Interval selection in the streaming model

Abstract

A set of intervals is independent when the intervals are pairwise disjoint. In the interval selection problem, we are given a set II of intervals and we want to find an independent subset of intervals of largest cardinality, denoted α(II).We discuss the estimation of α(II) in the streaming model, where we only have one-time, sequential access to II, the endpoints of the intervals lie in {1,..., n}, and the amount of the memory is constrained. For intervals of different sizes, we provide an algorithm that computes an estimate (image found) of α(II) that, with probability at least 2/3, satisfies 1/2 (1 − ε)α(II) ≤(image found) ≤ α(II). For same-length intervals, we provide another algorithm that computes an estimate (image found) of α(II) that, with probability at least 2/3, satisfies 2/3 (1 − ε)α(II) ≤(image found) ≤ α(II). The space used by our algorithms is bounded by a polynomial in ε−1 and log n. We also show that no better estimations can be achieved using o(n) bits.