5 Search Results for "Seelbach Benkner, Louisa"

Document
DOI: 10.4230/LIPIcs.AofA.2024.2
Enumeration and Succinct Encoding of AVL Trees

Authors: Jeremy Chizewer, Stephen Melczer, J. Ian Munro, and Ava Pun

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We use a novel decomposition to create succinct data structures - supporting a wide range of operations on static trees in constant time - for a variety of tree classes, extending results of Munro, Nicholson, Benkner, and Wild. Motivated by the class of AVL trees, we further derive asymptotics for the information-theoretic lower bound on the number of bits needed to store tree classes whose generating functions satisfy certain functional equations. In particular, we prove that AVL trees require approximately 0.938 bits per node to encode.
Cite as

Jeremy Chizewer, Stephen Melczer, J. Ian Munro, and Ava Pun. Enumeration and Succinct Encoding of AVL Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chizewer_et_al:LIPIcs.AofA.2024.2,
  author =	{Chizewer, Jeremy and Melczer, Stephen and Munro, J. Ian and Pun, Ava},
  title =	{{Enumeration and Succinct Encoding of AVL Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{2:1--2:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.2},
  URN =		{urn:nbn:de:0030-drops-204376},
  doi =		{10.4230/LIPIcs.AofA.2024.2},
  annote =	{Keywords: AVL trees, analytic combinatorics, succinct data structures, enumeration}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.13
On the Number of Distinct Fringe Subtrees in Binary Search Trees

Authors: Stephan Wagner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.
Cite as

Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wagner:LIPIcs.AofA.2024.13,
  author =	{Wagner, Stephan},
  title =	{{On the Number of Distinct Fringe Subtrees in Binary Search Trees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{13:1--13:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.13},
  URN =		{urn:nbn:de:0030-drops-204482},
  doi =		{10.4230/LIPIcs.AofA.2024.13},
  annote =	{Keywords: Fringe subtrees, binary search trees, tree compression, minimal DAG, asymptotics}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.45
Compression by Contracting Straight-Line Programs

Authors: Moses Ganardi

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size g in linear time into an equivalent SLP of size 𝒪(g) so that the height of the unique derivation tree is 𝒪(log N) where N is the length of the represented string (FOCS 2019). We introduce a new class of balanced SLPs, called contracting SLPs, where for every rule A → β₁ … β_k the string length of every variable β_i on the right-hand side is smaller by a constant factor than the string length of A. In particular, the derivation tree of a contracting SLP has the property that every subtree has logarithmic height in its leaf size. We show that a given SLP of size g can be transformed in linear time into an equivalent contracting SLP of size 𝒪(g) with rules of constant length. This result is complemented by a lower bound, proving that converting SLPs into so called α-balanced SLPs or AVL-grammars can incur an increase by a factor of Ω(log N). We present an application to the navigation problem in compressed unranked trees, represented by forest straight-line programs (FSLPs). A linear space data structure by Reh and Sieber (2020) supports navigation steps such as going to the parent, left/right sibling, or to the first/last child in constant time. We extend their solution by the operation of moving to the i-th child in time 𝒪(log d) where d is the degree of the current node. Contracting SLPs are also applied to the finger search problem over SLP-compressed strings where one wants to access positions near to a pre-specified finger position, ideally in 𝒪(log d) time where d is the distance between the accessed position and the finger. We give a linear space solution for the dynamic variant where one can set the finger in 𝒪(log N) time, and then access symbols or move the finger in time 𝒪(log d + log^(t) N) for any constant t where log^(t) N is the t-fold logarithm of N. This improves a previous solution by Bille, Christiansen, Cording, and Gørtz (2018) with access/move time 𝒪(log d + log log N).
Cite as

Moses Ganardi. Compression by Contracting Straight-Line Programs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ganardi:LIPIcs.ESA.2021.45,
  author =	{Ganardi, Moses},
  title =	{{Compression by Contracting Straight-Line Programs}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.45},
  URN =		{urn:nbn:de:0030-drops-146263},
  doi =		{10.4230/LIPIcs.ESA.2021.45},
  annote =	{Keywords: grammar-based compression, balancing, finger search}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.70
Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima

Authors: J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of 2n+o(n) bits, but the space drops to 1.736n + o(n) bits on average for random permutations of n elements, and 2lg binom{n}{r} + o(n) for arrays with r increasing runs, respectively. Both results are optimal; the former answers an open problem of Davoodi et al. (2014) and Golin et al. (2016). Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that they simultaneously achieve the optimal space usage to within lower order terms for a wide range of distributions over tree shapes, including: binary search trees (BSTs) generated by insertions in random order / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees.
Cite as

J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild. Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 70:1-70:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{munro_et_al:LIPIcs.ESA.2021.70,
  author =	{Munro, J. Ian and Nicholson, Patrick K. and Benkner, Louisa Seelbach and Wild, Sebastian},
  title =	{{Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{70:1--70:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.70},
  URN =		{urn:nbn:de:0030-drops-146512},
  doi =		{10.4230/LIPIcs.ESA.2021.70},
  annote =	{Keywords: analysis of algorithms, universal source code, compressed trees, succinct data structure, succinct trees, tree covering, random binary search trees, range-minimum queries}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.16
Average Case Analysis of Leaf-Centric Binary Tree Sources

Authors: Louisa Seelbach Benkner and Markus Lohrey

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
We study the average size of the minimal directed acyclic graph (DAG) with respect to so-called leaf-centric binary tree sources as studied by Zhang, Yang, and Kieffer. A leaf-centric binary tree source induces for every n >= 2 a probability distribution on all binary trees with n leaves. We generalize a result shown by Flajolet, Gourdon, Martinez and Devroye according to which the average size of the minimal DAG of a binary tree that is produced by the binary search tree model is Theta(n / log n).
Cite as

Louisa Seelbach Benkner and Markus Lohrey. Average Case Analysis of Leaf-Centric Binary Tree Sources. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{seelbachbenkner_et_al:LIPIcs.MFCS.2018.16,
  author =	{Seelbach Benkner, Louisa and Lohrey, Markus},
  title =	{{Average Case Analysis of Leaf-Centric Binary Tree Sources}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.16},
  URN =		{urn:nbn:de:0030-drops-95982},
  doi =		{10.4230/LIPIcs.MFCS.2018.16},
  annote =	{Keywords: Directed acylic graphs, average case analysis, tree compression}
}
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