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The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdo&huml;s over the past half century. Here I will explore a particular set of problems - all dealing(More)
The frequency moments of a sequence containing m i elements of type i, for 1 ≤ i ≤ n, are the numbers F k = n i=1 m k i. We consider the space complexity of randomized algorithms that approximate the numbers F k , when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F 0 , F 1 and F 2 can be(More)
Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform(More)
Let P be a property of graphs. An-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than n 2 edges(More)
A simple parallel randomized algorithm to find a maximal independent set in a graph G = (V, E) on n vertices is presented. Its expected rmming time on a concurrent-read concurrent-write PRAM with 0(1 E 1 d,,) processors is O(log n), where d,, denotes the maximum degree. On an exclusive-read exclusive-write PRAM with 0(1 El) processors the algorithm runs in(More)
The Regularity Lemma of Szemer edi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we rst demonstrate the computational diiculty of nding a regular partition; we show that deciding if a given partition of an input graph satisses the(More)
A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a (G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a (G) ≤ 16∆(G) for any graph G (see(More)