4 research outputs found

    An easy subexponential bound for online chain partitioning

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    Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width ww into w14lgww^{14\lg w} chains. We improve this to w6.5lgw+7w^{6.5 \lg w + 7} with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

    On-line partitioning of width w posets into w^O(log log w) chains

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    An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width ww into wO(loglogw)w^{O(\log{\log{w}})} chains. This improves over previously best known algorithms using wO(logw)w^{O(\log{w})} chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in wO(w)nw^{O(\sqrt{w})}n time, where ww is the width and nn is the size of a presented poset.Comment: 16 pages, 10 figure

    A subexponential upper bound for the on-line chain partitioning problem

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