9 Search Results for "Chang, Hsien-Chih"

Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.8
Clustering Under Perturbation Stability in Near-Linear Time

Authors: Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, and Emo Welzl

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is α-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most α. Our main contribution is in presenting efficient exact algorithms for α-stable clustering instances whose running times depend near-linearly on the size of the data set when α ≥ 2 + √3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when α ≥ 2 + √3 + ε for any constant ε > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for α > 5 in any fixed dimension; and for α ≥ 2 + √3 in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.
Cite as

Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, and Emo Welzl. Clustering Under Perturbation Stability in Near-Linear Time. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agarwal_et_al:LIPIcs.FSTTCS.2020.8,
  author =	{Agarwal, Pankaj K. and Chang, Hsien-Chih and Munagala, Kamesh and Taylor, Erin and Welzl, Emo},
  title =	{{Clustering Under Perturbation Stability in Near-Linear Time}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.8},
  URN =		{urn:nbn:de:0030-drops-132492},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.8},
  annote =	{Keywords: clustering, stability, local search, dynamic programming, coreset, polyhedral metric, trapezoid decomposition, range query}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.2
Dynamic Geometric Set Cover and Hitting Set

Authors: Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue

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

Abstract
We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ¹ and ℝ². The first framework uses bootstrapping and gives a (1+ε)-approximate data structure for dynamic interval set cover in ℝ¹ with O(n^α/ε) amortized update time for any constant α > 0; in ℝ², this method gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set with O(n^(1/2+α)) amortized update time. The second framework uses local modification, and leads to a (1+ε)-approximate data structure for dynamic interval hitting set in ℝ¹ with Õ(1/ε) amortized update time; in ℝ², it gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.
Cite as

Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic Geometric Set Cover and Hitting Set. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2020.2,
  author =	{Agarwal, Pankaj K. and Chang, Hsien-Chih and Suri, Subhash and Xiao, Allen and Xue, Jie},
  title =	{{Dynamic Geometric Set Cover and Hitting Set}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{2:1--2:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.2},
  URN =		{urn:nbn:de:0030-drops-121604},
  doi =		{10.4230/LIPIcs.SoCG.2020.2},
  annote =	{Keywords: Geometric set cover, Geometric hitting set, Dynamic data structures}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.45
A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread

Authors: Kyle Fox and Jiashuai Lu

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

Abstract
The geometric transportation problem takes as input a set of points P in d-dimensional Euclidean space and a supply function μ : P → ℝ. The goal is to find a transportation map, a non-negative assignment τ : P × P → ℝ_{≥ 0} to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., ∑_{r ∈ P} τ(q, r) - ∑_{p ∈ P} τ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, ∑_{(p, q) ∈ P × P} τ(p, q) ⋅ ||q - p||₂. We describe the first algorithm for this problem that returns, with high probability, a (1 + ε)-approximation to the optimal transportation map in O(n poly(1 / ε) polylog n) time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of P and the magnitude of its real-valued supplies.
Cite as

Kyle Fox and Jiashuai Lu. A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2020.45,
  author =	{Fox, Kyle and Lu, Jiashuai},
  title =	{{A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{45:1--45:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.45},
  URN =		{urn:nbn:de:0030-drops-122034},
  doi =		{10.4230/LIPIcs.SoCG.2020.45},
  annote =	{Keywords: Transportation map, earth mover’s distance, shape matching, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.81
Spectral Aspects of Symmetric Matrix Signings

Authors: Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, and Alexandra Kolla

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of finding symmetric signings of matrices with natural spectral properties. Our results are the following: 1) We characterize matrices that have an invertible signing: a symmetric matrix has an invertible symmetric signing if and only if the support graph of the matrix contains a perfect 2-matching. Further, we present an efficient algorithm to search for an invertible symmetric signing. 2) We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing. 3) We show NP-completeness of the following problems: verifying whether a given matrix has a symmetric signing that is singular or has bounded eigenvalues. However, we also illustrate that the complexity could differ substantially for input matrices that are adjacency matrices of graphs. We use combinatorial techniques in addition to classic results from matching theory.
Cite as

Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, and Alexandra Kolla. Spectral Aspects of Symmetric Matrix Signings. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carlson_et_al:LIPIcs.MFCS.2019.81,
  author =	{Carlson, Charles and Chandrasekaran, Karthekeyan and Chang, Hsien-Chih and Kakimura, Naonori and Kolla, Alexandra},
  title =	{{Spectral Aspects of Symmetric Matrix Signings}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{81:1--81:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.81},
  URN =		{urn:nbn:de:0030-drops-110258},
  doi =		{10.4230/LIPIcs.MFCS.2019.81},
  annote =	{Keywords: Spectral Graph Theory, Matrix Signing, Matchings}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.6
Efficient Algorithms for Geometric Partial Matching

