WADS 2015: Algorithms and Data Structures pp 127-139

Interval Selection in the Streaming Model

• Sergio Cabello
• Pablo Pérez-Lantero
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

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.

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