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 $\mathbb I$ of intervals and we want to find an independent subset of intervals of largest cardinality, denoted $\alpha (\mathbb I)$. We discuss the estimation of $\alpha (\mathbb I)$ in the streaming model, where we only have one-time, sequential access to $\mathbb I$, the endpoints of the intervals lie in $\{ 1,\dots ,n \}$, and the amount of the memory is constrained.

For intervals of different sizes, we provide an algorithm that computes an estimate $\hat{\alpha }$ of $\alpha (\mathbb I)$ that, with probability at least 2 / 3, satisfies $\tfrac{1}{2}(1-\varepsilon ) \alpha (\mathbb I) \le \hat{\alpha }\le \alpha (\mathbb I)$. For same-length intervals, we provide another algorithm that computes an estimate $\hat{\alpha }$ of $\alpha (\mathbb I)$ that, with probability at least 2 / 3, satisfies $\tfrac{2}{3}(1-\varepsilon ) \alpha (\mathbb I) \le \hat{\alpha }\le \alpha (\mathbb I)$. The space used by our algorithms is bounded by a polynomial in $\varepsilon ^{-1}$ and $\log n$. We also show that no better estimations can be achieved using o(n) bits.

The full version is online at the arXiv repository [2].

S. Cabello—Supported by the Slovenian Research Agency, program P1-0297, projects J1-4106 and L7-5459; by the ESF EuroGIGA project (project GReGAS) of the European Science Foundation. Part of the work was done while visiting Universidad de Valparaíso.

P. Pérez-Lantero—Supported by project Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 (Chile).