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Finding Range Minima in the Middle: Approximations and Applications

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Abstract

A Range Minimum Query asks for the position of a minimal element between two specified array-indices. We consider a natural extension of this, where our further constraint is that if the minimum in a query interval is not unique, then the query should return an approximation of the median position among all positions that attain this minimum. We present a succinct preprocessing scheme using Dn + o(n) bits in addition to the static input array (small constant D), such that subsequent “range median of minima queries” can be answered in constant time. This data structure can be built in linear time, with little extra space needed at construction time. We introduce several new combinatorial concepts such as Super-Cartesian Trees and Super-Ballot Numbers. We give applications of our preprocessing scheme in text indexes such as (compressed) suffix arrays and trees.

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Authors and Affiliations

  1. Center for Bioinformatics (ZBIT), Universität Tübingen, Sand 14, 72076, Tübingen, Germany

    Johannes Fischer

  2. Institut für Informatik, Ludwig-Maximilians-Universität München, Amalienstr. 17, 80333, Munich, Germany

    Volker Heun

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  1. Johannes Fischer
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  2. Volker Heun
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Correspondence to Johannes Fischer.

Additional information

Work on this article was partially funded by the German Research Foundation (DFG, Bioinformatics Initiative).

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Fischer, J., Heun, V. Finding Range Minima in the Middle: Approximations and Applications. Math.Comput.Sci. 3, 17–30 (2010). https://doi.org/10.1007/s11786-009-0007-8

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  • DOI: https://doi.org/10.1007/s11786-009-0007-8

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