4 Search Results for "Nova Fandina, Ora"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.38
Lower Bounds on Tree Covers

Authors: Yu Chen, Zihan Tan, and Hangyu Xu

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Given an n-point metric space (X,d_X), a tree cover 𝒯 is a set of |𝒯| = k trees on X such that every pair of vertices in X has a low-distortion path in one of the trees in 𝒯. Tree covers have been playing a crucial role in graph algorithms for decades, and the research focus is the construction of tree covers with small size k and distortion. When k = 1, the best distortion is known to be Θ(n). For a constant k ≥ 2, the best distortion upper bound is Õ(n^{1/k}) and the strongest lower bound is Ω(log_k n), leaving a gap to be closed. In this paper, we improve the lower bound to Ω(n^{1/(2^{k-1)}}). Our proof is a novel analysis on a structurally simple grid-like graph, which utilizes some combinatorial fixed-point theorems. We believe that they will prove useful for analyzing other tree-like data structures as well.
Cite as

Yu Chen, Zihan Tan, and Hangyu Xu. Lower Bounds on Tree Covers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2026.38,
  author =	{Chen, Yu and Tan, Zihan and Xu, Hangyu},
  title =	{{Lower Bounds on Tree Covers}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{38:1--38:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.38},
  URN =		{urn:nbn:de:0030-drops-253254},
  doi =		{10.4230/LIPIcs.ITCS.2026.38},
  annote =	{Keywords: Tree Covers, Combinatorial Fixed-Point Theorems}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.30
Light Spanners with Small Hop-Diameter

Authors: Sujoy Bhore and Lazar Milenković

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Lightness, sparsity, and hop-diameter are the fundamental parameters of geometric spanners. Arya et al. [STOC'95] showed in their seminal work that there exists a construction of Euclidean (1+ε)-spanners with hop-diameter O(log n) and lightness O(log n). They also gave a general tradeoff of hop-diameter k and sparsity O(α_k(n)), where α_k is a very slowly growing inverse of an Ackermann-style function. The former combination of logarithmic hop-diameter and lightness is optimal due to the lower bound by Dinitz et al. [FOCS'08]. Later, Elkin and Solomon [STOC'13] generalized the light spanner construction to doubling metrics and extended the tradeoff for more values of hop-diameter k. In a recent line of work [SoCG'22, SoCG'23], Le et al. proved that the aforementioned tradeoff between the hop-diameter and sparsity is tight for every choice of hop-diameter k. A fundamental question remains: What is the optimal tradeoff between the hop-diameter and lightness for every value of k? In this paper, we present a general framework for constructing light spanners with small hop-diameter. Our framework is based on tree covers. In particular, we show that if a metric admits a tree cover with γ trees, stretch t, and lightness L, then it also admits a t-spanner with hop-diameter k and lightness O(kn^{2/k}⋅ γ L). Further, we note that the tradeoff for trees is tight due to a construction in uniform line metric, which is perhaps the simplest tree metric. As a direct consequence of this framework, we obtain new tradeoffs between lightness and hop-diameter for doubling metrics.
Cite as

Sujoy Bhore and Lazar Milenković. Light Spanners with Small Hop-Diameter. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhore_et_al:LIPIcs.ICALP.2025.30,
  author =	{Bhore, Sujoy and Milenkovi\'{c}, Lazar},
  title =	{{Light Spanners with Small Hop-Diameter}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.30},
  URN =		{urn:nbn:de:0030-drops-234075},
  doi =		{10.4230/LIPIcs.ICALP.2025.30},
  annote =	{Keywords: Geometric Spanners, Lightness, Hop-Diameter, Recurrences, Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.31
Barriers for Faster Dimensionality Reduction

Authors: Ora Nova Fandina, Mikael Møller Høgsgaard, and Kasper Green Larsen

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
The Johnson-Lindenstrauss transform allows one to embed a dataset of n points in ℝ^d into ℝ^m, while preserving the pairwise distance between any pair of points up to a factor (1 ± ε), provided that m = Ω(ε^{-2} lg n). The transform has found an overwhelming number of algorithmic applications, allowing to speed up algorithms and reducing memory consumption at the price of a small loss in accuracy. A central line of research on such transforms, focus on developing fast embedding algorithms, with the classic example being the Fast JL transform by Ailon and Chazelle. All known such algorithms have an embedding time of Ω(d lg d), but no lower bounds rule out a clean O(d) embedding time. In this work, we establish the first non-trivial lower bounds (of magnitude Ω(m lg m)) for a large class of embedding algorithms, including in particular most known upper bounds.
Cite as

Ora Nova Fandina, Mikael Møller Høgsgaard, and Kasper Green Larsen. Barriers for Faster Dimensionality Reduction. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{novafandina_et_al:LIPIcs.STACS.2023.31,
  author =	{Nova Fandina, Ora and M{\o}ller H{\o}gsgaard, Mikael and Green Larsen, Kasper},
  title =	{{Barriers for Faster Dimensionality Reduction}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{31:1--31:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.31},
  URN =		{urn:nbn:de:0030-drops-176838},
  doi =		{10.4230/LIPIcs.STACS.2023.31},
  annote =	{Keywords: Dimensional reduction, Lower bound, Linear Circuits}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.13
Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures

Authors: Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [Yair Bartal et al., 2019] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1+ε average distortion for embedding into k-dimensional Euclidean space, where k = O(1/ε²), and for more general q-norms of distortion, k = O(max{1/ε²,q/ε}), whereas tight lower bounds were established only for large values of q via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 ≤ q ≤ O((log (2ε² n))/ε) and ε ≥ 1/(√n), where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ε then k = Ω(1/ε²), for any ε ≥ 1/(√n), matching the upper bounds of [Yair Bartal et al., 2019] and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.
Cite as

Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen. Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bartal_et_al:LIPIcs.SoCG.2022.13,
  author =	{Bartal, Yair and Fandina, Ora Nova and Larsen, Kasper Green},
  title =	{{Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.13},
  URN =		{urn:nbn:de:0030-drops-160219},
  doi =		{10.4230/LIPIcs.SoCG.2022.13},
  annote =	{Keywords: average distortion, practical dimensionality reduction, JL transform}
}
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