Document

**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

The B^ε-tree [Brodal and Fagerberg 2003] is a simple I/O-efficient external-memory-model data structure that supports updates orders of magnitude faster than B-tree with a query performance comparable to the B-tree: for any positive constant ε < 1 insertions and deletions take O(1/B^(1-ε) log_B N) time (rather than O(log_BN) time for the classic B-tree), queries take O(log_B N) time and range queries returning k items take O(log_B N + k/B) time. Although the B^ε-tree has an optimal update/query tradeoff, the runtimes are amortized. Another structure, the write-optimized skip list, introduced by Bender et al. [PODS 2017], has the same performance as the B^ε-tree but with runtimes that are randomized rather than amortized. In this paper, we present a variant of the B^ε-tree with deterministic worst-case running times that are identical to the original’s amortized running times.

Rathish Das, John Iacono, and Yakov Nekrich. External-Memory Dictionaries with Worst-Case Update Cost. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 21:1-21:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{das_et_al:LIPIcs.ISAAC.2022.21, author = {Das, Rathish and Iacono, John and Nekrich, Yakov}, title = {{External-Memory Dictionaries with Worst-Case Update Cost}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.21}, URN = {urn:nbn:de:0030-drops-173060}, doi = {10.4230/LIPIcs.ISAAC.2022.21}, annote = {Keywords: Data Structures, External Memory, Buffer Tree} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

We present the first succinct distance oracles for (unweighted) interval graphs and related classes of graphs, using a novel succinct data structure for ordinal trees that supports the mapping between preorder (i.e., depth-first) ranks and level-order (breadth-first) ranks of nodes in constant time. Our distance oracles for interval graphs also support navigation queries - testing adjacency, computing node degrees, neighborhoods, and shortest paths - all in optimal time. Our technique also yields optimal distance oracles for proper interval graphs (unit-interval graphs) and circular-arc graphs. Our tree data structure supports all operations provided by different approaches in previous work, as well as mapping to and from level-order ranks and retrieving the last (first) internal node before (after) a given node in a level-order traversal, all in constant time.

Meng He, J. Ian Munro, Yakov Nekrich, Sebastian Wild, and Kaiyu Wu. Distance Oracles for Interval Graphs via Breadth-First Rank/Select in Succinct Trees. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 25:1-25:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{he_et_al:LIPIcs.ISAAC.2020.25, author = {He, Meng and Munro, J. Ian and Nekrich, Yakov and Wild, Sebastian and Wu, Kaiyu}, title = {{Distance Oracles for Interval Graphs via Breadth-First Rank/Select in Succinct Trees}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {25:1--25:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.25}, URN = {urn:nbn:de:0030-drops-133693}, doi = {10.4230/LIPIcs.ISAAC.2020.25}, annote = {Keywords: succinct data structures, distance oracles, ordinal tree, level order, breadth-first order, interval graphs, proper interval graphs, succinct graph representation} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Under the word RAM model, we design three data structures that can be constructed in O(n √{lg n}) time over n points in an n × n grid. The first data structure is an O(n lg^ε n)-word structure supporting orthogonal range reporting in O(lg lg n+k) time, where k denotes output size and ε is an arbitrarily small constant. The second is an O(n lg lg n)-word structure supporting orthogonal range successor in O(lg lg n) time, while the third is an O(n lg^ε n)-word structure supporting sorted range reporting in O(lg lg n+k) time. The query times of these data structures are optimal when the space costs must be within O(n polylog n) words. Their exact space bounds match those of the best known results achieving the same query times, and the O(n √{lg n}) construction time beats the previous bounds on preprocessing. Previously, among 2d range search structures, only the orthogonal range counting structure of Chan and Pǎtraşcu (SODA 2010) and the linear space, O(lg^ε n) query time structure for orthogonal range successor by Belazzougui and Puglisi (SODA 2016) can be built in the same O(n √{lg n}) time. Hence our work is the first that achieve the same preprocessing time for optimal orthogonal range reporting and range successor. We also apply our results to improve the construction time of text indexes.

Younan Gao, Meng He, and Yakov Nekrich. Fast Preprocessing for Optimal Orthogonal Range Reporting and Range Successor with Applications to Text Indexing. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 54:1-54:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gao_et_al:LIPIcs.ESA.2020.54, author = {Gao, Younan and He, Meng and Nekrich, Yakov}, title = {{Fast Preprocessing for Optimal Orthogonal Range Reporting and Range Successor with Applications to Text Indexing}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {54:1--54:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.54}, URN = {urn:nbn:de:0030-drops-129202}, doi = {10.4230/LIPIcs.ESA.2020.54}, annote = {Keywords: orthogonal range search, geometric data structures, orthogonal range reporting, orthogonal range successor, sorted range reporting, text indexing, word RAM} }

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**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

We introduce the first index that can be built in o(n) time for a text of length n, and can also be queried in o(q) time for a pattern of length q. On an alphabet of size σ, our index uses O(n log σ) bits, is built in O(n log σ / √{log n}) deterministic time, and computes the number of occurrences of the pattern in time O(q/log_σ n + log n log_σ n). Each such occurrence can then be found in O(log n) time. Other trade-offs between the space usage and the cost of reporting occurrences are also possible.

