Miklós Ajtai

Sorting in c log n parallel steps

We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the algorithm we compare the contents of registersRj(i), andRk(i), wherej(i), k(i) are absolute constants then change their contents or not according to the result of the comparison.

On optimal matchings

Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.

A Dense Infinite Sidon Sequence

But the task of constructing a denser sequence has so far resisted all efforts, both constructive and random methods. Here we use a random construction for giving a sequence that is slightly denser than the above trivial one. (However Erdos conjectures that even fs(n» n!-e is possible.) Lemma 2 is of independent interest for a graph-theorist. We also remark that using random construction Erdos and Renyi [4] proved the existence of an infinite sequence S with fs(n) > cn l-e (for all n) such that the number of solutions of the equation

Largest random component of a k-cube

AbstractLetCk denote the graph with vertices (ɛ1, ...,ɛk),ɛi=0,1 and vertices adjacent if they differ in exactly one coordinate. We callCk thek-cube.LetG=Gk, p denote the random subgraph ofCk defined by letting

Extremal Uncrowded Hypergraphs

Prob(\{ i,j\} \in G) = p

Log n)sorting network

for alli, j ∈ Ck and letting these probabilities be mutually independent.We show that forp=λ/k, λ>1,Gk, p almost surely contains a connected component of sizec2k,c=c(λ). It is also true that the second largest component is of sizeo(2k).

There is no fast single hashing algorithm

A random graph with (1+ε)n/2 edges contains a path of lengthcn. A random directed graph with (1+ε)n edges contains a directed path of lengthcn. This settles a conjecture of Erdõs.

A Tribute to Paul Erdős: Generating expanders from two permutations

Abstract Let G be a (k + 1)-graph (a hypergraph with each hyperedge of size k + 1) with n vertices and average degreee t. Assume k ⪡ t ⪡ n. If G is uncrowded (contains no cycle of size 2, 3, or 4) then there exists and independent set of size c k ( n t )( ln t) 1 k .