Packing random intervals

Abstract

A packing of a collection of subintervals of [0, 1] is a pairwise disjoint subcollection of the intervals; its wasted space is the measure of the set of points not covered by the packing.

Consider n random intervals, $I_1, \dots, I_n$, chosen by selecting endpoints independently from the uniform distribution. We strengthen and simplify the results of Coffman, Poonen and Winkler, and we show that, for some universal constant K and for each $t \geq 1$, with probability greater than or equal to $1 - 1/n_t$, there is a packing with wasted space less than or equal to $Kt (\log n)^2 /n$.