1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2010 Project

2. Project Deliverables Proposal & Lead Class Discussion 4/15 10% on Related Paper Status Report 4/296% Final Project Report & Class Presentation*5/13 19% * in OS 311 * in OS 311 35% of course grade Deliverable Due Date Grade %

3. Project Guidelines: Proposal ä Objective: State the goal of the project ä State topic/research question ä Provide motivation: why do you find this topic interesting? ä Scope it to be doable in approximately one month ä Plan: List the tasks you need to accomplish ä Resources: What do you need? ä Specialized equipment, language, OS? ä Specialized software/libraries? ä Additional research papers, books? ä More background in some area? ä Assessment Checklist: Characterize your project (see next 2 slides)

4. Guidelines: Proposal (continued) ä Assessment Checklist: ä Characterize your project’s theoretical aspects: ä Algorithmic Paradigm Design ä Analysis Technique Design ä Algorithm Design ä Data Structure Design ä Algorithm and/or Data Structure Analysis ä correctness ä running time and/or space ä Observations/Conjectures Difficulty Creativity Clarity Organization Correctness Scope Impact

5. Guidelines: Proposal (continued) ä Assessment Checklist: ä Characterize your project’s implementation aspects: ä Reuse of existing Code/Libraries ä New Code ä Experimental Design ä Test Suites ä Degenerate/boundary cases ä Numerical robustness Creativity Clarity Impact Difficulty Organization Correctness Scope

6. Guidelines: Class Discussion ä 20 minutes per student ä Briefly state your project’s topic/research question ä Present (with slides) some interesting aspect of what you’ve learned so far from background/related work investigation ä Prepare several questions or observations to use as discussion points ä Lead a class discussion ä Provide handouts: copies of relevant paper from the literature ä Send to me prior to class (preferably several days in advance) and I will have copies made

7. Guidelines: Final Report ä Abstract ä Introduction ä Theoretical Results ä Algorithm ä Implementation ä Results ä Summary & Conclusion ä Future Work ä References Well- written final submissions with research content may be eligible for publishing as UMass Lowell CS technical reports. With some additional work (beyond class) some projects could be polished into CCCG conference submissions.

8. Guidelines: Final Report (continued) ä Abstract: Concise overview (at most 1 page) ä Introduction: ä Motivation: Why did you choose this project? ä State Topic / research question ä Background people need in order to understand project ä Related Work: Context with respect to literature ä Conference, journal papers, web sites ä Summary of Results ä Overview of paper’s organization

9. Guidelines: Final Report (continued) ä Theoretical Results: ä Clear, concise statements of definitions, lemmas, theorems and proofs ä Notation guidelines ä Algorithm: ä High-level algorithm description (& example) ä Algorithmic paradigm ä Data structures ä Pseudocode ä Analysis: ä Correctness ä Solutions generated by algorithm are correct ä account for degenerate/boundary/special cases ä If a correct solution exists, algorithm finds it ä Control structures (loops, recursions,...) terminate correctly ä Asymptotic Running Time and Space Usage

10. Guidelines: Final Report (continued) ä Experimental Design & Implementation: ä Enough of the right kind of information to allow other researchers to duplicate your work ä Resources & environment: ä What language did you code in? ä What existing code did you use? ( software libraries, etc.) ä What equipment did you use? ( machine (& processor speed), OS, compiler) ä Assumptions ä Parameter values ä Treatment of special issues, such as numerical robustness ä How did you decide what kinds of measurements would be meaningful? ä Randomness: statistical significance ä Test cases ä Representative examples ä Controlled tests to establish correctness ä Boundary/extreme cases ä Benchmarks, if available

11. Guidelines: Final Report (continued) ä Results: ä Experimental analysis ä Randomness: statistical analysis ä Test cases ä Tables ä Figures ä Graphs and Charts ä Comparison with benchmarks ä Meaningful measurements: ä CPU time? ä Combinatorial size of output? ä Effect of decisions on issues, such as numerical robustness ä Drawing appropriate conclusions ä Subjective? ä Objective? ä Were the results what you expected?

12. Guidelines: Final Report (continued) ä Summary: ä Summarize what you did ä Conclusion: ä Summarize results & impact ä Future Work: ä What would you do if you had more time? ä References: Bibliography ä Papers, books, web sites that you used ä Consistent format ä All work not your own must be cited! ä Others’ exact words must be quoted!

