algorithms. Theoretical Computer Science 79, 1 (Feb. 1991), 37--109.

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P. Flajolet, B. Salvy, and P. Zimmermann. Automatic Average-Case Analysis of Algorithms. Theor. Comp. Sci., (79):37--109, 1991.

No context found.

P. Flajolet, B. Salvy, and P. Zimmermann. Automatic Average-Case Analysis of Algorithms. Theor. Comp. Sci., (79):37-109, 1991.

No context found.

Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average-case analysis of algorithms. Theoretical Computer Science 79, 1 (1991), 37--109.

No context found.

Flajolet, P., Salvy, B., Zimmermann, P.: Automatic Average-Case Analysis of Algorithms. Theoretical Computer Science 79 (1991) 37-109

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Flajolet, P., Salvy, B., and Zimmermann, P. Automatic average-case analysis of algorithms. Theoretical Computer Science 79, 1 (1991), 37--109.

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