1 Proximity via Graph Spanners

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1 CS273: Algorithms for Structure Handout # 11 and Motion in Biology Stanford University Tuesday, 4 May 2003 Lecture #11: 4 May 2004 Topics: Proximity via Graph Spanners Geometric Models of Molecules, I Scribe: David Lowsky Note: This is a two-part lecture. 1 Proximity via Graph Spanners 1.1 Introduction In order to simulate molecular interaction forces, it is important to be able to maintain information about proximity between atoms as the position and shape of the molecule changes. Most interaction forces occur are close range, when atoms are 10-20Å apart, and beyond that threshold distance there is effectively zero force. In any basic model of proximity detection, it is necessary to express objects as a collection of elements of the same size and shape. A black box integrator moves the elements each iteration, and upon wake up the proximity information must be updated. This scenario, in which all element positions may be updated in a given iteration, is different from Monte Carlo simulation in which only a few of the positions are updated per iteration. The crucial question then becomes: What do you want to maintain? Is there some way to maintain the proximity information that is simpler and more computationally tractable than an exhaustive neighbor list? Can we store something that captures proximity information that enables fast queries but which will not change too much as the positions of atoms change? (This is referred to as the Faustian Dilemma. ) 1.2 Traditional methods Hashed Voxel Grids One traditional method to maintain proximity information is using a voxel grid. The molecular space is partitioned into a grid of voxels and each atom is assigned to the voxel where it lies. To find an atom s neighbors, the adjacent voxels are searched. A hashtable is used to store only non-empty voxels since many voxels will not contain any atoms. Because rehashing is required when proximity information is desired, the voxel grid method works well once folding has reached its endgame and atoms motion is limited, but is inefficient in the early stages where atoms move a lot between cells. Additionally, efficiency is highly dependent on choosing the right grid size.

2 2 CS273: Handout # Bounding Volume Hierarchies Another method to maintain proximity information is Bounding Volume Hierarchies. BVH have been used successfully to detect collisions between rigid objects. However when objects deform, the hierarchy must be updated upon each deformation which becomes computationally demanding. The additional drawbacks of hierarchies are that they use significant space and require recursive traversal of the hierarchy. 1.3 Geometric Spanners Introduction Geometric spanners enable efficient proximity computation between elements of a deformable structure by replacing expensive geometric search with a lightweight combinatorial graph search. Originally devised for use in communications networks, spanners can be applied to proximity computation between atoms of a deformable molecule. While the field of graph spanners is old and the literature is substantial, this is a unique approach to building spanners that are efficiently maintainable throughout motion of its elements Definition Let G be a full Euclidean graph. An α-spanner is a sparse subgraph G of graph G that approximates true distance, where the shortest path distance in G between two particular nodes is at most α times that of the equivalent shortest path in G. α is known as the spanning ratio. The aspect ratio (G) of graph G is the ratio of the longest distance between any two points in G to the shortest distance between any two points. This ratio is polynomial in the number of nodes in G Proximity queries To find all neighbors of a node u that lie within Euclidean distance d in G, simply traverse the α-spanner G from node u to all nodes up to a distance α d Spanner construction The spanner construction is based on the concept of discrete centers. Given a set of points, the discrete centers of radius r are samplings from the original point set such that every point is within a distance of r from some discrete center, and that no two discrete centers are within a distance of r of each other. Note that a point may be within a distance of r from many centers. We select an arbitrary but fixed center and call it the parent of the point. Assuming that the minimum separation distance between the points is 1, we construct a hierarchy of discrete centers as follows. The bottom level, level 0, is the original point

3 CS273: Handout # 11 3 Figure 1: 1st level centers. Figure 2: 1st level with parent-child relationships drawn. Figure 3: 2nd level centers. Figure 4: Top level centers. Figure 5: 2nd level spanner edges. Figure 6: Parent chain from 1st level node to top level center. set. Given level i 1, level i is simply a set of discrete centers of radius 2 i of the points in level i 1. We add all the parent-child edges in the hierarchy, and also, in each level i, add all center-center edges of length at most c 2 i to graph G, where c = /ɛ.

4 4 CS273: Handout # Parent chain Because all parent-child edges are contained in G, a node p is connected to its ancestor at level i by the parent chain whose total length is 2 i+1. The total path from p to its i-level ancestor is the sum of the intermediate paths between levels: Spanner theorem P ath(p, ancestor) i 2 i+1 The Spanner theorem states that G is a (1 + ɛ)-spanner, where ɛ = 16. Stated another (c 4) way, there exists a path in G between any two points p and q with length at most (1 + ɛ) pq. This theorem is proven as follows (see Figure 7): pq > p i 1 q i i > c 2 i i > c 2 i i 1 > (c 4) 2 i 1 pq c 4 > 2i 1 pq > p i q i 4 2 i pq i > p i q i path(p, q) p i q i i pq i i pq i pq i 1 pq + 16 c 4 pq pq + ɛ pq < (1 + ɛ) pq Maintenance As atoms in a molecule move around, the graph spanner must be modified to reflect the new atom positions. There are four main types of maintenance that must be performed:

