Jie Gao, Leonidas Guibas, An Nguyen Computer Science Department Stanford University

1. Flexible Spanners:A Proximity and Collision Detection Tool for Molecules and Other Deformable Objects Jie Gao, Leonidas Guibas, An Nguyen Computer Science Department Stanford University

2. Most forces in nature are short range: e.g.,neighbor lists in MD Self-collision detection Proximity Maintenance in Physical Simulations Cut-offs for Lennard-Jones potentials

3. Models of Deformable Objects • Deformable objects are modeled as connected collections of small elements of comparable size • Focus mostly on “linear” objects (macromolecules, but also vines, ropes, chains) • Each atom (element) moves a small distance after each simulation time step (discrete integration)

4. How much to know about object state? • The integrator (MD) gives us the new positions of the atoms. • How can we efficiently maintain self-proximity information, after small displacements of the atoms? • To do so, we want to capture and maintain information that is useful for proximity detection yet is relatively stable and easy to update at the same time.

5. Traditional: Hashed Voxel Grids • Partition space into a voxel grid and apportion the atoms of the molecule into cells of that grid • Use a hash table, since many of the cells will be empty • To find the neighbors of a given atom, search nearby cells • Efficiency highly dependent on voxel grid size chosen • For the right size grid, performs quite well for neighbor maintenance • Usually requires re-hashing all the atoms when proximity information is desired • Incremental updating can be good for end-game in folding, when atoms move only little • Bad for unfolded phases, when there are few proximities but large motions cause many atoms to change cells

6. Traditional: Bounding Volume Hierarchies • Bounding volume hierarchies (BVH), using spheres, bounding boxes, etc., have been successfully used for collision checking of rigid objects • Rigid bounding volume hierarchies are good for rigid objects, but • Use a fair amount of space • Collision checking requires a hierarchy traversal • Hierarchy must be updated when deformation occurs

7. The graph distance is within a constant factor of the Euclidean distance. New: Molecular Spanners A notion from graph theory/communication networks Spanning ratioα

8. Graph setting: Replace a dense graph with a sparse subgraph (the spanner), while approximately preserving shortest paths. Geometry setting: Approximate all distances between points using shortest paths on a graph defined by a sparse set of edges (the spanner). Graph and Geometric Spanners

9. Spanners for Continuous Objects Add a sparse set of shortcuts, sufficient to guarantee the spanning property 3HVT A protein example with α = 3

10. Spanners for Proximity Maintenance • Lightweight, sparse, combinatorial structure • Highly non-canonical • Can be chosen so as to be very stable under smooth deformations (redundancy helps) • To find all points at distance d from p, find all points within distance αd of p along the object and its shortcuts. • Before two elements p and q on a deformable object can collide, there has to be a shortcut between them.

11. Discrete Centers with Radius r parent-child

12. Spanner Graph Construction(step 1/2) Construct a hierarchy of levels • R0 is the original point set • Ri is a set of discrete centers of Ri-1 with radius 2i All parent-child edges are in the spanner

13. c 2i Spanner Graph Construction(step 2/2) • In each level Ri, put in the spanner all edges of length · c ¢ 2i, where c, c > 4, is a parameter

14. Spanner Theorem • G is a (1+e)-spanner [expansion factor α = 1+e], where e = 16/(c-4). Level i · 2i · 2i Level i-1 > c ¢2i-1 · 2i · 2i Level 0 p q path(p,q) · |pq| + 8 2i|pq| > (c - 4) 2i-1

15. More Spanner Observations • In level Ri • Nodes are at least 2i apart • Edges have length at most c ¢2i • Number of nodes is O(n/2i) • Each node is incident to at most O(1) edges • Number of edges is O(n/2i) • Total length of all edges is O(n)

16. Spanner Quality • Linear size: O(n) edges • Total length of all edges: O(n log n) • Maximum degree of a node: O(log n)

17. Maintenance under Deformation The correctness of the spanner can be certified by O(n) KDS-style certificates.These are all simple distance inequalities on the lengths of spanner edges. After each integration step we must verify and update the spanner – may need to deal with multiple certificate failures: • Verify and update the hierarchy • Verify and update spanner edges • We can exploit conservative bounds on failure times of certificates derived from prior node velocity bounds to avoid checking all certificates at each step.

18. ?? ? Updating the Hierarchy • In Ri • If two nodes come within 2i, remove one of the two nodes from Ri [node demotion] • If a node is not covered by any of the centers in Ri+1, add the node to Ri+1 [node promotion]

19. Updating Edges: Which Node Pairs Moved Near or Apart? • In Ri • If an edge has length > c ¢2i, remove it [easy] • Find new pairs of nodes within c ¢ 2i [interesting] • Sufficient to consider only “cousin” pairs Distant pairs coming close are always the issue ?? ?

20. Molecular Dynamics • Each frame corresponds to a hundred actual MD steps (Tinker data) • Only 2-4% of spanner edges change between frames • Spanner is quite stable, except for bottom-level edges (high frequency atomic vibrations) green = edge birth, red = edge death, gray = steady state

21. More Data (Erik Lindahl) 10 MD steps between frames

22. Additional Spanner Properties, I • Can be used to compute neighbor lists within any distance constraint, in an output-sensitive manner • Number of pairs of atoms examined is proportional to the number of pairs reported

23. Additional Spanner Properties, II • Our spanner defines a well-separated pair decomposition for the atoms • Electrostatic interactions can be modeled as dipole-dipole • Provides an n-body style approximation, but with a structure attached to the molecule and not to the ambient space

24. Well Separated Pair Decomposition Theorem • G induces a s-WSPD, where s = c/4 - 1. Level i P Q Level i-1 > c ¢2i-1 · 2i · 2i Level 0 p q |PQ| > (c - 4) 2i-1 Diam(P) < 2¢2i

25. Final Remarks • Unlike bounding volume hierarchies, the spanner is a lightweight, combinatorial structure • Unlike voxel grids, the spanner is essentially unchanged for mostly rigid molecular motions • We are also using it for macro-level simulations (rope, chain, filament) A Cosserat rope