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Randomness and Pseudo-Randomness in Discrete Mathematics The discovery, demonstrated in the early work of Paley, Zygmund, Erdos, Kac, Turan, Shannon,
 

Summary: Randomness and Pseudo-Randomness in Discrete Mathematics
Noga Alon
The discovery, demonstrated in the early work of Paley, Zygmund, Erdos, Kac, TurŽan, Shannon,
Szele and others, that deterministic statements can be proved by probabilistic reasoning, led already
in the first half of the century to several striking results in Analysis, Number Theory, Combinatorics
and Information Theory. It soon became clear that the method, which is now called the probabilis-
tic method, is a very powerful tool for proving results in Discrete Mathematics. The early results
combined combinatorial arguments with fairly elementary probabilistic techniques, whereas the de-
velopment of the method in recent years required the application of more sophisticated tools from
probability. The books [10], [54] are two recent texts dealing with the subject.
Most probabilistic proofs are existence, non-constructive arguments. The rapid development of
theoretical Computer Science, and its tight connection to Combinatorics, stimulated the study of the
algorithmic aspects of these proofs. In a typical probabilistic proof, one establishes the existence of a
combinatorial structure satisfying certain properties by considering an appropriate probability space
of structures, and by showing that a randomly chosen point of this space is, with positive probability,
a structure satisfying the required properties. Can we find such a structure efficiently, that is, by
a (deterministic or randomized) polynomial time algorithm ? In several cases the probabilistic
proof provides such a randomized efficient algorithm, and in other cases the task of finding such an
algorithm requires additional ideas. Once an efficient randomized algorithm is found, it is sometimes
possible to derandomize it and convert it into an efficient deterministic one. To this end, certain

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics