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- W1628674930 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 $hatalpha$ of $alpha(mathbb{I})$ that, with probability at least $2/3$, satisfies $tfrac 12(1-varepsilon) alpha(mathbb{I}) le hatalpha le alpha(mathbb{I})$. For same-length intervals, we provide another algorithm in the data stream model that computes an estimate $hatalpha$ of $alpha(mathbb{I})$ that, with probability at least $2/3$, satisfies $tfrac 23(1-varepsilon) alpha(mathbb{I}) le hatalpha le alpha(mathbb{I})$. 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 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." @default.
- W1628674930 created "2016-06-24" @default.
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- W1628674930 date "2015-01-09" @default.
- W1628674930 modified "2023-09-27" @default.
- W1628674930 title "Interval Selection in the Streaming Model" @default.
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