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 I of intervals and we want to find an independent subset of intervals of largest cardinality. Let α(I) denote the cardinality of an optimal solution. We discuss the estimation of α(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 given ε∈(0,1/2) computes an estimate αˆ of α(I) that, with probability at least 2/3, satisfies 12(1−ε)α(I)≤αˆ≤α(I). For same-length intervals, we provide another algorithm in the data stream model that given ε∈(0,1/2) computes an estimate αˆ of α(I) that, with probability at least 2/3, satisfies 23(1−ε)α(I)≤αˆ≤α(I). The space used by our algorithms is bounded by a polynomial in ε−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 the intervals have real endpoints and we want to report a feasible solution, that use O(α(I)) space. Our algorithms for the interval selection problem match the optimal results by Emek, Halldórsson and Rosén [Space-Constrained Interval Selection, TALG 2016], but are much simpler.