5 CS273: Handout # 11 5 Figure 7: Diagram of distances between two nodes. Maintaining parent-child edges 1. Node promotion 2. Node demotion Maintaining within-level spanner edges 3. Edge deletion 4. Edge insertion Node promotion Imagine a node was previously the child of an existing center, but then it moves so that it is no longer within a distance r of that center, and furthermore its new position is not within r of any other existing center. Then this node must be promoted to be a new independent center. Node demotion Imagine a node was previously a center, but then it drifts to a new location within radius r of another existing center. Since no two centers can lie within a distance r of each other, the node must be demoted to a child of the other center. This may have the secondary effect of requiring other nodes that were previously children to this node to be promoted to independent centers, as shown below, or they may become children of other centers. Edge deletion Imagine two level i centers are connected by a spanner edge of level i. Then one of the centers drifts away so that the new distance between the centers exceeds c 2 i. Then this spanner edge must be deleted from G. Edge insertion Imagine two level i centers are unconnected, but in a subsequent iteration they move to within c 2 i of each other. Then a spanner edge that connects them must be added to G. Note that there are potentially quadratically many number of pairs of points in level i that are more than c 2 i apart, and we cannot afford to

6 6 CS273: Handout # 11 Figure 8: Node promotion: Initially a child node; upon leaving parent s radius, node is promoted. Figure 9: Node demotion: Initially a center node; upon entering another center s radius, node is demoted to child and it s previous children become centers. Figure 10: Edge deletion: When a center strays too far from another center, the edge connecting them is removed.

7 CS273: Handout # 11 7 Figure 11: Edge insertion: When a center approaches within a distance c 2 i of another center, a connecting edge is added. check them all at every step. Luckily, it is sufficient to only consider cousin pairs when searching for new required spanner edges. It can be shown that before two nodes are connected, their parents must already be connected Spanner quality and running time The spanner contains O(n) edges total and O(log n) edges per node, and the hierarchy of the spanner has log (G) = O(log n) levels. Node insertion and deletion take O(log n) time. Initial construction takes O(n log n) time, since n nodes are inserted at O(log n) time each. The total number of maintenance changes that the spanner will incur is O(n 2 log n), when the nodes follow semi-algebraic trajectories Performance During performance tests that simulated molecular motion, the geometric spanner underwent few changes. Only 2-4% of spanner edges changed between frames, and most of these changes occurred at the lowest levels in the spanner hierarchy. High-level edges rarely changed. Each frame represented 100 molecular dynamic steps. When changes in atom position require that the spanner be repaired, a repair algorithm is invoked. However repairs do not necessarily have to occur after every frame. The number of frames between calls to the repair algorithm can be set to minimize unnecessary changes and to improve efficiency. Additionally, nodes in the higher levels of the hierarchy may be checked less frequently than those in the lower levels. Updates can be made even more efficient when specific information is known about the physics of the molecular interactions, e.g. knowledge of an upper bound on how fast atoms can move.

8 8 CS273: Handout # Capabilities The following are additional capabilities of the geometric spanner. Neighbor lists To find all neighbors of a node p that lie within Euclidean distance d in G, simply traverse the (1 + ɛ)-spanner from node p to all nodes up to a distance (1 + ɛ)d in the spanner. Since this may have netted some additional nodes that are not within Euclidean distance d of p, these nodes must be filtered out. The cost of the entire procedure is O(n + k), where k is the number of neighbors found. Approximate nearest neighbor The spanner can return an approximate nearest neighbor even faster. Given a node p, the spanner can return some neighbor q that lies within (1 + ɛ) times the distance of the true nearest neighbor q: pq (1 + ɛ) pq. This is accomplished by inserting p into the spanner, then taking its shortest edge. The running time of this procedure is O(log n). Well-separated pair decomposition The spanner can produce a well-separated pair decomposition of the element atoms, which allows the interactions between the atoms to be partitioned into clusters. This is useful when approximating molecular forces as a series of dipole interactions. Each cluster represents the sum of forces of all its atoms and contributes a single net force to the system. Geometric k-centers In addition, the spanner can produce an 8-approximation of the optimal k-center of a collection of points. 1.4 Summary The geometric spanner is a one-stop shopping data structure for proximity information involving elements under motion. It translates a costly geometric search algorithm into an efficient combinatorial algorithm. While it appears that the element positions may be changing drastically, the geometric spanner demonstrates that it is possible to maintain proximity information with a data structure that changes minimally. 2 Geometric Models of Molecules 2.1 Overview: Space-filling diagrams Space-filling diagrams are used to model the space occupied by a molecule. The following models for molecular surface are examined: - van der Walls surface - solvent accessible surface - molecular (Connolly) surface

