5 Search Results for "Amani, Mahdi"

Document
DOI: 10.4230/LIPIcs.ESA.2025.49
Efficient Top-Down Updates in AVL Trees

Authors: Vincent Jugé

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Since AVL trees were invented in 1962, two major open questions about rebalancing operations, which found positive answers in other balanced binary search trees, were left open: can these operations be performed top-down (with a fixed look-ahead), and can they use an amortised constant number of write operations per update? We propose an algorithm that solves both questions positively.
Cite as

Vincent Jugé. Efficient Top-Down Updates in AVL Trees. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{juge:LIPIcs.ESA.2025.49,
  author =	{Jug\'{e}, Vincent},
  title =	{{Efficient Top-Down Updates in AVL Trees}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.49},
  URN =		{urn:nbn:de:0030-drops-245172},
  doi =		{10.4230/LIPIcs.ESA.2025.49},
  annote =	{Keywords: AVL trees, data structures, amortised complexity}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.40
Grandchildren-Weight-Balanced Binary Search Trees

Authors: Vincent Jugé

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We revisit weight-balanced trees, also known as trees of bounded balance. Invented by Nievergelt and Reingold in 1972, these trees are obtained by assigning a weight to each node and requesting that the weight of each node should be quite larger than the weights of its children, the precise meaning of "quite larger" depending on a real-valued parameter γ. Blum and Mehlhorn then showed how to maintain them in a recursive (bottom-up) fashion when 2/11 ⩽ γ ⩽ 1-1/√2, their algorithm requiring only an amortised constant number of tree rebalancing operations per update (insertion or deletion). Later, in 1993, Lai and Wood proposed a top-down procedure for updating these trees when 2/11 ⩽ γ ⩽ 1/4. Our contribution is two-fold. First, we strengthen the requirements of Nievergelt and Reingold, by also requesting that each node should have a substantially larger weight than its grandchildren, thereby obtaining what we call grandchildren-balanced trees. Grandchildren-balanced trees are not harder to maintain than weight-balanced trees, but enjoy a smaller node depth, both in the worst case (with a 6 % decrease) and on average (with a 1.6 % decrease). In particular, unlike standard weight-balanced trees, all grandchildren-balanced trees with n nodes are of height less than 2 log₂(n). Second, we adapt the algorithm of Lai and Wood to all weight-balanced trees, i.e., to all parameter values γ such that 2/11 ⩽ γ ⩽ 1-1/√2. More precisely, we adapt it to all grandchildren-balanced trees for which 1/4 < γ ⩽ 1 - 1/√2. Finally, we show that, except in limit cases (where, for instance, γ = 1 - 1/√2), all these algorithms result in making a constant amortised number of tree rebalancing operations per tree update.
Cite as

Vincent Jugé. Grandchildren-Weight-Balanced Binary Search Trees. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 40:1-40:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{juge:LIPIcs.WADS.2025.40,
  author =	{Jug\'{e}, Vincent},
  title =	{{Grandchildren-Weight-Balanced Binary Search Trees}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{40:1--40:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.40},
  URN =		{urn:nbn:de:0030-drops-242710},
  doi =		{10.4230/LIPIcs.WADS.2025.40},
  annote =	{Keywords: Data structures, Balanced binary trees}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.26
Geometric Spanners of Bounded Tree-Width

Authors: Kevin Buchin, Carolin Rehs, and Torben Scheele

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Given a point set P in the Euclidean space, a geometric t-spanner G is a graph on P such that for every pair of points, the shortest path in G between those points is at most a factor t longer than the Euclidean distance between those points. The value t ≥ 1 is called the dilation of G. Commonly, the aim is to construct a t-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation t. Let d be a fixed integer and P ⊂ ℝ^d be a point set with n points. We give a first algorithm to compute an 𝒪(n/k^{d/(d-1)})-spanner on P with tree-width at most k. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width k: We show that there is a set of n points such that every spanner of tree-width k has dilation 𝒪(n/k^{d/(d-1)}). We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in ℝ², a plane spanner with tree-width at most k and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of n equally spaced points on a circle.
Cite as

