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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

For any epsilon in (0,1), a (1+epsilon)-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor (1+epsilon) smaller than the true mode. For this problem, we design a data structure occupying O(n/epsilon) bits of space to answer queries in O(lg(1/epsilon)) time. This is an encoding data structure which does not require access to the input sequence; the space cost of this structure is asymptotically optimal for constant epsilon as we also prove a matching lower bound. Furthermore, our solution improves the previous best result of Greve et al. (Cell Probe Lower Bounds and Approximations for Range Mode, ICALP'10) by saving the space cost by a factor of lg n while achieving the same query time. In dynamic settings, we design an O(n)-word data structure that answers queries in O(lg n /lg lg n) time and supports insertions and deletions in O(lg n) time, for any constant epsilon in (0,1); the bounds for non-constant epsilon = o(1) are also given in the paper. This is the first result on dynamic approximate range mode; it can also be used to obtain the first static data structure for approximate 3-sided range mode queries in two dimensions.
Another problem we consider is approximate range selection. For any alpha in (0,1/2), an alpha-approximate range selection query asks for the position of an element whose rank in the query range is in [k - alpha s, k + alpha s], where k is a rank given by the query and s is the size of the query range. When alpha is a constant, we design an O(n)-bit encoding data structure that can answer queries in constant time and prove this space cost is asymptotically optimal. The previous best result by Krizanc et al. (Range Mode and Range Median Queries on Lists and Trees, Nordic Journal of Computing, 2005) uses O(n lg n) bits, or O(n) words, to achieve constant approximation for range median only. Thus we not only improve the space cost, but also provide support for any arbitrary k given at query time. We also analyse our solutions for non-constant alpha.

Hicham El-Zein, Meng He, J. Ian Munro, Yakov Nekrich, and Bryce Sandlund. On Approximate Range Mode and Range Selection. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{elzein_et_al:LIPIcs.ISAAC.2019.57, author = {El-Zein, Hicham and He, Meng and Munro, J. Ian and Nekrich, Yakov and Sandlund, Bryce}, title = {{On Approximate Range Mode and Range Selection}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {57:1--57:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.57}, URN = {urn:nbn:de:0030-drops-115531}, doi = {10.4230/LIPIcs.ISAAC.2019.57}, annote = {Keywords: data structures, approximate range query, range mode, range median} }

Document

**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Given an array A of n elements, we wish to support queries for the most frequent and least frequent element in a subrange [l, r] of A. We also wish to support updates that change a particular element at index i or insert/ delete an element at index i. For the range mode problem, our data structure supports all operations in O(n^{2/3}) deterministic time using only O(n) space. This improves two results by Chan et al. [Timothy M. Chan et al., 2014]: a linear space data structure supporting update and query operations in O~(n^{3/4}) time and an O(n^{4/3}) space data structure supporting update and query operations in O~(n^{2/3}) time. For the range least frequent problem, we address two variations. In the first, we are allowed to answer with an element of A that may not appear in the query range, and in the second, the returned element must be present in the query range. For the first variation, we develop a data structure that supports queries in O~(n^{2/3}) time, updates in O(n^{2/3}) time, and occupies O(n) space. For the second variation, we develop a Monte Carlo data structure that supports queries in O(n^{2/3}) time, updates in O~(n^{2/3}) time, and occupies O~(n) space, but requires that updates are made independently of the results of previous queries. The Monte Carlo data structure is also capable of answering k-frequency queries; that is, the problem of finding an element of given frequency in the specified query range. Previously, no dynamic data structures were known for least frequent element or k-frequency queries.

