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Documents authored by Le, Hung

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DOI: 10.4230/LIPIcs.SoCG.2026.29
Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above

Authors: Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: 1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.
Cite as

Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2026.29,
  author =	{Chan, Timothy M. and Chang, Hsien-Chih and Gao, Jie and Kisfaludi-Bak, S\'{a}ndor and Le, Hung and Zheng, Da Wei},
  title =	{{Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.29},
  URN =		{urn:nbn:de:0030-drops-258357},
  doi =		{10.4230/LIPIcs.SoCG.2026.29},
  annote =	{Keywords: Graph Diameter, Geometric Intersection Graphs, Unit Ball Graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.68
Optimal Bounds for Spanners and Tree Covers in Doubling Metrics

Authors: An La, Hung Le, Shay Solomon, Cuong Than, Vinayak, Shuang Yang, and Tianyi Zhang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
It is known that any n-point set in the d-dimensional Euclidean space ℝ^d, for d = O(1), admits: 1) A (1+ε)-spanner with maximum degree Õ(ε^{-d+1}) and with lightness Õ(ε^{-d}), for any ε > 0. 2) A (1+ε)-tree cover with Õ(n ⋅ ε^{-d+1}) trees and maximum degree of O(1) in each tree. Moreover, all the parameters in these constructions are optimal: For any 2 ≤ d = O(1), there exists an n-point set in ℝ^d, for which any (1+ε)-spanner has Ω̃(n⋅ε^{-d+1}) edges and lightness Ω̃(ε^{-d}). The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in ℝ^d, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has Õ(n⋅ε^{-d}) edges, and it could be transformed into a spanner of maximum degree Õ(ε^{-d}) and lightness Õ(n⋅ε^{-(d+1)}) by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a (1+ε)-tree cover with Õ(ε^{-d}) trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) A surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters. 2) A new construction of (1+ε)-tree covers with Õ(n⋅ε^{-d}) trees, with maximum degree O(1) in each tree. This construction is optimal with respect to the number of trees and maximum degree.
Cite as

An La, Hung Le, Shay Solomon, Cuong Than, Vinayak, Shuang Yang, and Tianyi Zhang. Optimal Bounds for Spanners and Tree Covers in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 68:1-68:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{la_et_al:LIPIcs.SoCG.2026.68,
  author =	{La, An and Le, Hung and Solomon, Shay and Than, Cuong and Vinayak and Yang, Shuang and Zhang, Tianyi},
  title =	{{Optimal Bounds for Spanners and Tree Covers in Doubling Metrics}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{68:1--68:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.68},
  URN =		{urn:nbn:de:0030-drops-258756},
  doi =		{10.4230/LIPIcs.SoCG.2026.68},
  annote =	{Keywords: doubling metrics, doubling spanners, Euclidean spanners, tree cover}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.70
Tree-Like Shortcuttings of Trees

Authors: Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Sparse shortcuttings of trees - equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter - have been studied extensively (under different names and settings), since the pioneering works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for n-node trees with sparsity O(log^* n). Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity Ω(log n)), which is a significant drawback for many applications. We initiate a systematic study of constant-hop tree shortcuttings that are "tree-like". We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold. - New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to O(log log n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((log log n)²) for all values of hop-diameter, resolving an open question of [Arnold Filtser and Hung Le, 2022; H. Le, 2023]. - Applications of these bounds, focusing on low-dimensional Euclidean and doubling metrics. A seminal work of Arya et al. [S. Arya et al., 1995] presented a (1+ε)-spanner with constant hop-diameter and sparsity O(log^* n), but with large arboricity. We show that constant hop-diameter is sufficient to achieve arboricity O(log^*{n}). Furthermore, we present a (1+ε)-stretch routing scheme in the fixed-port model with 3 hops and a local memory of O(log²n / log log n) bits, resolving an open question of [Omri Kahalon et al., 2022].
Cite as

Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Tree-Like Shortcuttings of Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{le_et_al:LIPIcs.SoCG.2026.70,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
  title =	{{Tree-Like Shortcuttings of Trees}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{70:1--70:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.70},
  URN =		{urn:nbn:de:0030-drops-258776},
  doi =		{10.4230/LIPIcs.SoCG.2026.70},
  annote =	{Keywords: spanner, tree shortcutting, arboricity, treewidth}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.71
Approximating Euclidean Shallow-Light Trees