13. Guidelines: Final Presentation ä 20 minute class presentation ä Explain to the class what you did. ä Structure it any way you like. ä Some ideas: ä slides (electronic or transparency) ä demos ä handouts

14. Project Topics

15. Sample Prior Project Topics ä Multiple robotic arm reachability implementation ä Coreset algorithm implementation for approximate clustering ä Geometric modeling implementation: Marching Cubes ä 2D polygonal covering implementation: ä Constrained triangulation for improved subdivision ä Recursive algorithm for 2-contact group generation ä Orthotopes in 2D and higher dimensions ä Algorithms for approximate 3D convex hull construction ä Parallel coordinates for high-dimensional visualization ä Thrackle reduction theoretical results ä Splines: convex hull of planar splines ä Creating skeletons for complex Chinese characters using stable medial axis method ä New “glow clustering” heuristic inspired by physics of light and images of spiral galaxies ä Real-time map labelling for mobile applications ä Motion planning through a set of static obstacles ä Implementation of a convex polygon intersection algorithm

16. Project Topics (some possibilities) ä Extend a Part I assignment (or a deBerg et al. exercise) ä Work on a problem from an “open problems” list ä Open Problem Project (O’Rourke, Demaine, Mitchell) ä http://maven.smith.edu/~orourke/TOPP/ http://maven.smith.edu/~orourke/TOPP/ ä Many conference, journal papers pose open problems ä Symposium on Computational Geometry ä Computational Geometry: Theory and Applications ä Journal of Experimental Algorithmics ä Algorithm Engineering and Experiments ä Some conferences hold open problem sessions ä Canadian Conference on Computational Geometry

17. Project Topics (some possibilities) Investigate a topic not covered in class ä “Parallel” Computational Geometry (or multiple threads) ä Reference: Parallel Computational Geometry by Akl, Lyons, 1993 ä Parallel 3D meshing ä Randomized Computational Geometry algorithms ä Reference: Computational Geometry: An Introduction Through Randomized Algorithms by Mulmuley, 1994 ä Dynamic Computational Geometry ä Reference: Kinetic Data Structures: A State-of-the-Art Report by Guibas, Proc. 3 rd Workshop on Algorithmic Foundations of Robotics, 1998 ä Specialized Computational Geometry Application Areas: ä Nanomanufacturing: Lattice packings ä Video Games: Graphics ä CGAL library: Add a new feature or experiment with a complex existing feature.

18. Covering: 2D B-Spline Covering [CORS/INFORMS2004, UMass Lowell Student Research Symposium 2004, Computers Graphics Forum, 2006] With graduate student C. Neacsu Supported under NSF/DARPA CARGO program

19. Covering: Box Covering [12 th WSEAS Int. Conf. on Computers, 2008] ä Goal: Translate boxes to cover another box ä Orthotope (box) covering in 2D, 3D, … With Masters student B. England Partial cover (red part uncovered) Full cover 2D views of 3D covering Supported under NSF/DARPA CARGO program

20. Geometric Modeling: Mesh Generation for Finite Element Modeling [Research Note for 17 th Int. Meshing Roundtable, 2008; also presented at 2009 Fall CG Workshop] Doctoral student S. Ye ä Needed for signal integrity in printed circuit board interconnect routing ä 2D constrained Delaunay triangulation is extruded into 3D to form triangular prism mesh Courtesy of Cadence Design Systems

21. Computational Geometry: Thrackle Extensibility [CCCG 2006] ä Thrackle: ä Drawing of a simple graph on the plane: ä each edge drawn as a smooth arc with distinct end-points, ä every two edges have exactly one common point, ä endpoints of each edge are two vertices; ä no edge crosses itself. ä Conway’s thrackle conjecture: ä Number of edges for n vertices is at most n. With graduate student W. Li and Math Prof. Rybnikov

22. Sample Future Packing and Covering Topics ä Packing/Layout: ä 3D translational lattice packings for groups of shapes ä 3D containment using extensions to CGAL ä 3D constrained layout/packing of rectilinear objects ä Covering: ä Explore “phase transitions” for 2D translational covering ä Rotational 2D covering ä Union formulation: ä Target shape does not intersect complement of union of covering shapes ä Combinatorial union equivalence for pairwise Minkowski “difference” ä Regions of Minkowski “difference” that maintain coverage ä Useful for dynamic covering? ä Convex relaxation & linear programming? ä Necessary but not sufficient condition for coverage: ä Target inside complement of convex hull of union