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. Let $α(\mathbb{I})$ denote the cardinality of an optimal solution. We discuss the estimation of $α(\mathbb{I})$ in the streaming model, where we only have one-time, sequential access to the input intervals, 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 in the data stream model that computes an estimate $\hatα$ of $α(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 12(1-\varepsilon) α(\mathbb{I}) \le \hatα\le α(\mathbb{I})$. For same-length intervals, we provide another algorithm in the data stream model that computes an estimate $\hatα$ of $α(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 23(1-\varepsilon) α(\mathbb{I}) \le \hatα\le α(\mathbb{I})$. 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 of storage. We also develop new, approximate solutions to the interval selection problem, where we want to report a feasible solution, that use $O(α(\mathbb{I}))$ space. Our algorithms for the interval selection problem match the optimal results by Emek, Halld{ó}rsson and Ros{é}n [Space-Constrained Interval Selection, ICALP 2012], but are much simpler.