Data Structures for Orthogonal Range Queries

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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

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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.

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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.

Contact search.
Contact check.
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Contact Detection Depends on ORQs

• 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.

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Data Structures for ORQs

• 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.

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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.

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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.

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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.

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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.

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Sparse Cell Arrays

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

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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.

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List of Methods
Label Description
seq. scan sequential scan
projection projection
pt-in-box point-in-box
kd-tree kd-tree
kd-tree d. kd-tree with domain checking
octree octree
cell cell array
sparse cell sparse cell array
cell b. s. cell array with binary search
cell f. s. cell array with forward search
cell f. s. k. cell array with forward search on keys
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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.

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Computational Complexity Table
Method Expected Complexity of ORQ Storage
seq. scan N N
projection K log NN1-1/K K N
pt-in-box K log NN1-1/K (2K1) N
kd-tree log NI N
kd-tree d. log NI N
octree log NI (D1) N
cell (R1)K(11/R)K I MN
sparse cell (R1)K-1 log(M1/K)(R1)K(11/R)K I M1-1/KN
cell b. s. (R1)K-1 log(N/M1-1/K)(11/R)K-1 I M1-1/KN
cell f. s. (R1)K-1N/Q(11/R)K-1 I M1-1/KN
cell f. s. k. (R1)K-1N/Q(11/R)K-1 I M1-1/KN
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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.

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Execution Time Random Points
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Memory Usage Random Points
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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.