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

Transcription

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

2 Project Deliverables Deliverable Due Date Grade % Proposal & Lead Class Discussion 4/11 7% on Related Paper Status Report 4/25 3% Status Report & Lead Class Discussion 5/2 7% on Project Topic Status Report 5/9 3% Final Project Report & Class Presentation 5/16 15% 35% of course grade

3 Project Guidelines: Proposal Objective: State the goal of the project State topic/research question Scope it to be doable in 6 weeks Plan: List the tasks you need to accomplish Resources: What do you need? Specialized equipment, language, OS? Specialized software/libraries? Additional research papers, books? More background in some area? Assessment Checklist: Characterize your project (see next 2 slides)

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

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

6 Guidelines: Class Discussion 30 minutes per student Briefly state your project s topic/research question Present (with slides) some interesting aspect of what you ve learned so far from background/related work investigation Prepare several questions or observations to use as discussion points Lead a class discussion Provide handouts: copies of relevant paper from the literature

7 Guidelines: Final Report Abstract Introduction Theoretical Results Algorithm Implementation Results Summary & Conclusion Future Work References Well- written final submissions with research content may be eligible for publishing as UMass Lowell CS technical reports.

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

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

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

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

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

13 Guidelines: Final Presentation 30 minute class presentation Explain to the class what you did. Structure it any way you like. Some ideas: slides (electronic or transparency) demos handouts

14 Project Topics

15 Sample Prior Project Topics Multiple robotic arm reachability implementation Coreset algorithm implementation for approximate clustering Geometric modeling implementation: Marching Cubes 2D polygonal covering implementation: Constrained triangulation for improved subdivision Recursive algorithm for 2-contact group generation Orthotopes in 2D and higher dimensions Algorithms for approximate 3D convex hull construction Parallel coordinates for high-dimensional visualization Thrackle reduction theoretical results Splines: convex hull of planar splines

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

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

18 Input: Covering: 2D Polygonal Covering [CCCG 2001,CCCG2003] Covering polygons Q = {Q 1, Q 2,..., Q m } Target polygons (or point-sets) P = {P 1, P 2,..., P n } Output: Supported under NSF/DARPA CARGO program Translations g = {g 1, g 2,..., g m } such that P 1 jm g ( j Q j ) P 2 Translational 2D Polygon Covering P 2 P 1 Q 1 Q 2 Q 3 Q 3 P 1 Q 2 Q 1 Sample P and Q Translated Q Covers P With graduate students R. Inkulu, A. Mathur, C.Neacsu, & UNH professor R. Grinde

19 Covering: 2D B-Spline Covering [CORS/INFORMS2004, UMass Lowell Student Research Symposium 2004, Computers Graphics Forum, 2006] Supported under NSF/DARPA CARGO program T 1 Out E S I In T 2 With graduate student C. Neacsu

20 Covering: Box Covering Goal: Translate boxes to cover another box Orthotope (box) covering in 2D, 3D, 2D views of 3D covering Partial cover (red part uncovered) Full cover With Masters student B. England

21 Covering: Covering Web Site With graduate student C. Neacsu and undergraduate A. Hussin

22 Sample Future Packing and Covering Topics Packing/Layout: 3D translational lattice packings for groups of shapes 3D constrained layout/packing of rectilinear objects Covering: Explore phase transitions for 2D translational covering Rotational 2D covering Union formulation: Target shape does not intersect complement of union of covering shapes Combinatorial union equivalence for pairwise Minkowski difference Regions of Minkowski difference that maintain coverage Useful for dynamic covering? Convex relaxation & linear programming? Necessary but not sufficient condition for coverage: 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 Euler characteristic Surface area Supported under NSF/DARPA CARGO program heart MRI data Stanford bunny

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

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