Hicham El-Zein, Meng He, J. Ian Munro, and Bryce Sandlund. Improved Time and Space Bounds for Dynamic Range Mode. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{elzein_et_al:LIPIcs.ESA.2018.25, author = {El-Zein, Hicham and He, Meng and Munro, J. Ian and Sandlund, Bryce}, title = {{Improved Time and Space Bounds for Dynamic Range Mode}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {25:1--25:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.25}, URN = {urn:nbn:de:0030-drops-94886}, doi = {10.4230/LIPIcs.ESA.2018.25}, annote = {Keywords: dynamic data structures, range query, range mode, range least frequent, range k-frequency} }

Document

**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

In this paper we present a succinct data structure for the dynamic one-dimensional range reporting problem. Given an interval [a,b] for some a,b in [m], the range reporting query on an integer set S subseteq [m] asks for all points in S cap [a,b]. We describe a data structure that answers reporting queries in optimal O(k+1) time, where k is the number of points in the answer, and supports updates in O(lg^epsilon m) expected time. Our data structure uses B(n,m) + o(B(n,m)) bits where B(n,m) is the minimum number of bits required to represent a set of size n from a universe of m elements. This is the first dynamic data structure for this problem that uses succinct space and achieves optimal query time.

Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct Dynamic One-Dimensional Point Reporting. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 17:1-17:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{elzein_et_al:LIPIcs.SWAT.2018.17, author = {El-Zein, Hicham and Munro, J. Ian and Nekrich, Yakov}, title = {{Succinct Dynamic One-Dimensional Point Reporting}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {17:1--17:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.17}, URN = {urn:nbn:de:0030-drops-88438}, doi = {10.4230/LIPIcs.SWAT.2018.17}, annote = {Keywords: Succinct Data Structures, Range Searching, Computational Geometry} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

In this paper we study succinct data structures for one-dimensional color reporting and color counting problems.
We are given a set of n points with integer coordinates in the range [1,m] and every point is assigned a color from the set {1,...\sigma}.
A color reporting query asks for the list of distinct colors that occur in a query interval [a,b] and a color counting query asks for the number of distinct colors in [a,b].
We describe a succinct data structure that answers approximate color counting queries in O(1) time and uses \mathcal{B}(n,m) + O(n) + o(\mathcal{B}(n,m)) bits,
where \mathcal{B}(n,m) is the minimum number of bits required to represent an arbitrary set of size n from a universe of m elements. Thus we show, somewhat counterintuitively,
that it is not necessary to store colors of points in order to answer approximate color counting queries.
In the special case when points are in the rank space (i.e., when n=m), our data structure needs only O(n) bits.
Also, we show that \Omega(n) bits are necessary in that case.
Then we turn to succinct data structures for color reporting.
We describe a data structure that uses \mathcal{B}(n,m) + nH_d(S) + o(\mathcal{B}(n,m)) + o(n\lg\sigma) bits and answers queries in O(k+1) time,
where k is the number of colors in the answer, and nH_d(S) (d=\log_\sigma n) is the d-th order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates. Our dynamic data structure uses nH_d(S)+o(n\lg\sigma) bits and supports queries in O(k+1) time.

Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct Color Searching in One Dimension. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{elzein_et_al:LIPIcs.ISAAC.2017.30, author = {El-Zein, Hicham and Munro, J. Ian and Nekrich, Yakov}, title = {{Succinct Color Searching in One Dimension}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {30:1--30:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.30}, URN = {urn:nbn:de:0030-drops-82096}, doi = {10.4230/LIPIcs.ISAAC.2017.30}, annote = {Keywords: Succinct Data Structures, Range Searching, Computational Geometry} }

Document

**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Given a permutation of n elements, stored as an array, we address the problem of replacing the permutation by its kth power. We aim to perform this operation quickly using o(n) bits of extra storage. To this end, we first present an algorithm for inverting permutations that uses O(lg^2 n) additional bits and runs in O(n lg n) worst case time. This result is then generalized to the situation in which the permutation is to be replaced by its kth power. An algorithm whose worst case running time is O(n lg n) and uses O(lg^2 n + min{k lg n, n^{3/4 + epsilon}}) additional bits is presented.

Hicham El-Zein, J. Ian Munro, and Matthew Robertson. Raising Permutations to Powers in Place. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 29:1-29:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{elzein_et_al:LIPIcs.ISAAC.2016.29, author = {El-Zein, Hicham and Munro, J. Ian and Robertson, Matthew}, title = {{Raising Permutations to Powers in Place}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {29:1--29:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.29}, URN = {urn:nbn:de:0030-drops-67992}, doi = {10.4230/LIPIcs.ISAAC.2016.29}, annote = {Keywords: Algorithms, Combinatorics, Inplace, Permutations, Powers} }

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