Hot topics and Open problems in Computational Geometry. My (limited) perspective. Class lecture for CSE 546,

Transcription

1 Hot topics and Open problems in Computational Geometry. My (limited) perspective Class lecture for CSE 546, Some slides from this talk are from Jack Snoeyink and L. Kavrakis

2 Key trends on Computational Geometry 10 years ago Hot topics today Focus on discrete problems Points, lines or line segments Real Data: Messy, dirty, and lots of it. Low dimensional problems. Many algorithms have complexities such as O(2 d log n). Now proteins, images, robots with legs all define problems with tens to thousands of dimensions Focus on exact solutions. or approximate w/ bounded error Heuristic approaches often used in graphics/protein modeling, Changing definition of dynamic Dynamic means removing or adding a point. Dynamic means maintaining a data structure as objects move around.

3 Example problem: Nearest Neighbor Search Given: a set P of n points in R d Goal: a data structure, which given a query point q, finds the nearest neighbor p of q in P q p

4 2-D solution Quadtrees. (works for 3D too, oct-trees ). (can t work in 4-D because there is no greek root for 16). Simplest spatial structure on Earth! Split the space into 2 d equal subsquares Repeat until done: only one pixel left only one point left only a few points left Variants: split only one dimension at a time k-d-trees (in a moment)

5 Complaint: time is exponential in dimension What does it mean exactly? Unless we do something really stupid, query time is at most dn (could compute distance to each point and choose the min). Therefore, the actual query time is Min[ dn, exponential(d) ] This is still quite bad though, when the dimension is around Unfortunately, it seems inevitable (both in theory and practice)

6 Approximate nearest neighbor for exact queries, we can use binary search trees or hashing can we adapt hashing to nearest neighbor search?

7 Locality-Sensitive Hashing [Indyk-Motwani 98] Hash functions are locality-sensitive, if, for a random hash random function h, for any pair of points p,q we have: Pr[h(p)=h(q)] is high if p is close to q Pr[h(p)=h(q)] is low if p is far from q

8 Do such functions exist? Consider the hypercube, i.e., points from {0,1} d Hamming distance D(p,q)= # positions on which p and q differ Define hash function h by choosing a set I of k random coordinates, and setting h(p) = projection of p on I

9 Example Take d=10, p= k=2, I={2,5} Then h(p)=11 10-d point. Just worry about 2 dimensions, in particular, the 2 nd and 5 th. For large k, need to be very close for hash to be the same (most bits need to be the same). For large k, don t need to be so close Can randomly choose which 2 dimensions to defeat an adversary

10 How can we use LSH? Choose several h 1..h l Initialize a hash array for each h i Store each point p in the bucket h i (p) of the i-th hash array, i=1...l In order to answer query q for each i=1..l, retrieve points in a bucket h i (q) return the closest point found

11 The LSH algorithm Therefore, we can solve (approximately) the near neighbor problem with given parameter r Worst-case analysis guarantees dn 1/(1+ε) query time Practical evaluation indicates much better behavior [GIM 99,HGI 00,Buh 00,BT 00] Drawbacks: works best for Hamming distance (although can be generalized to Euclidean space) requires radius r to be fixed in advance

12 More High-D problems path planning (used to be a low-d problem).

13 2-D motion CB = B A = {b - a a in A, b in B}

14 Rigid Robot Translating and Rotating in 2-D

15 C-Obstacle for Articulated Robot

16 Motion Planning as a Computational Problem Goal: Characterize the connectivity of a space (e.g., the collision-free subset of configuration space) High computational complexity: Requires time exponential in number of degrees of freedom, or number of moving obstacles, or etc Two main algorithmic approaches: Planning by random sampling Planning by extracting criticalities

17 Principle of Randomized (Probabilistic Roadmap) Planning free space milestone qg qb (Video) [Kavraki, Svetska, Latombe,Overmars,, 95]

18 Why Does it Work? [Kavraki, Latombe, Motwani, Raghavan, 95]

19 Streaming Computation of Delaunay Triangulations (example of modern challenges with real data). Martin Isenburg UC Berkeley Yuanxin Liu UNC Chapel Hill Jonathan Shewchuk UC Berkeley Jack Snoeyink UNC Chapel Hill

20 The 2D Delaunay Triangulation important tool for interpolating points graphics, GIS, finite elements, a triangulation in which every triangle has an empty circumscribing circle

21 Billions of Points courtesy of Neuse-River Basin Elevation map 0.5 billion points 11 GB Contour map

22 input from disk Streaming Computation another small buffer small buffer small buffer in main memory in main memory output to disk finalization tags finalization tags external memory restrict computations to data in buffer process data in the order it arrives output results early and reuse the memory do not use external memory (Video)

23 2006 SOCG papers. Streaming Computation of Delaunay Triangulations. Martin Isenburg, (just did ) Ray Shooting and Intersection Searching Amidst Fat Convex Polyhedra in 3- Space, Boris Aronov,

24 Conclusion, of sorts (for a different lecture this might be a pun). Resurgence of research in computational geometry Sensor networks New problems in nearest neighbor queries and density estimates come from machine learning and data mining applications. Not all high-d data is the same finding the structure of a high-d data set is key to successful applications (robot motion planning and the PRM, large image data sets and manifold learning).