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Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

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Abstract

We present \(O(n)\)-space data structures to support various range frequency queries on a given array \(A[0:n-1]\) or tree \(T\) with \(n\) nodes. Given a query consisting of an arbitrary pair of pre-order rank indices \((i,j)\), our data structures return a least frequent element, mode, \(\alpha \)-minority, or top-\(k\) colors (values) of the multiset of elements in the unique path with endpoints at indices \(i\) and \(j\) in \(A\) or \(T\). We describe a data structure that supports range least frequent element queries on arrays in \(O(\sqrt{n / w})\) time, improving the \({\varTheta }(\sqrt{n})\) worst-case time required by the data structure of Chan et al. (SWAT 2012), where \(w \in {\varOmega }(\log n)\) is the word size in bits. We describe a data structure that supports path mode queries on trees in \(O(\log \log n \sqrt{n / w})\) time, improving the \({\varTheta }(\sqrt{n} \log n)\) worst-case time required by the data structure of Krizanc et al. (ISAAC 2003). We describe the first data structures to support path least frequent element queries, path \(\alpha \)-minority queries, and path top-\(k\) color queries on trees in \(O(\log \log n \sqrt{n/w}),\,O(\alpha ^{-1} \log \log n)\), and \(O(k)\) time, respectively, where \(\alpha \in [0,1]\) and \(k \in \{1,\ldots , n\}\) are specified at query time.

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Acknowledgments

The authors thank the anonymous reviewers as well as Djamal Belazzougui for their helpful suggestions. Part of this work was done while the fourth author was visiting the University of Manitoba in July 2012 and February 2013.

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Authors and Affiliations

  1. University of Manitoba, Winnipeg, Canada

    Stephane Durocher & Matthew Skala

  2. Louisiana State University, Baton Rouge, USA

    Rahul Shah

  3. Georgia Institute of Technology, Atlanta, USA

    Sharma V. Thankachan

Authors
  1. Stephane Durocher
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  2. Rahul Shah
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  3. Matthew Skala
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  4. Sharma V. Thankachan
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Corresponding author

Correspondence to Stephane Durocher.

Additional information

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and National Science Foundation (NSF) Grants CCF–1017623 (R. Shah) and CCF–1218904 (R. Shah). Preliminary versions of some of these results appeared at the International Symposium on Mathematical Foundations of Computer Science (MFCS) [18] and the String Processing and Information Retrieval Symposium (SPIRE) [19].

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Durocher, S., Shah, R., Skala, M. et al. Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees. Algorithmica 74, 344–366 (2016). https://doi.org/10.1007/s00453-014-9947-8

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  • DOI: https://doi.org/10.1007/s00453-014-9947-8

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