Kevin Buchin, Carolin Rehs, and Torben Scheele. Geometric Spanners of Bounded Tree-Width. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2025.26,
  author =	{Buchin, Kevin and Rehs, Carolin and Scheele, Torben},
  title =	{{Geometric Spanners of Bounded Tree-Width}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.26},
  URN =		{urn:nbn:de:0030-drops-231786},
  doi =		{10.4230/LIPIcs.SoCG.2025.26},
  annote =	{Keywords: Computational Geometry, Geometric Spanner, Tree-width}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.48
Exact Algorithms for Minimum Dilation Triangulation

Authors: Sándor P. Fekete, Phillip Keldenich, and Michael Perk

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set P of n points in the plane, find a triangulation T, such that a shortest Euclidean path in T between any pair of points increases by the smallest possible factor compared to their straight-line distance. No polynomial-time algorithm is known for the problem; moreover, evaluating the objective function involves computing the sum of (possibly many) square roots. On the other hand, the problem is not known to be NP-hard. (1) We provide practically robust methods and implementations for computing an MDT for benchmark sets with up to 30,000 points in reasonable time on commodity hardware, based on new geometric insights into the structure of optimal edge sets. Previous methods only achieved results for up to 200 points, so we extend the range of optimally solvable instances by a factor of 150. (2) We develop scalable techniques for accurately evaluating many shortest-path queries that arise as large-scale sums of square roots, allowing us to certify exact optimal solutions, with previous work relying on (possibly inaccurate) floating-point computations. (3) We resolve an open problem by establishing a lower bound of 1.44116 on the dilation of the regular 84-gon (and thus for arbitrary point sets), improving the previous worst-case lower bound of 1.4308 and greatly reducing the remaining gap to the upper bound of 1.4482 from the literature. In the process, we provide optimal solutions for regular n-gons up to n = 100.
Cite as

Sándor P. Fekete, Phillip Keldenich, and Michael Perk. Exact Algorithms for Minimum Dilation Triangulation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2025.48,
  author =	{Fekete, S\'{a}ndor P. and Keldenich, Phillip and Perk, Michael},
  title =	{{Exact Algorithms for Minimum Dilation Triangulation}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.48},
  URN =		{urn:nbn:de:0030-drops-232006},
  doi =		{10.4230/LIPIcs.SoCG.2025.48},
  annote =	{Keywords: dilation, minimum dilation triangulation, exact algorithms, algorithm engineering, experimental evaluation}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2016.25
A Plane 1.88-Spanner for Points in Convex Position

Authors: Mahdi Amani, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Anil Maheshwari, and Michiel Smid

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Abstract
Let S be a set of n points in the plane that is in convex position. For a real number t>1, we say that a point p in S is t-good if for every point q of S, the shortest-path distance between p and q along the boundary of the convex hull of S is at most t times the Euclidean distance between p and q. We prove that any point that is part of (an approximation to) the diameter of S is 1.88-good. Using this, we show how to compute a plane 1.88-spanner of S in O(n) time, assuming that the points of S are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was 1.998 (which, in fact, holds for any point set, i.e., even if it is not in convex position).
Cite as

Mahdi Amani, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Anil Maheshwari, and Michiel Smid. A Plane 1.88-Spanner for Points in Convex Position. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{amani_et_al:LIPIcs.SWAT.2016.25,
  author =	{Amani, Mahdi and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Maheshwari, Anil and Smid, Michiel},
  title =	{{A Plane 1.88-Spanner for Points in Convex Position}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.25},
  URN =		{urn:nbn:de:0030-drops-60474},
  doi =		{10.4230/LIPIcs.SWAT.2016.25},
  annote =	{Keywords: points in convex position, plane spanner}
}
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