J. Ian Munro, Gonzalo Navarro, and Yakov Nekrich. Text Indexing and Searching in Sublinear Time. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 24:1-24:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{munro_et_al:LIPIcs.CPM.2020.24, author = {Munro, J. Ian and Navarro, Gonzalo and Nekrich, Yakov}, title = {{Text Indexing and Searching in Sublinear Time}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.24}, URN = {urn:nbn:de:0030-drops-121497}, doi = {10.4230/LIPIcs.CPM.2020.24}, annote = {Keywords: data structures, string indexes} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We present a number of new results about range searching for colored (or "categorical") data:
1) For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,…,n}. Previous data structures require O((log n)/(log log n) + k) query time. Our result also implies improvements in higher constant dimensions.
2) Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log²n) query time.
3) For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O((log n)/(log log n) + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k (log n)/(log log n)) time.
Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.

Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further Results on Colored Range Searching. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 28:1-28:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2020.28, author = {Chan, Timothy M. and He, Qizheng and Nekrich, Yakov}, title = {{Further Results on Colored Range Searching}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.28}, URN = {urn:nbn:de:0030-drops-121868}, doi = {10.4230/LIPIcs.SoCG.2020.28}, annote = {Keywords: Range searching, geometric data structures, randomized incremental construction, random sampling, word RAM} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+ε} n + k log^ε n) time, where k is the number of reported points, n is the number of points in the data structure, and ε is an arbitrarily small positive constant. Our second data structure uses O(n log^ε n) space and answers queries in O(log n+k) time.
These are the first data structures for this problem that use linear (resp. O(n log^ε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^ε n)-space data structure supports queries in O(n^ε + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ≥ 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+ε} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time.

Yakov Nekrich. Four-Dimensional Dominance Range Reporting in Linear Space. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 59:1-59:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{nekrich:LIPIcs.SoCG.2020.59, author = {Nekrich, Yakov}, title = {{Four-Dimensional Dominance Range Reporting in Linear Space}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {59:1--59:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.59}, URN = {urn:nbn:de:0030-drops-122170}, doi = {10.4230/LIPIcs.SoCG.2020.59}, annote = {Keywords: Range searching, geometric data structures, word RAM} }

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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 129, 35th International Symposium on Computational Geometry (SoCG 2019)

In this paper we describe a fully-dynamic data structure for the planar point location problem in the external memory model. Our data structure supports queries in O(log_B n(log log_B n)^3)) I/Os and updates in O(log_B n(log log_B n)^2)) amortized I/Os, where n is the number of segments in the subdivision and B is the block size. This is the first dynamic data structure with almost-optimal query cost. For comparison all previously known results for this problem require O(log_B^2 n) I/Os to answer queries. Our result almost matches the best known upper bound in the internal-memory model.

J. Ian Munro and Yakov Nekrich. Dynamic Planar Point Location in External Memory. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 52:1-52:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{munro_et_al:LIPIcs.SoCG.2019.52, author = {Munro, J. Ian and Nekrich, Yakov}, title = {{Dynamic Planar Point Location in External Memory}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {52:1--52:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.52}, URN = {urn:nbn:de:0030-drops-104566}, doi = {10.4230/LIPIcs.SoCG.2019.52}, annote = {Keywords: Data Structures, Dynamic Data Structures, Planar Point Location, External Memory} }

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**Published in:** LIPIcs, Volume 127, 22nd International Conference on Database Theory (ICDT 2019)

In this paper, we consider a variant of the color range reporting problem called color reporting with frequencies. Our goal is to pre-process a set of colored points into a data structure, so that given a query range Q, we can report all colors that appear in Q, along with their respective frequencies. In other words, for each reported color, we also output the number of times it occurs in Q. We describe an external-memory data structure that uses O(N(1+log^2D/log N)) words and answers one-dimensional queries in O(1 +K/B) I/Os, where N is the total number of points in the data structure, D is the total number of colors in the data structure, K is the number of reported colors, and B is the block size.
Next we turn to an approximate version of this problem: report all colors sigma that appear in the query range; for every reported color, we provide a constant-factor approximation on its frequency. We consider color reporting with approximate frequencies in two dimensions. Our data structure uses O(N) space and answers two-dimensional queries in O(log_B N +log^*B + K/B) I/Os in the special case when the query range is bounded on two sides. As a corollary, we can also answer one-dimensional approximate queries within the same time and space bounds.