Authors: Hung Le, Shay Solomon, Cuong Than, Csaba D. Tóth, and Tianyi Zhang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
For a weighted graph G = (V, E, w) and a designated source vertex s ∈ V, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source s and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an (α, β)-SLT of G w.r.t. s ∈ V is a spanning tree of G with root-stretch α (preserving all distances between s and all other vertices up to a factor of α) and lightness β (its weight is at most β times the weight of a minimum spanning tree of G). It was shown in the early 1990s that (1) for any graph, any source, and any ε > 0, there is a (1 + ε, O(1/ε))-SLT, and (2) there exist graphs for which β = Ω(1/ε) for any (1+ε,β)-SLT. The focus of this work is on SLTs in low-dimensional Euclidean spaces, which are of special interest for some applications of SLTs, in geometric network optimization problems. The aforementioned existential lower bound applies to Euclidean plane, as well. It was shown more than a decade ago that (1) by using Steiner points, one can reduce the lightness bound from O(1/ε) to O(√{1/ε}), and (2) there exist point sets in the plane for which β = Ω(√{1/ε}) for any Steiner (1+ε,β)-SLT. These tight existential bounds for the Euclidean case yield approximation factors of O(1/ε) and O(√{1/ε}) on the minimum weight of any non-Steiner and Steiner tree with root-stretch 1+ε, respectively. Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards resolving this question by presenting two bicriteria approximation algorithms. For any ε > 0, a set P of n points in constant-dimensional Euclidean space and a source s ∈ P, our first (respectively, second) algorithm returns, in O(n log n ⋅ polylog(ε^{-1})) time, a non-Steiner (resp., Steiner) tree with root-stretch 1+O(ε log ε^{-1}) and weight at most O(opt_ε ⋅ log² ε^{-1}) (resp., O(opt_ε ⋅ log ε^{-1})), where opt_ε denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch 1+ε.
Cite as

Hung Le, Shay Solomon, Cuong Than, Csaba D. Tóth, and Tianyi Zhang. Approximating Euclidean Shallow-Light Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{le_et_al:LIPIcs.SoCG.2026.71,
  author =	{Le, Hung and Solomon, Shay and Than, Cuong and T\'{o}th, Csaba D. and Zhang, Tianyi},
  title =	{{Approximating Euclidean Shallow-Light Trees}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{71:1--71:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.71},
  URN =		{urn:nbn:de:0030-drops-258789},
  doi =		{10.4230/LIPIcs.SoCG.2026.71},
  annote =	{Keywords: geometric network design, optimization, shallow-light tree, Steiner point}
}
Document
DOI: 10.4230/DagRep.15.5.134
Metric Sketching and Dynamic Algorithms for Geometric and Topological Graphs (Dagstuhl Seminar 25212)

Authors: Sujoy Bhore, Jie Gao, Hung Le, Csaba D. Tóth, and Lazar Milenković

Published in: Dagstuhl Reports, Volume 15, Issue 5 (2025)

Abstract
Sketching is a basic technique to handle big data: Compress a big input dataset into a small dataset, called a sketch, that (approximately) preserves the important information in the input dataset. A metric space is often given as a distance matrix with Ω(n²) entries, and metric sketching techniques aim to reduce the space to linear. One goal of this Dagstuhl Seminar was to understand different sketching techniques and metric spaces that admit small sketches. Another common approach to handling big datasets is dynamic algorithms. Typically, large datasets do not arrive in a single batch; instead, they are updated over time in small increments. The objective of dynamic algorithms is to respond to data updates quickly, ideally with an update time that is polylogarithmic in the size of the whole dataset. In this Dagstuhl Seminar "Metric Sketching and Dynamic Algorithms for Geometric and Topological Graphs" (25212), we considered sketching and dynamic algorithms in the context of geometric intersection graphs and topological graphs. Geometric intersection graphs have been used to model many real-world massive graphs, such as wireless networks. Topological graphs, including planar graphs, have been used in applications such as geographic information systems and motion planning. While geometric intersection graphs and topological graphs are seemingly different, they have common structural properties that allow the transfer of algorithmic techniques between the two domains, which was the motivation of this seminar: Uncovering deeper connections between metric sketching, dynamic algorithms, geometric intersection graphs, and topological graphs. More concretely, we studied: (1) the construction of sketching structures, such as spanners, tree covers, distance oracles, and emulators with optimal parameters for various metrics and graphs, including geometric and topological graphs; (2) dynamic problems in geometric intersections graphs, including connectivity, spanners, shortest paths; and (3) dynamic maintenance of metric sketching structures in topological graphs.
Cite as

