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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)

We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results. 1. Introduction. The problem of… (More)

- Noga Alon, Assaf Naor
- STOC
- 2004

The <i>cut-norm</i> ||A||<inf>C</inf> of a real matrix A=(a<inf>ij</inf>)<inf>i∈ R,j∈S</inf> is the maximum, over all I ⊂ R, J ⊂ S of the quantity | Σ<inf>i ∈ I, j ∈ J</inf> a<inf>ij</inf>|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here… (More)

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: If G is a directed graph with maximum outdegree d, and if the number of Eulerian subgraphs of G with an even number of edges differs from the number of Eulerian subgraphs with an odd number… (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)

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that lect cliques in graphs. In particular, Razborov showed that detecting cliques of size s in a graph dh m vertices requires monotone circuits of size .Q(m-'/(log m) ~') for fixed s, and size rn ao°~') for ,. :[log ml4J. In this paper we modify the arguments of Razborov to… (More)