Jean Françon

Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'euler et nombres de Genocchi

We study permutations whose type is given, the type being the sets of the values of the peaks, throughs, doubles rises and double falls. We show that the type of a permutation on n letters is caracterized by a map @c[n]->[n]; the number of possible types is the Catalan number; the number of permutations whose type is associated with @c is the product @c(1)@c(2).@c(n). This result is a corollary of an explicit bijection between permutations and pairs (@c, @?) where @? is a map dominated by @c. Specifying this bijection tG various classes of permutations provides enumerative formulas for classical numbers, e.g. Euler and Genocchi numbers. It has been proved recently that each enumerative formula of this work is equivalent to a continued fraction expansion of a generating serie.

On the topology of an arithmetic plane.

An arithmetic plane is the set of points (x, y, z) in Z3 satisfying the inequalities p < LZX + by + cz < p + w, where all parameters are integers and w > 0. We show that for w = I a ( + 1 b ( + 1 c 1, a plane can be furnished with a canonical structure of a two-dimensional, connected, orientable combinatoric manifold without boundary, whose faces are quadrangles and whose vertices are points on the plane. This result is of interest in three-dimensional computer imaging. 0. Summary in English An arithmetic plane is the set of points (x, y, z) of Z3 satisfying the inequalities p < ax + by + cz < p + w, where all parameters are integers and w > 0. For c = 0 it defines an arithmetic line in H2. The parameter w is called the (arithmetic) width of the plane or the line. These lines are basic in the arithmetical geometry introduced and studied by Jean-Pierre Reveilles [16]. He has shown that the topological properties of a line L are controlled by its * Email.francon@dpt-info.u-strasbg.fr. 0304-3975/96/

Description and analysis of an efficient priority queue representation

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Algebraic Specification and Development in Geometric Modeling

We present a new data-structure for representing priority queues, the pagoda. A detailed analysis shows that the pagoda provides a very efficient implementation of priority queues, where our measure of efficiency is the average run time of the various algorithms. It handles an arbitrary sequence of n primitive operations chosen from MIN, INSERT, UNION, EXTRACT and EXTRACTMIN in time o(n log n). The constant factors affecting these asymptotic run time are small enough to make the pagoda competitive with any other priority queue, including structures which cannot handle UNION or EXTRACT. The given algorithms process an arbitrary sequence of n operations MIN, INSERT and EXTRACT in linear average time O(n), and a sequence of n INSERT in linear worst case time O(n).

Topological 3D-manifolds: a statistical study of the cells

For several years now, the Geometric Modeling Group of Strasbourg has been working on new formal concepts and tools for describing and manipulating the boundary representation of geometric objects. In a large project of an interactive modeller for volumic objects, the description of which is based on generalized maps, it attempts to cover the whole process from mathematical modeling to efficient implementation, via a complete algebraic specification. Basic concepts and results of this experiment in horizontal and vertical software specification and development are presented along with several illustrations. Advances in algebraic specification methodology are highlighted, specially hierarchical construction of ordered sorts and operations.

Elliptic functions, continued fractions and doubled permutations

In the field of Geometric Modelling, as well as in theoretical physics, 2- and 3-combinatorial manifolds are often manipulated. Statistics on the cells of these manifolds are necessary in Geometric Modeling for the complexity analysis of data structures and algorithms dealing with these manifolds. These statistics are known in the 2D case. We study here the 3D case. We consider the set of combinatorial manifolds of dimension 3, without boundary, and with a fixed number V of vertices, E of edges, F of faces, and W of volumes (number of cells of different dimensions), and the average number of edges (resp. faces, volumes) by vertex, the average number of volumes by edge, the average number of vertices by face, and the average number of vertices (resp. of edges, faces) by volume. These quantities are shown to be sufficient to determine all the other quantities studied in this paper. We give some relations between these quantities. We give several expressions of the total number of cells, and the distribution of number of cells in relation with their dimension, with respect to some of these quantities. For the 3-G-map representations of these manifolds we also express the number of darts with respect to these quantities. And we add some hypothesis : H1 : all vertices are orientable and have the same genus G′ (i.e. the dual of each vertex has a genus G′); H2 : all volumes are orientable and have the same genus G. We study the consequences of H1 and H2 on the above relations. Particularly, we obtain a general Euler formula: V(1−G′)−E+F−W(1−G)=0. At last, we study various particular cases of manifolds without boundary, quite regular, and their barycentric triangulations. We also show the following experimental law : when one represents a 3-manifold without boundary by a 3-G-map, the number of darts is about or exactly 6 times the number of cells.

Computing integrated costs of sequences of operations with application to dictionaries

The Taylor coefficients of the Jacobian elliptic functions are shown to count classes of permutations with a simple repetitive order pattern. The proof relies on the use of enumerative properties of continued fractions, and on a mapping between path diagrams and permutations.

Towards analysing sequences of operations for dynamic data structures

We introduce a notion of integrated cost of a dictionary, as average cost of sequences of search, insert and delete operations. We express generating functions of these sequences in terms of continued fractions; from this we derive an explicit integral expression of integrated costs for three common representations of dictionaries.

Dynamic Data Structures with Finite Population: A Combinatorial Analysis

This paper presents the average case performance analysis of dynamic data structures subjected to arbitrary sequences of insert, delete and query operations. To such sequences of operations are associated, for each data type, a specific continued fraction and a familly of orthogonal polynomials : Tchebycheff for stacks, Laguerre for dictionaries, Hermite for priority queues, Meixner for linear lists and Charlier for symbol tables. We define a notion of integrated cost of a data structure as the average cost over all possible sequences of operations. Our main result is an explicit expression, for each of these data structures, of the generating function for integrated costs as a linear integral transform of the generating functions for individual operation costs. We use the result to explicitly compute integrated costs of various efficient data structure implementations.

Analysis of Dynamic Algorithms in D. E. Knuth's Model

This paper analyzes the average behaviour of algorithms that operate on dynamically varying data structures subject to insertions I, deletions D, positive (resp. negative) queries Q+ (resp.Q−) under the following assumptions: i) the universe of keys is finite: U [N]={1, 2, 3,..., N} ii) if the size of the data structure is k (k≤N), then the number of possibilities for the operations D and Q+ is k, whereas the number of possibilities for the i-th insertion or negative query is equal to N-i+1 for i≤N.