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This report is a contributed chapter to the Handbook of Theoretical Computer Science (North-Holland, 1990). Its aim is to describe the main mathematical methods and applications in the average-case analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact "analytic models" can be derived.

This paper is concerned with the time-analysis of functional programs. Techniques which enable us to reason formally about a program's execution costs have had relatively little attention in the study of functional programming. We concentrate here on the construction of equations which compute the time-complexity of expressions in a lazy higher-order language. The problem with higher-order functions is that complexity is dependent on the cost of applying functional parameters. Structures called cost-closures are introduced to allow us to model both functional parameters and the cost of their application. The problem with laziness is that complexity is dependent on context. Projections are used to characterise the context in which an expression is evaluated, and cost-equations are parameterised by this context-description to give a compositional time-analysis. Using this form of context information we introduce two types of time-equation: sufficient-time equations and nece...

We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2-connected and has approximately 2n edges. We present a rst attack on the problem of describing what the random planar graph looks like.

. Lambda-Upsilon-Omega, \Upsilon\Omega , is a system designed to perform automatic analysis of well-defined classes of algorithms operating over "decomposable" data structures. It consists of an `Algebraic Analyzer' System that compiles algorithms specifications into generating functions of average costs, and an `Analytic Analyzer' System that extracts asymptotic informations on coefficients of generating functions. The algebraic part relies on recent methodologies in combinatorial analysis based on systematic correspondences between structural type definitions and counting generating functions. The analytic part makes use of partly classical and partly new correspondences between singularities of analytic functions and the growth of their Taylor coefficients. The current version \Upsilon\Omega 0 of \Upsilon\Omega implements as basic data types, term trees as encountered in symbolic algebra systems. The analytic analyzer can treat large classes of functions with explicit expressio...

Abstract. Any tree can be represented in a max/ma//y compact form as a directed acyclic graph where common subtrees are factored and shared, being represented only once. Such a compaction can be effected in linear time. It is used to save storage in implementations of functional programming languages, as well as in symbolic manipulation and computer a/gebra systems. In compiling, the compaction problem is known as the "common subexpression problem " and it plays a central r61e in register allocation, code generation and optimisation. We establish here that, under a variety of probabilistic models, a tree of size n has a compacted form of expected size asymptotically n C.--ogWi- ' where the constant C is explicitly related to the type of trees to be compacted and to the statistical model reflecting tree usage. In particular the savings in storage approach 100 % on average for large structures, which overperforms the commonly used form of sharing that is restmcted to leaves (atoms). Introduction. A tree can be compacted by representing occurrences of repeated subtrees only once. In that case, several pointers will point to the representation of any common subtree,

This paper concerns the uniform random generation and the approximate counting of combinatorial structures admitting an ambiguous description. We propose a general framework to study the complexity of these problems and present some applications to specic classes of languages. In particular, we give a uniform random generation algorithm for nitely ambiguous context-free languages of the same time complexity of the best known algorithm for the unambiguous case. Other applications include a polynomial time uniform random generator and approximation scheme for the census function of (i) languages recognized in polynomial time by one-way nondeterministic auxiliary pushdown automata of polynomial ambiguity and (ii) polynomially ambiguous rational trace languages. Keywords: uniform random generation, approximate counting, contextfree languages, auxiliary pushdown automata, rational trace languages, inherent ambiguity. 1 Introduction In this work we propose a general framewo...

The effect of updating (deletions/insertions) on binary search trees has been an interesting research topic for almost three decades, but in the last five years there have been a few contributions, due partially to the intrinsic difficulty of the involved analysis. Since the problem is quite difficult to be solved in a general fashion, we have restricted ourselves to solve a simpler problem, which shall be considered as an important and necessary basis for further developments. In this paper, we have faced the systematization of the study of deletion algorithms, and deduced the effect that a single random deletion produces in the probability distribution of binary trees of arbitrary size for a wide variety of deletion algorithms. Furthermore, to carry on the analysis, some new tools have been introduced, such as the concepts of strong and weak invariance of probability functions induced by an algorithm. Among others, we have been able to derive interesting results such as an extension ...

In this paper we give a survey of the symbolic operator methods to do statistics on random trees. We present some examples and apply the techniques to find their asymptotic behaviour. 1 Introduction Let us consider a class E of combinatorial objects, let A be an algorithm defined over the class E, and let denote the complexity measure we are interested in. Such a class E of combinatorial objects consists on a set, usually denoted by the same name as the class, and a size measure j \Delta j E : E \Gamma! IN. The subscript E in j \Delta j E will be dropped whenever it is clear from the context. We shall denote by E n the set of objects in E of size n. To analyze the average behaviour of A on an input e 2 E n with respect to measure means to compute A (n) = EfA (e) j e 2 E n g; (1:1) where EfXg denotes the expectation of the random variable X [Knu68, VF90]. By definition of expectation, Equation (1.1) can be written as A (n) = X k k PrfA (e) = k j e 2 E n g = X e2En Prfeg ...

The development of digital systems is particularly challenging, if their correctness depends on the right timing of operations. One approach to enhance the reliability of such systems is model-based development. This allows for a formal analysis throughout all stages of design. Model-based