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Documents authored by Zheng, Da Wei


Document
Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane

Authors: Timothy M. Chan, Pingan Cheng, and Da Wei Zheng

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


Abstract
Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries: 1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.) 2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.) 3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.)

Cite as

Timothy M. Chan, Pingan Cheng, and Da Wei Zheng. Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{chan_et_al:LIPIcs.SoCG.2024.33,
  author =	{Chan, Timothy M. and Cheng, Pingan and Zheng, Da Wei},
  title =	{{Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{33:1--33:15},
  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.2024.33},
  URN =		{urn:nbn:de:0030-drops-199785},
  doi =		{10.4230/LIPIcs.SoCG.2024.33},
  annote =	{Keywords: Computational geometry, range searching, intersection searching, semialgebraic sets, data structures, polynomial partitioning}
}
Document
Track A: Algorithms, Complexity and Games
Faster Submodular Maximization for Several Classes of Matroids

Authors: Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges.

Cite as

Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster Submodular Maximization for Several Classes of Matroids. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{henzinger_et_al:LIPIcs.ICALP.2023.74,
  author =	{Henzinger, Monika and Liu, Paul and Vondr\'{a}k, Jan and Zheng, Da Wei},
  title =	{{Faster Submodular Maximization for Several Classes of Matroids}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{74:1--74:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.74},
  URN =		{urn:nbn:de:0030-drops-181267},
  doi =		{10.4230/LIPIcs.ICALP.2023.74},
  annote =	{Keywords: submodular optimization, dynamic data structures, matching algorithms}
}
Document
CG Challenge
Conflict-Based Local Search for Minimum Partition into Plane Subgraphs (CG Challenge)

Authors: Jack Spalding-Jamieson, Brandon Zhang, and Da Wei Zheng

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


Abstract
This paper examines the approach taken by team gitastrophe in the CG:SHOP 2022 challenge. The challenge was to partition the edges of a geometric graph, with vertices represented by points in the plane and edges as straight lines, into the minimum number of planar subgraphs. We used a simple variation of a conflict optimizer strategy used by team Shadoks in the previous year’s CG:SHOP to rank second in the challenge.

Cite as

Jack Spalding-Jamieson, Brandon Zhang, and Da Wei Zheng. Conflict-Based Local Search for Minimum Partition into Plane Subgraphs (CG Challenge). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 72:1-72:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{spaldingjamieson_et_al:LIPIcs.SoCG.2022.72,
  author =	{Spalding-Jamieson, Jack and Zhang, Brandon and Zheng, Da Wei},
  title =	{{Conflict-Based Local Search for Minimum Partition into Plane Subgraphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{72:1--72:6},
  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.2022.72},
  URN =		{urn:nbn:de:0030-drops-160807},
  doi =		{10.4230/LIPIcs.SoCG.2022.72},
  annote =	{Keywords: local search, planar graph, graph colouring, geometric graph, conflict optimizer}
}
Document
CG Challenge
Coordinated Motion Planning Through Randomized k-Opt (CG Challenge)

Authors: Paul Liu, Jack Spalding-Jamieson, Brandon Zhang, and Da Wei Zheng

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)


Abstract
This paper examines the approach taken by team gitastrophe in the CG:SHOP 2021 challenge. The challenge was to find a sequence of simultaneous moves of square robots between two given configurations that minimized either total distance travelled or makespan (total time). Our winning approach has two main components: an initialization phase that finds a good initial solution, and a k-opt local search phase which optimizes this solution. This led to a first place finish in the distance category and a third place finish in the makespan category.

Cite as

Paul Liu, Jack Spalding-Jamieson, Brandon Zhang, and Da Wei Zheng. Coordinated Motion Planning Through Randomized k-Opt (CG Challenge). In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 64:1-64:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{liu_et_al:LIPIcs.SoCG.2021.64,
  author =	{Liu, Paul and Spalding-Jamieson, Jack and Zhang, Brandon and Zheng, Da Wei},
  title =	{{Coordinated Motion Planning Through Randomized k-Opt}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{64:1--64:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.64},
  URN =		{urn:nbn:de:0030-drops-138635},
  doi =		{10.4230/LIPIcs.SoCG.2021.64},
  annote =	{Keywords: motion planning, randomized local search, path finding}
}
Document
CG Challenge
Computing Low-Cost Convex Partitions for Planar Point Sets with Randomized Local Search and Constraint Programming (CG Challenge)

Authors: Da Wei Zheng, Jack Spalding-Jamieson, and Brandon Zhang

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


Abstract
The Minimum Convex Partition problem (MCP) is a problem in which a point-set is used as the vertices for a planar subdivision, whose number of edges is to be minimized. In this planar subdivision, the outer face is the convex hull of the point-set, and the interior faces are convex. In this paper, we discuss and implement the approach to this problem using randomized local search, and different initialization techniques based on maximizing collinearity. We also solve small instances optimally using a SAT formulation. We explored this as part of the 2020 Computational Geometry Challenge, where we placed first as Team UBC.

Cite as

Da Wei Zheng, Jack Spalding-Jamieson, and Brandon Zhang. Computing Low-Cost Convex Partitions for Planar Point Sets with Randomized Local Search and Constraint Programming (CG Challenge). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 83:1-83:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{zheng_et_al:LIPIcs.SoCG.2020.83,
  author =	{Zheng, Da Wei and Spalding-Jamieson, Jack and Zhang, Brandon},
  title =	{{Computing Low-Cost Convex Partitions for Planar Point Sets with Randomized Local Search and Constraint Programming}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{83:1--83:7},
  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.2020.83},
  URN =		{urn:nbn:de:0030-drops-122412},
  doi =		{10.4230/LIPIcs.SoCG.2020.83},
  annote =	{Keywords: convex partition, randomized local search, planar point sets}
}