Endre Szemerédi

Storing a Sparse Table with 0 (1) Worst Case Access Time

We describe a data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time. Both the data structure and the query algorithm are easy to implement.

An 0(n log n) sorting network

The purpose of this paper is to describe a sorting network of size 0(n log n) and depth 0(log n). A natural way of sorting is through consecutive halvings: determine the upper and lower halves of the set, proceed similarly within the halves, and so on. Unfortunately, while one can halve a set using only 0(n) comparisons, this cannot be done in less than log n (parallel) time, and it is known that a halving network needs (½)n log n comparisons. It is possible, however, to construct a network of 0(n) comparisons which halves in constant time with high accuracy. This procedure (ε-halving) and a derived procedure (ε-nearsort) are described below, and our sorting network will be centered around these elementary steps.

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.

Many hard examples for resolution

For every choice of positive integers <italic>c</italic> and <italic>k</italic> such that <italic>k</italic> ≥ 3 and <italic>c</italic>2<supscrpt>-<italic>k</italic></supscrpt> ≥ 0.7, there is a positive number ε such that, with probability tending to 1 as <italic>n</italic> tends to ∞, a randomly chosen family of <italic>cn</italic> clauses of size <italic>k</italic> over <italic>n</italic> variables is unsatisfiable, but every resolution proof of its unsatisfiability must generate at least (1 + ε)<supscrpt><italic>n</italic></supscrpt> clauses.

Extremal problems in discrete geometry

AbstractIn this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdõs, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc1 so that when

Crossing-Free Subgraphs

\sqrt n \leqq t \leqq \left( {_2^n } \right)

On the second eigenvalue of random regular graphs

, the number of incidences betweenn points andt lines is less thanc1n2/3t2/3. Using this result, it follows immediately that there exists an absolute constantc2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc2n2/k3. We then prove that there exists an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points. Finally, we show that there is an absolute constantc4 so that there are less than exp (c4 √n) sequences 2≦y1≦y2≦...≦yr for which there is a set ofn points and a setl1,l2, ...,lt oft lines so thatlj containsyj points.

On The Complexity Of Matrix Group Problems I

Abstract Upper bounds are found for the Ramsey function. We prove R(3, x) cx 2 ln x and, for each k ⩾ 3, R(k, x) c k x k − 1 ( ln x) k − 2 asymptotically in x .