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The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = ∑n i=1m k i . We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F0, F1 and F2 can be… (More)
We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, in particular,… (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… (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)
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)
Query optimizers rely on fast high quality estimates of re sult sizes in order to select between various join plans Self join sizes of relations provide bounds on the join size of any pairs of such relations It also indicates the degree of skew in the data and has been advocated for several estimation procedures Exact computation of the self join size… (More)
We present a general algebraic technique and discuss some of its numerous applications in Combinatorial Number Theory, in Graph Theory and in Combinatorics. These applications include results in additive number theory and in the study of graph coloring problems. Many of these are known results, to which we present unified proofs, and some results are new.
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.