Arnab Ganguly, J. Ian Munro, Yakov Nekrich, Rahul Shah, and Sharma V. Thankachan. Categorical Range Reporting with Frequencies. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 9:1-9:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ganguly_et_al:LIPIcs.ICDT.2019.9, author = {Ganguly, Arnab and Munro, J. Ian and Nekrich, Yakov and Shah, Rahul and Thankachan, Sharma V.}, title = {{Categorical Range Reporting with Frequencies}}, booktitle = {22nd International Conference on Database Theory (ICDT 2019)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-101-6}, ISSN = {1868-8969}, year = {2019}, volume = {127}, editor = {Barcelo, Pablo and Calautti, Marco}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2019.9}, URN = {urn:nbn:de:0030-drops-103115}, doi = {10.4230/LIPIcs.ICDT.2019.9}, annote = {Keywords: Data Structures, Range Reporting, Range Counting, Categorical Range Reporting, Orthogonal Range Query} }

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

An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost.
In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element e_i does not exceed min(log(W/w_i),log n)+O(1) where w_i is the number of times e_i was accessed and W is the total length of the access sequence.
Our techniques can also be used to encode a sequence of m symbols with a dynamic alphabetic code in O(m) time so that the encoding length is bounded by m(H+O(1)), where H is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol.

Mordecai Golin, John Iacono, Stefan Langerman, J. Ian Munro, and Yakov Nekrich. Dynamic Trees with Almost-Optimal Access Cost. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 38:1-38:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golin_et_al:LIPIcs.ESA.2018.38, author = {Golin, Mordecai and Iacono, John and Langerman, Stefan and Munro, J. Ian and Nekrich, Yakov}, title = {{Dynamic Trees with Almost-Optimal Access Cost}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {38:1--38:14}, 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.2018.38}, URN = {urn:nbn:de:0030-drops-95017}, doi = {10.4230/LIPIcs.ESA.2018.38}, annote = {Keywords: Data Structures, Binary Search Trees, Adaptive Alphabetic Coding} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.
- Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time.
- Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing.
For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.

Timothy M. Chan, Yakov Nekrich, Saladi Rahul, and Konstantinos Tsakalidis. Orthogonal Point Location and Rectangle Stabbing Queries in 3-d. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2018.31, author = {Chan, Timothy M. and Nekrich, Yakov and Rahul, Saladi and Tsakalidis, Konstantinos}, title = {{Orthogonal Point Location and Rectangle Stabbing Queries in 3-d}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.31}, URN = {urn:nbn:de:0030-drops-90352}, doi = {10.4230/LIPIcs.ICALP.2018.31}, annote = {Keywords: geometric data structures, orthogonal point location, rectangle stabbing, pointer machines, I/O model, word RAM model} }

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**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} }

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**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 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We introduce a compressed suffix array representation that, on a text T of length n over an alphabet of size \sigma, can be built in O(n) deterministic time, within O(n\log\sigma) bits of working space, and counts the number of occurrences of any pattern P in T in time O(|P| + \log\log_w \sigma) on a RAM machine of w=\Omega(\log n)-bit words. This new index outperforms all the other compressed indexes that can be built in linear deterministic time, and some others. The only faster indexes can be built in linear time only in expectation, or require \Theta(n\log n) bits.

J. Ian Munro, Gonzalo Navarro, and Yakov Nekrich. Fast Compressed Self-Indexes with Deterministic Linear-Time Construction. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 57:1-57:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{munro_et_al:LIPIcs.ISAAC.2017.57, author = {Munro, J. Ian and Navarro, Gonzalo and Nekrich, Yakov}, title = {{Fast Compressed Self-Indexes with Deterministic Linear-Time Construction}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {57:1--57:12}, 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.2017.57}, URN = {urn:nbn:de:0030-drops-82328}, doi = {10.4230/LIPIcs.ISAAC.2017.57}, annote = {Keywords: Succinct data structures, Self-indexes, Suffix arrays, Deterministic construction} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

In this paper we describe compressed indexes that support pattern matching queries for strings with wildcards. For a constant size alphabet our data structure uses O(n.log^e(n)) bits for any e>0 and reports all occ occurrences of a wildcard string in O(m+s^g.M(n)+occ) time, where M(n)=o(log(log(log(n)))), s is the alphabet size, m is the number of alphabet symbols and g is the number of wildcard symbols in the query string. We also present an O(n)-bit index with O((m+s^g+occ).log^e(n)) query time and an O(n{log(log(n))}^2)-bit index with O((m+s^g+occ).log(log(n))) query time. These are the first non-trivial data structures for this problem that need o(n.log(n)) bits of space.

Moshe Lewenstein, Yakov Nekrich, and Jeffrey Scott Vitter. Space-Efficient String Indexing for Wildcard Pattern Matching. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 506-517, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{lewenstein_et_al:LIPIcs.STACS.2014.506, author = {Lewenstein, Moshe and Nekrich, Yakov and Vitter, Jeffrey Scott}, title = {{Space-Efficient String Indexing for Wildcard Pattern Matching}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {506--517}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.506}, URN = {urn:nbn:de:0030-drops-44838}, doi = {10.4230/LIPIcs.STACS.2014.506}, annote = {Keywords: compressed data structures, compressed indexes, pattern matching} }

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