Sujoy Bhore, Jie Gao, Hung Le, Csaba D. Tóth, and Lazar Milenković. Metric Sketching and Dynamic Algorithms for Geometric and Topological Graphs (Dagstuhl Seminar 25212). In Dagstuhl Reports, Volume 15, Issue 5, pp. 134-157, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{bhore_et_al:DagRep.15.5.134,
  author =	{Bhore, Sujoy and Gao, Jie and Le, Hung and T\'{o}th, Csaba D. and Milenkovi\'{c}, Lazar},
  title =	{{Metric Sketching and Dynamic Algorithms for Geometric and Topological Graphs (Dagstuhl Seminar 25212)}},
  pages =	{134--157},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2025},
  volume =	{15},
  number =	{5},
  editor =	{Bhore, Sujoy and Gao, Jie and Le, Hung and T\'{o}th, Csaba D. and Milenkovi\'{c}, Lazar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.15.5.134},
  URN =		{urn:nbn:de:0030-drops-252753},
  doi =		{10.4230/DagRep.15.5.134},
  annote =	{Keywords: geometric spanners, geometric intersection graphs, planar metrics, metric covering, computational geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.37
Optimal Euclidean Tree Covers

Authors: Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set. In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.
Cite as

Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal Euclidean Tree Covers. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2024.37,
  author =	{Chang, Hsien-Chih and Conroy, Jonathan and Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
  title =	{{Optimal Euclidean Tree Covers}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.37},
  URN =		{urn:nbn:de:0030-drops-199828},
  doi =		{10.4230/LIPIcs.SoCG.2024.37},
  annote =	{Keywords: Tree cover, spanner, Steiner point, routing, bounded-degree, quadtree, net-tree}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.38
Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs

Authors: Hsien-Chih Chang, Jie Gao, and Hung Le

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs is one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary. In this paper, we show a truly-subquadratic algorithm of running time O^~(n^{2-1/18}), for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 2. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension - either of k-hop balls or the distance encoding vectors - is 4. This contrasts to the VC dimension of the pseudo-disks themselves as geometric ranges (which is known to be 3). Second, we introduce a clique-based r-clustering for geometric intersection graphs, which is an analog of the r-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and O(1) query time. The results naturally extend to unit L₁ or L_∞-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly subquadratic algorithm to find the exact diameter.
Cite as

Hsien-Chih Chang, Jie Gao, and Hung Le. Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2024.38,
  author =	{Chang, Hsien-Chih and Gao, Jie and Le, Hung},
  title =	{{Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{38:1--38:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.38},
  URN =		{urn:nbn:de:0030-drops-199833},
  doi =		{10.4230/LIPIcs.SoCG.2024.38},
  annote =	{Keywords: Unit-disk graph, pseudo-disks, r-division, VC-dimension, distance oracle, clique-based separator}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.47
Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function

