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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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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics, FMFUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Escuela de Ingeniería Civil en InformáticaUniversidad de ValparaísoValparaísoChile

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