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 $\alpha(\mathbb{I})$ denote the cardinality of an optimal solution. We discuss the estimation of $\alpha(\mathbb{I})$ 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 $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $$\tfrac 12(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$$. For same-length intervals, we provide another algorithm in the data stream model that computes an estimate $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $$\tfrac 23(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$$. The space used by our algorithms is bounded by a polynomial in $\varepsilon<sup>{-1}$ 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 $O(\alpha(\mathbb{I}))$ space. Our algorithms for the interval selection problem match the optimal results by Emek, Halld{\'o}rsson and Ros{\'e}n [Space-Constrained Interval Selection, ICALP 2012], but are much simpler.