Authors: Hung Le, Lazar Milenković, and Shay Solomon

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
In STOC'95 [S. Arya et al., 1995] Arya et al. showed that any set of n points in ℝ^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter k with O(n α_k(n)) edges, for any k ≥ 2. The function α_k is the inverse of a certain Ackermann-style function, where α₀(n) = ⌈n/2⌉, α₁(n) = ⌈√n⌉, α₂(n) = ⌈log n⌉, α₃(n) = ⌈log log n⌉, α₄(n) = log^* n, α₅(n) = ⌊ 1/2 log^*n ⌋, …. Roughly speaking, for k ≥ 2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Despite a large body of work on spanners of bounded hop-diameter, the fundamental question of whether this tradeoff between size and hop-diameter of Euclidean (1+ε)-spanners is optimal has remained open, even in one-dimensional spaces. Three lower bound tradeoffs are known: - An optimal k versus Ω(n α_k(n)) by Alon and Schieber [N. Alon and B. Schieber, 1987], but it applies to stretch 1 (not 1+ε). - A suboptimal k versus Ω(nα_{2k+6}(n)) by Chan and Gupta [H. T.-H. Chan and A. Gupta, 2006]. - A suboptimal k versus Ω(n/(2^{6⌊k/2⌋}) α_k(n)) by Le et al. [Le et al., 2022]. This paper establishes the optimal k versus Ω(n α_k(n)) lower bound tradeoff for stretch 1+ε, for any ε > 0, and for any k. An important conceptual contribution of this work is in achieving optimality by shaving off an extremely slowly growing term, namely 2^{6⌊k/2⌋} for k ≤ O(α(n)); such a fine-grained optimization (that achieves optimality) is very rare in the literature. To shave off the 2^{6⌊k/2⌋} term from the previous bound of Le et al., our argument has to drill much deeper. In particular, we propose a new way of analyzing recurrences that involve inverse-Ackermann style functions, and our key technical contribution is in presenting the first explicit construction of concave versions of these functions. An important advantage of our approach over previous ones is its robustness: While all previous lower bounds are applicable only to restricted 1-dimensional point sets, ours applies even to random point sets in constant-dimensional spaces.
Cite as

Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{le_et_al:LIPIcs.SoCG.2023.47,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
  title =	{{Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{47:1--47:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.47},
  URN =		{urn:nbn:de:0030-drops-178976},
  doi =		{10.4230/LIPIcs.SoCG.2023.47},
  annote =	{Keywords: Euclidean spanners, Ackermann functions, convex functions, hop-diameter}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.54
Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

Authors: Hung Le, Lazar Milenković, and Shay Solomon

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

Abstract
In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.
Cite as

Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.SoCG.2022.54,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
  title =	{{Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{54:1--54:15},
  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.2022.54},
  URN =		{urn:nbn:de:0030-drops-160629},
  doi =		{10.4230/LIPIcs.SoCG.2022.54},
  annote =	{Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.96
Dynamic Matching Algorithms Under Vertex Updates

Authors: Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
Dynamic graph matching algorithms have been extensively studied, but mostly under edge updates. This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges. A basic setting arising in online algorithms and studied by Bosek et al. [FOCS'14] and Bernstein et al. [SODA'18] is that of dynamic approximate maximum cardinality matching (MCM) in bipartite graphs in which one side is fixed and vertices on the other side either arrive or depart via vertex updates. In the BASIC-incremental setting, vertices only arrive, while in the BASIC-decremental setting vertices only depart. When vertices can both arrive and depart, we have the BASIC-dynamic setting. In this paper we also consider the setting in which both sides of the bipartite graph are dynamic. We call this the MEDIUM-dynamic setting, and MEDIUM-decremental is the restriction when vertices can only depart. The GENERAL-dynamic setting is when the graph is not necessarily bipartite and the vertices can both depart and arrive. Denote by K the total number of edges inserted and deleted to and from the graph throughout the entire update sequence. A well-studied measure, the recourse of a dynamic matching algorithm is the number of changes made to the matching per step. We largely focus on Maximal Matching (MM) which is a 2-approximation to the MCM. Our main results are as follows. - In the BASIC-dynamic setting, there is a straightforward algorithm for maintaining a MM, with a total runtime of O(K) and constant worst-case recourse. In fact, this algorithm never removes an edge from the matching; we refer to such an algorithm as irrevocable. - For the MEDIUM-dynamic setting we give a strong conditional lower bound that even holds in the MEDIUM-decremental setting: if for any fixed η > 0, there is an irrevocable decremental MM algorithm with a total runtime of O(K ⋅ n^{1-η}), this would refute the OMv conjecture; a similar (but weaker) hardness result can be achieved via a reduction from the Triangle Detection conjecture. - Next, we consider the GENERAL-dynamic setting, and design an MM algorithm with a total runtime of O(K) and constant worst-case recourse. We achieve this result via a 1-revocable algorithm, which may remove just one edge per update step. As argued above, an irrevocable algorithm with such a runtime is not likely to exist. - Finally, back to the BASIC-dynamic setting, we present an algorithm with a total runtime of O(K), which provides an (e/(e-1))-approximation to the MCM. To this end, we build on the classic "ranking" online algorithm by Karp et al. [STOC'90]. Beyond the results, our work draws connections between the areas of dynamic graph algorithms and online algorithms, and it proposes several open questions that seem to be overlooked thus far.
Cite as

Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams. Dynamic Matching Algorithms Under Vertex Updates. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 96:1-96:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.ITCS.2022.96,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Vassilevska Williams, Virginia},
  title =	{{Dynamic Matching Algorithms Under Vertex Updates}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{96:1--96:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.96},
  URN =		{urn:nbn:de:0030-drops-156923},
  doi =		{10.4230/LIPIcs.ITCS.2022.96},
  annote =	{Keywords: maximal matching, approximate matching, dynamic algorithm, vertex updates}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.67
Light Euclidean Spanners with Steiner Points

Authors: Hung Le and Shay Solomon

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

Abstract
The FOCS'19 paper of Le and Solomon [Hung Le and Shay Solomon, 2019], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ε)-spanner in ℝ^d is Õ(ε^{-d}) for any d = O(1) and any ε = Ω(n^{-1/(d-1)}) (where Õ hides polylogarithmic factors of 1/ε), and also shows the existence of point sets in ℝ^d for which any (1+ε)-spanner must have lightness Ω(ε^{-d}). Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in ℝ² with lightness O(ε^{-1} log Δ), where Δ is the spread of the point set. In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications ε often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ε, this upper bound provides a quadratic improvement over the non-Steiner bound of [Hung Le and Shay Solomon, 2019], We then demonstrate that such a light spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ε^{-(d+1)/2} + ε^{-2} log Δ) for any 3 ≤ d = O(1) and any ε = Ω(n^{-1/(d-1)}).
Cite as

Hung Le and Shay Solomon. Light Euclidean Spanners with Steiner Points. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{le_et_al:LIPIcs.ESA.2020.67,
  author =	{Le, Hung and Solomon, Shay},
  title =	{{Light Euclidean Spanners with Steiner Points}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{67:1--67:22},
  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.2020.67},
  URN =		{urn:nbn:de:0030-drops-129331},
  doi =		{10.4230/LIPIcs.ESA.2020.67},
  annote =	{Keywords: Euclidean spanners, Steiner spanners, light spanners}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2016.8
Optimal Dynamic Program for r-Domination Problems over Tree Decompositions

Authors: Glencora Borradaile and Hung Le

Published in: LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Abstract
There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems is optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to r-domination problems. In r-dominating set, one wishes to find a minimum subset S of vertices such that every vertex of G is within r hops of some vertex in S. In connected r-dominating set, one additionally requires that the set induces a connected subgraph of G. We give a O((2r+1)^tw n) time algorithm for r-dominating set and a randomized O((2r+2)^tw n^{O(1)}) time algorithm for connected r-dominating set in n-vertex graphs of treewidth tw. We show that the running time dependence on r and tw is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.
Cite as

Glencora Borradaile and Hung Le. Optimal Dynamic Program for r-Domination Problems over Tree Decompositions. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{borradaile_et_al:LIPIcs.IPEC.2016.8,
  author =	{Borradaile, Glencora and Le, Hung},
  title =	{{Optimal Dynamic Program for r-Domination Problems over Tree Decompositions}},
  booktitle =	{11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
  pages =	{8:1--8:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-023-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{63},
  editor =	{Guo, Jiong and Hermelin, Danny},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.8},
  URN =		{urn:nbn:de:0030-drops-69199},
  doi =		{10.4230/LIPIcs.IPEC.2016.8},
  annote =	{Keywords: r-dominating set, Exponential Time Hypothesis, Dynamic Programming}
}
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