Authors: Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Let A and B be two point sets in the plane of sizes r and n respectively (assume r <= n), and let k be a parameter. A matching between A and B is a family of pairs in A x B so that any point of A cup B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = sum_{(a, b) in M}||a-b||_p^q where ||*||_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B. We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2)polylog n) time, and a (1 + epsilon)-approximation algorithm that runs in O((n + k sqrt{k})polylog n * log epsilon^{-1}) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O(min{n^2, rn^{3/2}}polylog n) time.
Cite as

Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao. Efficient Algorithms for Geometric Partial Matching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.6,
  author =	{Agarwal, Pankaj K. and Chang, Hsien-Chih and Xiao, Allen},
  title =	{{Efficient Algorithms for Geometric Partial Matching}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.6},
  URN =		{urn:nbn:de:0030-drops-104109},
  doi =		{10.4230/LIPIcs.SoCG.2019.6},
  annote =	{Keywords: partial matching, transportation, optimal transport, minimum-cost flow, bichromatic closest pair}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.25
Lower Bounds for Electrical Reduction on Surfaces

Authors: Hsien-Chih Chang, Marcos Cossarini, and Jeff Erickson

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We strengthen the connections between electrical transformations and homotopy from the planar setting - observed and studied since Steinitz - to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the number of electrical transformations required to reduce an n-vertex graph on surface in the worst case [SOCG 2016] in two different directions. Our previous Omega(n^{3/2}) lower bound applies only to facial electrical transformations on plane graphs with no terminals. First we provide a stronger Omega(n^2) lower bound when the planar graph has two or more terminals, which follows from a quadratic lower bound on the number of homotopy moves in the annulus. Our second result extends our earlier Omega(n^{3/2}) lower bound to the wider class of planar electrical transformations, which preserve the planarity of the graph but may delete cycles that are not faces of the given embedding. This new lower bound follow from the observation that the defect of the medial graph of a planar graph is the same for all its planar embeddings.
Cite as

Hsien-Chih Chang, Marcos Cossarini, and Jeff Erickson. Lower Bounds for Electrical Reduction on Surfaces. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2019.25,
  author =	{Chang, Hsien-Chih and Cossarini, Marcos and Erickson, Jeff},
  title =	{{Lower Bounds for Electrical Reduction on Surfaces}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{25:1--25:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.25},
  URN =		{urn:nbn:de:0030-drops-104294},
  doi =		{10.4230/LIPIcs.SoCG.2019.25},
  annote =	{Keywords: electrical transformation, Delta-Y-transformation, homotopy, tight, defect, SPQR-tree, smoothings, routing set, 2-flipping}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.16
Near-Optimal Distance Emulator for Planar Graphs

Authors: Hsien-Chih Chang, Pawel Gawrychowski, Shay Mozes, and Oren Weimann

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator. We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3}).
Cite as

Hsien-Chih Chang, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Near-Optimal Distance Emulator for Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chang_et_al:LIPIcs.ESA.2018.16,
  author =	{Chang, Hsien-Chih and Gawrychowski, Pawel and Mozes, Shay and Weimann, Oren},
  title =	{{Near-Optimal Distance Emulator for Planar Graphs}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.16},
  URN =		{urn:nbn:de:0030-drops-94796},
  doi =		{10.4230/LIPIcs.ESA.2018.16},
  annote =	{Keywords: planar graphs, shortest paths, metric compression, distance preservers, distance emulators, distance oracles}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.29
Untangling Planar Curves

Authors: Hsien-Chih Chang and Jeff Erickson

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.
Cite as

Hsien-Chih Chang and Jeff Erickson. Untangling Planar Curves. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2016.29,
  author =	{Chang, Hsien-Chih and Erickson, Jeff},
  title =	{{Untangling Planar Curves}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.29},
  URN =		{urn:nbn:de:0030-drops-59218},
  doi =		{10.4230/LIPIcs.SoCG.2016.29},
  annote =	{Keywords: computational topology, homotopy, planar graphs, Delta-Y transformations, defect, Reidemeister moves, tangles}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.689
From Proximity to Utility: A Voronoi Partition of Pareto Optima

Authors: Hsien-Chih Chang, Sariel Har-Peled, and Benjamin Raichel

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We present an extension of Voronoi diagrams where not only the distance to the site is taken into account when considering which site the client is going to use, but additional attributes (i.e., prices or weights) are also considered. A cell in this diagram is then the loci of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest.
Cite as

Hsien-Chih Chang, Sariel Har-Peled, and Benjamin Raichel. From Proximity to Utility: A Voronoi Partition of Pareto Optima. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 689-703, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chang_et_al:LIPIcs.SOCG.2015.689,
  author =	{Chang, Hsien-Chih and Har-Peled, Sariel and Raichel, Benjamin},
  title =	{{From Proximity to Utility: A Voronoi Partition of Pareto Optima}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{689--703},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.689},
  URN =		{urn:nbn:de:0030-drops-50925},
  doi =		{10.4230/LIPIcs.SOCG.2015.689},
  annote =	{Keywords: Voronoi diagrams, expected complexity, backward analysis, Pareto optima, candidate diagram, Clarkson-Shor technique}
}
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