Data Structures for Orthogonal Range Queries

1. Data Structures for Orthogonal Range Queries A New Data Structure and Comparison to Previous Work. Application to Contact Detection in Solid Mechanics. Sean Mauch Caltech April, 2003

2. Terminology: Orthogonal Range Queries • An orthogonal range query, (ORQ), determines the points in a set which lie within a given window (bounding box). • To the right is a set of points representing the location and population of cities. • We do an orthogonal range query on the cities We show the window and its projections onto the coordinate planes. 2

3. Example Application: Contact Detection • Algorithm for solid mechanics simulation with contact detection: • Search for potential contacts between nodes and surfaces. Which nodes are close to each face? • Do detailed contact check among potential contacts. • Enforce contacts by computing force required to remove overlap. 3 Contact search. Contact check.

4. Contact Detection Depends on ORQ’s • Searches for potential contacts are done with orthogonal range queries. • The capture box contains the nodes that may contact the face. • An ORQ finds the nodes in the capture box. • Detecting possible contacts is computationally expensive, typically half of total solid simulation execution time. • Good performance depends on choosing an efficient data structure and search algorithm for the orthogonal range queries. 4

5. Data Structures for ORQ’s • Octrees and kd-trees. • Tried and tested data structures from computational geometry. • Point in box method, (Swegle, et al.). • Developed at Sandia National Laboratories for doing parallel contact. • Cell arrays. • A bucket sort in 3-D. • Optimal computational complexity when properly tuned. • Dense arrays may have a high storage overhead. • Cell arrays coupled with binary and forward searching. • New data structure. 5

6. Kd-trees • Kd-trees recursively divide the domain by choosing the median in the chosen coordinate direction. • Depth is determined only by the number of points. 6

7. Octrees • Quadtrees (2-D) and Octrees (3-D) recursively divide space into quadrants or octants. • Depth is determined by the number of points and point spacing. 7

8. Point-in-box Method (Swegle et. al.) • Sorts and ranks the points in each coordinate direction. • There are three slices for a window. Choose the slice with the least points and see if those points are in the window. • The rank array allows integer instead of floating point comparisons which is much faster on some processors. 8

9. Cell Arrays • The computational domain is spanned by an array of cells. Each cell contains a set of points. • Constant time access to cells, but perhaps large storage overhead because of empty cells. 9

10. Sparse Cell Arrays • Array is sparse in one dimension. • Store only non-empty cells. • Trade reduced storage for increased cell access time. 10

11. Cell Arrays Coupled with Searching • Cell array does not span one dimension. Produces long, thin cells. • Compared to sparse cell array, we search on records instead of on cells. • Binary search on records is similar to sparse cell array. • If we sort the queries we can do forward searching in each cell. 11

12. List of Methods 12

13. Parameters • N: The number of records. • K: The records are in K-dimensional space. • M: The number of cells. • I: The number of records in the query. • R: The ratio of the length of a query range to the length of a cell. • Q: The number of queries. 13

14. Computational Complexity Table 14

15. Test Problem: Randomly Distributed Points • Records are randomly distributed points in the unit cube. • Number of records varies from 100 to 1,000,000. • ORQ on a box containing approximately 10 records is performed around each point. 15

16. Execution Time – Random Points 16

17. Memory Usage – Random Points 17

18. Conclusions • Projection methods like Point-In-Box perform poorly. • The kd-tree does not yield the best performance, but it performs fairly well over a wide range of problems. • Cell methods offer the best performance when they are well suited to the problem at hand. • Dense cell arrays work well for uniformly distributed data. • Sparse cell arrays and cell arrays coupled with binary searching offer good performance for most problems. • Cell arrays coupled with forward searching offers the best performance when doing multiple queries. 18