Packing Random Intervals

Let n random intervals I 1 ; : : : ; I n be chosen by selecting end-points independently from the uniform distribution on 0; 1]. A packing is a pairwise disjoint subset of the intervals; its wasted space is the Lebesgue measure of the points of 0; 1] not covered by the packing. In any set of intervals the packing with least wasted space is computa-tionally easy to nd; but its expected wasted space in the random case is not obvious. We show that with high probability for large n, this \best" packing has wasted space O(log 2 n n). It turns out that if the endpoints 0 and 1 are identiied, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced to O(1=n). Interestingly, there is a striking diierence between the sizes of the best packings: about log n intervals in the unit interval case, but usually only one or two arcs in the circle case.