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New Results
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Section: New Results

Analysis of Splitting Algorithms

Participants : Hanène Mohamed, Philippe Robert.

Algorithms with an underlying tree structure are quite common in computer science and communication networks. Splitting algorithms are examples of such algorithms.

A splitting algorithm is a procedure that divides recursively into subgroups an initial group of nitems until each of the subgroups obtained has a cardinality strictly less than some fixed number D. A common problem is, given an initial number nof requests, to estimate the time it takes to complete the algorithm. In the language of trees, it amounts to give an asymptotic expression of the number Rn of nodes of the corresponding tree.

Dynamic tree algorithm

This is a dynamic version of a class of algorithms analyzed by Mohamed and Robert  [8] . The splitting procedure is the same, but a phenomenon of immigration has to be considered : on every leaf of the associated tree, every time unit, new messages arrive following a Poisson process of parameter $ \lambda$ . Contrary to the « static case », the boundary conditions turn out to complicate a lot the resolution of the problem. A new probabilistic tool has to be used ; an auto-regressive process whose invariant density plays an important role to determine the asymptotic behavior of the cost of the algorithm.

Leader election algorithm

A related algorithm, a leader election algorithm, has been analyzed. It has been previously investigated by Janson and Szpankowski with analytic methods. This algorithm is used in the context of a distributed system of nstations sharing a common channel of communication that can transmit only one message by unit of time. We assume that every station which sends a message to network can listen at the same time to the channel and so discern one of three possible information on the state of this one ; a collision when it there at least two tries of transmission, a silence when none of the stations tried to send its message or a success when exactly one station tries transmission. Question is then how these stations can, by using the same protocol, identify one of them as a leader to coordinate the whole system ?

Such algorithm based on a process of random elimination has varied applications in distributed systems field such as cell telephones and networks of wireless communications. The problem of leader election in networks computer science is fundamental to assure communications and synchronization of the different components of the system. This problem was also studied in the context of radio networks.

Formally, the algorithm of leader election divides an initial group of nitems into two subgroups, eliminates one of two and continues the same process until finding one leader . If at a given level kall items are eliminated, algorithm starts again from the previous level k-1 .

Our study, based on probabilistic techniques, allows to simplify the analysis of such algorithm and especially to eliminate the implicit dependency of its asymptotic behavior as it is the case in the expression established by Janson and Szpankowski. Besides, an explicit representation of the associated oscillation phenomenon has also been obtained. These results are obtained via a careful analysis of the following probabilistic functional equation

Im15 ${h(x)=h{(px)}+h{(qx)}~e^{-px}+{f(x)}=\#120124 \mfenced o=( c=) {h(Ax)~}e^{-p1_{{A=q}}}+{f(x),}}$

where Ais a random variable of distribution Im16 ${\#119986 =p~\#948 _pq~\#948 _q}$ and fa given function. The use of a simple iterative scheme gives an explicit expression of the average cost of the algorithm Im17 ${{\#120124 (}H_n{)}}$ . It is proved that the centered average cost of the algorithm Im18 ${H_n-\#8970 -log_p{(n)\#8971 }}$ is asymptotically identical to a periodical function F, whose explicit expression is known, of -log p( n)

Im19 ${{\#120124 (}H_n{)-\#8970 -}log_p{(n)\#8971 =F(-}log_p{(n))+O(1/n).}}$

Extensions to Stationary Sequences

The results of Mohamed and Robert  [8] have been extended to the case where the branching procedure are not independent but are driven by a dynamical system. These results are known (See Vallée and its co-workers) to hold for some dynamical systems generated by the iterations of some function on [0, 1] . Our approach gives a further extension to general dynamical systems. It uses a general version of the renewal theorem for stationary sequences together with a representation of the cost function as counting functional.


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