Dimitris Achlioptas

Learn More
A classic result of Johnson and Lindenstrauss asserts that any set of <i>n</i> points in <i>d</i>-dimensional Euclidean space can be embedded into <i>k</i>-dimensional Euclidean space where <i>k</i> is logarithmic in <i>n</i> and independent of <i>d</i> so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions(More)
A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space—where k is logarithmic in n and independent of d—so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the(More)
It is widely believed that the probability of satis"ability for random k-SAT formulae exhibits a sharp threshold as a function of their clauses-to-variables ratio. For the most studied case, k = 3, there have been a number of results during the last decade providing upper and lower bounds for the threshold’s potential location. All lower bounds in this vein(More)
In the last few years there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Quite intriguingly, experimental results with various models for generating random CSP instances suggest that the probability of such problems having a solution exhibits a “threshold–like”(More)
Understanding the graph structure of the Internet is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining this graph structure can be a surprisingly difficult task, as edges cannot be explicitly queried. For instance, empirical studies of the network of Internet Protocol (IP)(More)
For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random(More)
Let f k (n; p) denote the probability that the random graph G(n; p) is k-colorable. We show that for every k 3, there exists d k (n) such that for any > 0, lim n!1 f k (n; d k (n) ? n) = 1 and lim n!1 f k (n; d k (n) + n) = 0 : As a result we conclude that for any given value of n the the chromatic number of G(n; d=n) is concentrated in one value for all(More)