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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the dynamic range mode problem, we are given a sequence a of length bounded by N and asked to support element insertion, deletion, and queries for the most frequent element of a contiguous subsequence of a. In this work, we devise a deterministic data structure that handles each operation in worst-case Õ(N^0.655994) time, thus breaking the O(N^{2/3}) per-operation time barrier for this problem. The data structure is achieved by combining the ideas in Williams and Xu (SODA 2020) for batch range mode with a novel data structure variant of the Min-Plus product.

Bryce Sandlund and Yinzhan Xu. Faster Dynamic Range Mode. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 94:1-94:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{sandlund_et_al:LIPIcs.ICALP.2020.94, author = {Sandlund, Bryce and Xu, Yinzhan}, title = {{Faster Dynamic Range Mode}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {94:1--94:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.94}, URN = {urn:nbn:de:0030-drops-125018}, doi = {10.4230/LIPIcs.ICALP.2020.94}, annote = {Keywords: Range Mode, Min-Plus Product} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We consider the minimum cut problem in undirected, weighted graphs. We give a simple algorithm to find a minimum cut that 2-respects (cuts two edges of) a spanning tree T of a graph G. This procedure can be used in place of the complicated subroutine given in Karger’s near-linear time minimum cut algorithm [Karger, 2000]. We give a self-contained version of Karger’s algorithm with the new procedure, which is easy to state and relatively simple to implement. It produces a minimum cut on an m-edge, n-vertex graph in O(m log³ n) time with high probability, matching the complexity of Karger’s approach.

Nalin Bhardwaj, Antonio J. Molina Lovett, and Bryce Sandlund. A Simple Algorithm for Minimum Cuts in Near-Linear Time. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhardwaj_et_al:LIPIcs.SWAT.2020.12, author = {Bhardwaj, Nalin and Molina Lovett, Antonio J. and Sandlund, Bryce}, title = {{A Simple Algorithm for Minimum Cuts in Near-Linear Time}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {12:1--12:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.12}, URN = {urn:nbn:de:0030-drops-122594}, doi = {10.4230/LIPIcs.SWAT.2020.12}, annote = {Keywords: minimum cut, sparsification, near-linear time, packing} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

A lattice is a partially-ordered set in which every pair of elements has a unique meet (greatest lower bound) and join (least upper bound). We present new data structures for lattices that are simple, efficient, and nearly optimal in terms of space complexity.
Our first data structure can answer partial order queries in constant time and find the meet or join of two elements in O(n^{3/4}) time, where n is the number of elements in the lattice. It occupies O(n^{3/2}log n) bits of space, which is only a Θ(log n) factor from the Θ(n^{3/2})-bit lower bound for storing lattices. The preprocessing time is O(n²). This structure admits a simple space-time tradeoff so that, for any c ∈ [1/2, 1], the data structure supports meet and join queries in O(n^{1-c/2}) time, occupies O(n^{1+c}log n) bits of space, and can be constructed in O(n² + n^{1+3c/2}) time.
Our second data structure uses O(n^{3/2}log n) bits of space and supports meet and join in O(d (log n)/(log d)) time, where d is the maximum degree of any element in the transitive reduction graph of the lattice. This structure is much faster for lattices with low-degree elements.
This paper also identifies an error in a long-standing solution to the problem of representing lattices. We discuss the issue with this previous work.

J. Ian Munro, Bryce Sandlund, and Corwin Sinnamon. Space-Efficient Data Structures for Lattices. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{munro_et_al:LIPIcs.SWAT.2020.31, author = {Munro, J. Ian and Sandlund, Bryce and Sinnamon, Corwin}, title = {{Space-Efficient Data Structures for Lattices}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {31:1--31:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.31}, URN = {urn:nbn:de:0030-drops-122782}, doi = {10.4230/LIPIcs.SWAT.2020.31}, annote = {Keywords: Lattice, Partially-ordered set, Space-efficient data structure, Succinct data structure} }

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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} }

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**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} }

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