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On Cartesian Trees and Range Minimum Queries

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Abstract

We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cache-oblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the two-dimensional version of the range minimum query problem.

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Notes

  1. Two RMQ arrays are different if their range minimum is in different locations for some range.

  2. We follow Seidel [34]. The function α(⋅) is usually defined slightly differently, but all variants are equivalent up to an additive constant.

  3. log∗∗ n is the number of times log∗ function is applied to n to produce a constant, \(\alpha_{k}(n)=\log^{\overbrace{**\cdots*}^{k \mathrm{times}}}n\) and the inverse Ackerman function α(n) is the smallest k such that α k (n) is a constant.

  4. Since the (unknown) B can be greater than the length of S we don’t really scan S for s 2 times. Instead, we scan once a sequence of s copies of S.

  5. This sorting is in the most general comparison model. With integer weights one can use faster sorting algorithms such as Fusion trees [19] in O(lgn) time.

  6. Two matrices are different if their range minima are in different locations for some rectangular range.

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Acknowledgements

G.M. Landau is partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 347/09, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Erik D. Demaine

  2. Department of Computer Science, University of Haifa, Haifa, 31905, Israel

    Gad M. Landau & Oren Weimann

  3. Department of Computer Science and Engineering, NYU-Poly, New York, USA

    Gad M. Landau

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  1. Erik D. Demaine
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  2. Gad M. Landau
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Correspondence to Oren Weimann.

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Demaine, E.D., Landau, G.M. & Weimann, O. On Cartesian Trees and Range Minimum Queries. Algorithmica 68, 610–625 (2014). https://doi.org/10.1007/s00453-012-9683-x

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