As opposed to probabilistic algorithms (so-called Monte Carlo algorithms) randomized algorithms (also known as Las Vegas algorithms) give the exact solution of a deterministic problem and do not make any statistic assumption on the input data. Only their complexity depends on the random choices, and is evaluated as a probablistic expectation. Since the undesirable configurations are given a negligible statistic weight, randomization leads to simple and efficient algorithms, namely in geometry.
Randomization also became a method for proving combinatorial results. Finally, as a paradox, derandomizing randomized algorithms yields, in some cases, optimal deterministic algorithms.
The Prisme group particularly studied incremental randomized algorithms, and developed the Influence Graph. The Influence Graph is a data structure keeping the history of the construction of a given object. Using such a structure leads to semi-dynamic algorithms, able to insert on-line data that are not known in advance. For most fundamental problems, even a fully dynamic algorithm can be obtained, dealing with insertions as well as deletions of data. |
Another use of randomized techniques was recently done for searching neighbors in very high dimensional spaces in molecular algorithmic.
For most geometric problems, the size of the result varies a lot, not only as a function of the number of input data, but also as a function of the configuration of the input data. For instance, the convex hull of a set of n points in dimension d is a polytope, whose numebr of faces can range from (d+1) ! (for a (d+1)-simplex) to $\Omega (n^ { \lfloor {\frac{d+1}{2}} \rfloor }) $ (for a maximal polytope with n vertices); the number of intersections of a set of n segments in the plane can range from 0 to n (n-1) / 2. So, for these exemples, it is particularly interesting to have sensitive algorithms whose complexity depends on the size of the result. In other problems, other parameters can play a crucial role. The contribution of the Prisme group in this field essentially relates to convex hull and piercing problems (or of coverage, by duality).
Last modified: Mon Nov 29 11:09:39 MET 1999 | Monique Teillaud | Topics | Français |