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Proceedings in a Single Schedule of Talks 27th Annual Fall Workshop on Computational Geometry November 3–4, 2017 Stony Brook University Stony Brook, NY Sponsored by the National Science Foundation and the Department of Applied Mathematics and Statistics and the Department of Computer Science, College of Engineering and Applied Sciences, Stony Brook University Program Committee: GregAloupis(Tufts) PeterBrass(CCNY) WilliamRandolphFranklin(RPI) JieGao(co-chair,StonyBrook) MayankGoswami(QueensCollege) MatthewP.Johnson(Lehman College) Joseph S. B. Mitchell (co-chair, Stony Brook) Don Sheehy (University of Connecticut) Jinhui Xu (University at Buffalo) Friday, November 3, 2017: Computer Science (NCS), Room 120 9:20 Opening remarks Contributed Session 1 Chair: Joe Mitchell 9:30–9:45 “Sampling Conditions for Clipping-free Voronoi Meshing by the VoroCrust Algorithm”, Scott Mitchell, Ahmed Abdelkader, Ahmad Rushdi, Mohamed Ebeida, Ahmed Mahmoud, John Owens and Chandrajit Bajaj 9:45–10:00 “Quasi-centroidal Simplicial Meshes for Optimal Variational Methods”, Xiangmin Jiao 10:00–10:15 “A Parallel Approach for Computing Discrete Gradients on Mulfiltrations”, Federico Iuricich, Sara Scaramuccia, Claudia Landi and Leila De Floriani 10:15–10:30 “Parallel Intersection Detection in Massive Sets of Cubes”, W. Randolph Franklin and Salles V.G. Magalh˜aes 10:30–11:00 Break. Contributed Session 2 Chair: Jie Gao 11:00–11:15 “Dynamic Orthogonal Range Searching on the RAM, Revisited”, Timothy M. Chan and Konstantinos Tsakalidis 11:15–11:30 “Faster Algorithm for Truth Discovery via Range Cover”, Ziyun Huang, Hu Ding and Jinhui Xu 11:30–11:45 “An Efficient Sum Query Algorithm for Distance-based Locally Dominating Func- tions”, Ziyun Huang and Jinhui Xu 11:45–12:00 “Approximate Convex Hull of Data Streams”, Avrim Blum, Vladimir Braverman, Ananya Kumar, Harry Lang and Lin Yang 12:00–1:00 Lunch (provided) 1:00–2:00 Invited Talk: “The geometry of data structures”, John Iacono (NYU Tandon School of Engineering and UniversitLibre´ de Bruxelles) Abstract: A technique has been developed to study and analyze data structures called the geometric view. Several examples of the use of this view will be presented including, rotation- based binary search tree data structure, cache-oblivious persistence, and disjoint sets. In all of these examples the main theme is that by moving to the geometric view, details of the data structure that seem cumbersome and hard to approach are abstracted in such a way that allows a simple, yet equivalent, view. Contributed Session 3 Chair: Jinhui Xu 2:00–2:15 “Online Unit Covering in L2”, Anirban Ghosh 2:15–2:30 “An O(n log n)-Time Algorithm for the k-Center Problem in Trees”, Haitao Wang and Jingru Zhang 2:30–2:45 “Compact Data Structures for Abstract Order Types and Optimal Encodings for SUM Problems”, Jean Cardinal, Timothy Chan, John Iacono, Stefan Langerman and Aur´elien Ooms 2:45–3:00 “Lightweight Sketches for Mining Trajectory Data”, Maria Astefanoaei, Panagiota Kat- sikouli, Mayank Goswami and Rik Sarkar 3:00–3:30 Break Contributed Session 4 Chair: Don Sheehy 3:30–3:45 “Calculating the Dominant Guard Set of a Simple Polygon”, Eyup Serdar Ayaz and Alper Ungor 3:45–4:00 “Perfect Polygons”, Hugo A. Akitaya, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Joseph S.B. Mitchell and David Stalfa 4:00–4:15 “Efficient Approximations for the Online Dispersion Problem”, Jing Chen, Bo Li and Yingkai Li 4:15–4:30 “Edge Conflicts Can Increase Along Minimal Rotation-Distance Paths”, Sean Cleary and Roland Maio 4:30–4:45 “Optimal Safety Patrol Scheduling Using Randomized Traveling Salesman Tour”, Hao- Tsung Yang, Shih-Yu Tsai, Jie Gao and Shan Lin 5:00–6:00 Open Problem Session Saturday, November 4, 2017: Computer Science (NCS), Room 120 Contributed Session 5 Chair: Jie Gao 9:30–9:45 “On the Density of Triangles with Periodic Billiard Paths”, Ramona Charlton 9:45–10:00 “Nearest Neighbor Condensation with Guarantees”, Alejandro Flores Velazco and David Mount 10:00–10:15 “On the Complexity of Random Semi Voronoi Diagrams”, Chenglin Fan and Ben- jamin Raichel 10:15–10:30 “Efficient Algorithms for Computing a Minimal Homology Basis”, Tamal K. Dey, Tianqi Li and Yusu Wang 10:30–11:00 Break Contributed Session 6 Chair: Matt Johnson 11:00–11:15 “Cardiac Trabeculae Segmentation: an Application of Computational Topology”, Chao Chen, Dimitris Metaxas, Yusu Wang, Pengxiang Wu and Changhe Yuan 11:15–11:30 “Topological and Geometric Reconstruction of Metric Graphs in Rn”, Brittany Fasy, Rafal Komendarc´zyk, Sushovan Majhi and Carola Wenk 11:30–11:45 “Randomized Incremental Construction of Net-Trees”, Mahmoodreza Jahanseir and Donald Sheehy 11:45–12:00 “On Computing a Timescale Invariant Bottleneck Distance”, Nicholas J. Cavanna, Oliver Kisielius and Donald R. Sheehy 12:00–1:00 Lunch (provided) 1:00–2:00 Invited Talk: “Sublinear algorithms for outsourced data analysis”, Suresh Venkatasub- ramanian (University of Utah) Abstract: In the era of outsourcing, communication has replaced computation as an expensive