9 CS273: Handout # 11 9 Figure 12: Left: Van der Waals surface. Right: The outer surface is the solvent accessible surface after fattening, while the inner surface is the molecular surface. 2.2 Union of disks: van der Waals surface Union of disks is a simple model for the surface of a molecule. Given a set B of n disks, each disk b i = z i, r i where z i is the 2-D coordinate of the disk center and r i is the disk radius, the space filling diagram is the union of the disks. If r i is the van der Waals radii, the union of disks produces the van der Waals surface. 2-D Unions of Disks have linear complexity in the number of disks. 2.3 Fattened disks: Solvent accessible surface Fattened disks add an offset r to each disk radius, thereby fattening the outer surface of the molecule. Once fattened, a disk may be rolled along the fattened disk to remove a thin outside layer. However because the rolled disk is round and cannot penetrate sharp crevices, some of the fattened material will remain which has the effect of filling in these crevices in the original van der Waals surface. The outer surface represents the solvent accessible surface, while the inner layer is the molecular or Connolly surface. The advantage of the molecular surface is that it appears smoother and avoids many of the crevices of the van der Waals surface. However even the molecular surface may contain crevices that are unreachable by water molecules. One solution to this problem is to allow surfaces of atoms to intersect themselves, although this is usually not ideal because it creates unusual shapes. 2.4 Molecular sphere model The molecular sphere model is the 3-D version of the molecular surface model. It takes a union of spheres, fattens the spheres, then cuts out excess material from the resulting

10 10 CS273: Handout # 11 Figure 13: Left: van der Waals surface. Right: Molecular sphere model. molecule by rolling a sphere along the surface. The molecular sphere model has a smooth, appealing shape. In the worse case, when the balls vary greatly in radius, the complexity of a molecular model surface is quadratic in the number of balls. In certain scenarios, a hole penetrates through a molecule, but it is not large enough at some point to allow a water molecule to pass all the way through, e.g. the tunnel in the middle of a Gramicidin molecule. 2.5 Ball growth processes To model molecules of different sizes, a Voronoi diagram can be used, where each Voronoi cell in the diagram represents all points in the molecule that are closest to a particular atom A. The bisector surfaces between Voronoi cells are hyperboloids. Specifically: - Distance to ball d i (x) = x z i r i - Voronoi cell C i = {x R 3 d i (x) d j (x), j} - Bisector surfaces x z i x z j = r j r i As the balls are fattened, the diagram will change combinatorially. 2.6 Power distance Another way to model molecular growth is to allow each ball to grow at a different rate, where at time t, the i-th ball grows to radius ri 2 + t. In this case, larger balls grow

11 CS273: Handout # Figure 14: When large and small spheres are separated, equipower line lies between them. When the small sphere moves into the large sphere, equipower line lies to the side of both spheres. more slowly than small balls. For a point in space, the power distance takes the distance to the center of an atom squared, minus the radius of the atom squared, and therefore represents the distance to the tangent of the atom: Power distance = π i (x) = x z i 2 r i 2 An equipower line is the loci of all points that have the same power distance between two balls, and therefore divide two cells in the power diagram. When a small atom intersects with a large sphere, the equipower line will lie outside both atoms, whereas if the small sphere moves outsides the large sphere, the equipower line moves between them. This is illustrated in Figure Power Diagrams and Delaunay Triangulations Power diagrams are Voronoi diagrams in which the cells are defined by power distance. Each cell contains all points whose power distance to the disk in the cell is higher than to any other disk. The primary advantage of using power distance rather than standard distance is that the equipower lines are planes and not the more complex hyperboloids as in the Voronoi diagram above. Theoretically there may be disks in a power diagram with no associated cell. For example, this will occur when two disks share the same center. In this case, any point is closer to the larger disk than the small disk, so the small disk will have no points in the power diagram. A Delaunay Triangulation is a structure which connects each pair of atom centers in the power diagram whose cells are adjacent to each other.

12 12 CS273: Handout # 11 Figure 15: Left: Voronoi diagram with hyperboloid bisectors. Right: Power diagram with straight line bisectors. Figure 16: Delaunay Triangulation. Figure 17: Illustration of orthosphere. 2.8 Orthospheres Two balls b i = z i, r i and b j = z j, r j are orthogonal when the power distance between them is 0: π ij = 0. When this occurs, the intersecting tangents between the two balls intersect at a right angle. When the intersecting tangents between the two balls occurs at an angle less than 90, they are said to be less than orthogonal, while when the angle is greater than 90, they are said to be more than orthogonal. An orthosphere is a sphere that is centered on a Voronoi vertex and orthogonal to all the balls defining the Voronoi vertex. That is, it is orthogonal to all balls whose cells are adjacent to that vertex. Additionally, an orthosphere is less than orthogonal to all other spheres in the Voronoi diagram.

13 CS273: Handout # References [1] H. Edelsbrunner. CPS Lecture Notes: Bio-Geometric Modeling. [2] J. Gao, L. J. Guibas, and A. Nguyen. Deformable Spanners and Applications. To appear in Symposium on Computational Geometry, 2004.