1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 Project

2. Project Deliverables Proposal & Lead Class Discussion 4/11 7% on Related Paper Status Report 4/253% Status Report & Lead Class Discussion 5/2 7% on Project Topic on Project Topic Status Report5/93% Final Project Report & Class Presentation5/16 15% 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 Scope it to be doable in 6 weeks 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 30 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

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.

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 30 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

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 (multiple threads) ä Reference: Parallel Computational Geometry by Akl, Lyons, 1993 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

18. With graduate students R. Inkulu, A. Mathur, C.Neacsu, & UNH professor R. Grinde Covering: 2D Polygonal Covering [CCCG 2001,CCCG2003] Q3Q3 Q1Q1 Q2Q2 Sample P and Q P1P1 P2P2 Translated Q Covers P P1P1 Q1Q1 Q2Q2 Q3Q3 P2P2 Translational 2D Polygon Covering ä Input: ä Covering polygons Q = {Q 1, Q 2,..., Q m } ä Target polygons (or point-sets) P = {P 1, P 2,..., P n } ä Output:  Translations  = {  1,  2,...,  m } such that Supported under NSF/DARPA CARGO program

19. 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

20. Covering: Box Covering 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

21. Covering: Covering Web Site http://www.cs.uml.edu/~kdaniels/covering/covering.htm With graduate student C. Neacsu and undergraduate A. Hussin

22. Sample Future Packing and Covering Topics ä Packing/Layout: ä 3D translational lattice packings for groups of shapes 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

23. Geometric Modeling: Estimating Topological Properties from a Point Sample With graduate student C. Neacsu, UMass Amherst student B. Jones, UML Math Profs. Klain, Rybnikov, students N. Laflin, V. Durante ä Euler characteristic ä Surface area Supported under NSF/DARPA CARGO program heart MRI data Stanford bunny

24. 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

25. With Doctoral student S. Lee Improved Support Vector Clustering [ICBA2004, SIAM Data Mining 2006, UMass Lowell Student Research Symposium 2003 ] ä Goal: Find natural groupings of data points ä Support Vector Clustering based on machine learning method