resource. Researchers have proposed numerous models for communication-efficient computing that draw on classical ideas of interactive proofs, updated for our more “sublinear” world. I’ll talk about the model of streaming interactive proofs and how we can solve classic problems in data analysis like near neighbor search, minimum enclosing balls, and range searching in general with sublinear communication. Contributed Session 7 Chair: Mayank Goswami 2:00–2:15 “Improved Results for Minimum Constraint Removal”, Eduard Eiben, Jonathan Gem- mell, Iyad Kanj and Andrew Youngdahl 2:15–2:30 “Algorithm for Optimal Chance Constrained Knapsack with Applications to Multi- robot Teaming”, Fan Yang and Nilanjan Chakraborty 2:30–2:45 “Freeze Tag Awakening in 2D is NP-hard”, Hugo Akitaya, Jingjin Yu and Zachary Abel 2:45–3:00 “Freeze Tag is Hard in 3D”, Erik D. Demaine and Mikhail Rudoy; Matthew P. Johnson (“Easier Hardness for 3D Freeze-Tag”) 3:00–3:30 Break Contributed Session 8 Chair: W. Randolph Franklin 3:30–3:45 “Declarative vs Rule-based Control for Flocking Dynamics”, Usama Mehmood, Nicola Paoletti, Dung Phan, Radu Grosu, Shan Lin, Scott Stoller, Ashish Tiwari, Junxing Yang and Scott Smolka 3:45–4:00 “Realizing Minimum Spanning Trees from Random Embeddings”, Saad Quader, Alexan- der Russell and Ion Mandoiu 4:00–4:15 “The Minimum Road Trips Problem”, Samuel Micka and Brendan Mumey 4:15–4:30 “Harder Hardness of Approximation for 2D r-Gather”, Matthew P. Johnson Sampling Conditions for Clipping-free Voronoi Meshing by the VoroCrust Algorithm Scott Mitchell1, Ahmed Abdelkader2, Ahmad Rushdi1;3, Mohamed Ebeida1, Ahmed Mahmoud3, John Owens3 and Chandrajit Bajaj4 1 Sandia National Laboratories 2 University of Maryland, College Park 3 University of California, Davis 4 University of Texas, Austin For presentation at the Fall Workshop on Computational Geometry (FWCG 2017) Abstract Decomposing the volume bounded by a closed surface using simple cells is a key step in many applications. Although tetrahedral meshing is well-established with many successful implementations, many research questions remain open for polyhedral meshing, which promises significant advantages in some contexts. Unfortunately, current approaches to polyhedral meshing often resort to clipping cells near the boundary, which results in certain defects. In this talk, we present an analysis of the VoroCrust algorithm, which leverages ideas from α-shapes and the power crust algorithm to produce conforming Voronoi cells on both sides of the surface. We derive sufficient conditions for a weighted sampling to produce a topologically-correct and geometrically-accurate reconstruction, using the -sampling paradigm with a standard sparsity condition. The resulting surface reconstruction consists of weighted Delaunay triangles, except inside tetrahedra with a negative weighted circumradius where a Steiner vertex is generated close to the surface. Generating quality meshes is an important problem in computer graphics and geometric modeling. Polyhedral meshing offers higher degrees of freedom per element than tetrahedral or hex-dominant meshing, and is more efficient in filling a space, because it produces fewer elements for the same number of nodes. Within the class of polyhedral mesh elements, Voronoi cells are particularly useful in numerical simulations for their geometric properties, e.g., planar facets and positive Jacobians. A conforming mesh exhibits two desirable properties simultaneously: 1) a decomposition of the enclosed volume, and 2) a reconstruction of the bounding surface. A common technique for producing boundary-conforming decomposition from Voronoi cells relies on clipping, i.e., intersecting and truncating, each cell by the bounding surface [3]. An alternative to clipping is to locally mirror the Voronoi generators on either side of the surface [2]. VoroCrust can be viewed as a principled mirroring technique. Similar to the power crust [1], the reconstruction is composed of the facets shared by cells on the inside and outside of the manifold. However, VoroCrust uses pairs of unweighted generators tightly hugging the surface, which allows further decomposition of the interior without disrupting the surface reconstruction. VoroCrust can also be viewed as a special case of the weighted α-shape [4]. A description of the abstract VoroCrust algorithm we analyze is provided next. Figure1 illustrates the basic concepts in 2D. 1 The Abstract VoroCrust Algorithm 1. Take as input a weighted point sampling P = f(pi; wi)g of a closed 2-manifold M. p 2. Use weights to define a ball Bi of radius ri = wi centered at each sample pi. 3. Find intersecting triplet of balls to obtain one corner point on either side of M. 4. Collect the Voronoi generators G as the set of corner points outside all sample balls. 5. Optionally, include in G more generators far-inside M to further decompose its interior. 6. Output the Voronoi diagram of G as the desired decomposition, and the facets separating the inside and outside generators as the reconstructed surface M^ .
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