FLAJOLET-A

la modélisation de l'analyse combinatoire appliquée à la théorie de la décision

The computational complexity of algorithms (CEC) is a field of mathematics applied to the economy developed by the French mathematician Philippe Flajolet around the middle of the 20th century. Flajolet’s field of study was always that of discrete mathematics, i. e., mathematics that refer to the complementary aspects of the mathematics of the continuous. Objects, categories, integers, set elements, and points in the Cartesian plane are examples of components of discrete mathematics.

Binary systems, combinations, permutations and systematic counts of set elements and combinations of set elements are the subject of the CEC.

Other aspects covered in this field are the generation of random series and the study of their asymptotic properties, the distribution statistics of the elements of a finite set and their direct application to algorithm analysis.

Many people refer to Flajolet as a computational scientist dedicated to analyzing algorithms that took advantage of all the resources of combinatorial analysis.

For decision making problems, it is very useful to analyze comparatively the two ways that a problem can be solved: first, under the approach and the use of the methodologies belonging to the decade of the seventies, and later, in the light of the advances that the theory of Flajolet has meant to the field of algorithm analysis.

In the next set of equations, the first block corresponds to those relations that we would have had in the decade of the seventies; we are trying to get a count of one trajectory. This type of trajectories is called three steps, because the only way you can make a step is of a single unit, of two units, or otherwise, by not making any step at all, that is, step of zero units.

On the basis of analytic combinatorics it is considered that the direction of the steps can be positive or negative, as long as the lower quadrant of the Cartesian plane is not invaded.

For example, in the above trajectory we can see how the function starts with a type 1 step, i.e., a (1), succeeded by a zero type step, a(0), and, subsequently steps types a(-1) , a(1), a(-1), a(0), a(0), a(1), a(1), a(1), a(-1), a(1), a(-1), a(-1), a(-1), a(0).

Under this scheme, the relationships that can be established for the feasible trajectories are:

Recurrence relation:

a(n)=a(n+1)+∑_(k=0)^(n-2)〖a(k)a(n-k-2)〗

a(0)=1

Generating function:

A(z)=∑_(n≥0)〖a(n) z^n 〗

Functional equation

A(z)=1+zA(z)+z2A(z)2

Expression of the Generating Function

A(z)=(1-z-√((1+z)(1-3z)))/(2z^2 )

Expression of the series:

a(n)=∑_(k=0)^(n/2) n!/(k!(k+1)!(n-2k)!)

Asymptotic study of the sum

a(n) ᷉ (3√3)/(2√π)3nn-3/2

FLAJOLET-A deals with this kind of relationships in an very intuitive way.