The Rtree Index

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The R-tree Index

• Rula Alnaser
• Rob Bracero
• CIS 6930 Database Systems
• USF, Spring 09

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R-Tree

• Tree data structured index that handles spatial
  data
• Height balanced data structure
• Data entries are stored in leaf nodes
• Nodes can be kept 50 full (except root)

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• Search key is a collection of intervals with one
  interval per dimension
• Leaf entry ltn-dimension box, ridgt
• Non-leaf entry ltn-dimension box, pointer to
  child nodegt

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Search Queries

• May have to search several sub-trees at each node
• Improving search using convex polygons to
  approximate query region more accurately
• Can detect that there is no node overlap even
  when we have non-empty bounding box intersection
• Potential I/O cost saving (test node overlap is
  CPU cost)

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Insert Entry ltB, ptrgt

• Start at root and go down to best-fit leaf L.
• Go to child whose box needs least enlargement to
  cover B resolve ties by going to smallest area
  child.
• If best-fit leaf L has space, insert entry and
  stop. Otherwise, split L into L1 and L2.
• Adjust entry for L in its parent so that the box
  now covers (only) L1.
• Add an entry (in the parent node of L) for L2.
  (This could cause the parent node to recursively
  split.)

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Splitting a Node During Insertion

• The entries in node L plus the newly inserted
  entry must be distributed between L1 and L2.
• Goal is to reduce likelihood of both L1 and L2
  being searched on subsequent queries.

• Idea Redistribute so as to minimize area of L1
  plus area of L2.

GOOD SPLIT!
BAD!
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Delete Entry

• Deletion consists of searching for the entry to
  be deleted, removing it, and if the node becomes
  under-full, deleting the node and then
  re-inserting the remaining entries.

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Types of Queries

• Spatial Range Queries
• Find all hotels within a 10 mile radius.
• Nearest Neighbor Queries
• Find the 5 parks closest to the USF campus.
• Spatial Join Queries
• Find all cities within 100 miles of each other.

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Spatial Range Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
R7
R2
R6

• At each node, follow all children whose bounding
  box overlaps the query bounding box.

R13
R5
R12
R11
R14
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Spatial Range Queries
R1
R4
R10
Q
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
R7
R2
R6

• At each node, follow all children whose bounding
  box overlaps the query bounding box.

R13
R5
R12
R11
R14
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Spatial Range Queries
R1
R2
R1
R3
R4
R5
R6
R4
R10
Q
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
R7
R2
R6

• At each node, follow all children whose bounding
  box overlaps the query bounding box.

R13
R5
R12
R11
R14
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Spatial Range Queries
R1
R2
R1
R3
R4
R5
R6
R4
R10
Q
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
R7
R2
R6

• At each node, follow all children whose bounding
  box overlaps the query bounding box.

R13
R5
R12
R11
R14
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Spatial Range Queries
R1
R2
R1
R3
R4
R5
R6
R4
R10
Q
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
R7
R2
R6

• At each node, follow all children whose bounding
  box overlaps the query bounding box.

R13
R5
R12
R11
R14
N number of records in index M max entries
per node m min entries per node, m lt M/2

• Best case O(logm N)
• Worst Case O(N)
• In practice, only a small portion of the tree is
  searched.

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Nearest Neighbor Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
Q
R7
R2

• Calculate distance to all objects in leaf nodes
  that overlap the query region.

R6
R13
R5
R12
R11
R14
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Nearest Neighbor Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
Q
R7
R2

• Calculate distance to all objects in leaf nodes
  that overlap the query region.
• If no leaf nodes overlap the query region, expand
  the query bounding box and try again.

R6
R13
R5
R12
R11
R14
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Nearest Neighbor Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
R3
R8
Q
R7
R2

• Calculate distance to all objects in leaf nodes
  that overlap the query region.
• If no leaf nodes overlap the query region, expand
  the query bounding box and try again.

R6
R13
R5
R12
R11
R14

• Complexity is similar to range search. It is
  dependent on the data set and query.

• The algorithm may have to run multiple times
  until an overlapping leaf node is encountered.

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Nearest Neighbor Queries

• Alternative Depth-First
• Starting at root, move to the child with minimum
  distance to query region.
• Repeat recursively until an object is reached and
  store the distance to that object.
• Move back up the tree and visit only those
  children that have a distance from the query
  region that is smaller than the smallest distance
  to an object found so far.
• Once no such child node exists, the algorithm
  terminates.

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Nearest Neighbor Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
Q
R3
R8
R7

• Depth-first search would visit R1, R3, R7, R4,
  R9.

R2
R6
R13
R5
R12
R11
R14
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Nearest Neighbor Queries

• Alternative Best-First
• Starting at the root node, insert all children of
  current node into a priority queue order by
  distance to the query region.
• Choose the node at the top of the priority queue
  (with smallest distance to query region) and
  repeat recursively.
• Once a node at the leaf level is chosen, the
  algorithm terminates.
• Better performance than depth-first search, but
  larger space requirements.

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Nearest Neighbor Queries
R1
R4
R10
R9
R7
R8
R9
R10
R11
R12
R13
R14
Q
R3
R8
R7

• Best-first search would visit R1, R3, R4, R9.

R2
R6
R13
R5
R12
R11
R14
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R-Tree Variants

• R-Tree
• R-Tree

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R-Tree

• Avoids overlap by inserting entries into multiple
  leaves if necessary.
• Entries of internal nodes have no overlap.
• Point query searches now only follow one path
  along tree.
• Redundant leaf node entries and larger tree
  height.

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R-Tree
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R-Tree

• Uses forced re-inserts to try to minimize overlap
  of minimum bounding rectangles.
• On an insert, instead of splitting an overflowed
  node, remove some entries and re-insert them.
• Entries may fit in existing nodes and eliminate
  the need for a split.
• Minimize box areas, perimeters, and overlap
  rather than only box areas when splitting.

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References

• R. Ramakrishnan and J. Gehrke. Database
  Management Systems. 3rd Edition, McGraw-Hill,
  2003.
• Guttman, A. 1984. R-trees a dynamic index
  structure for spatial searching. In Proceedings
  of the 1984 ACM SIGMOD international Conference
  on Management of Data (Boston, Massachusetts,
  June 18 - 21, 1984). SIGMOD '84. ACM, New York,
  NY, 47-57.
• Sellis, T. K., Roussopoulos, N., and Faloutsos,
  C. 1987. The R-Tree A Dynamic Index for
  Multi-Dimensional Objects. In Proceedings of the
  13th international Conference on Very Large Data
  Bases (September 01 - 04, 1987). P. M. Stocker,
  W. Kent, and P. Hammersley, Eds. Very Large Data
  Bases. Morgan Kaufmann Publishers, San Francisco,
  CA, 507-518.
• Beckmann, N., Kriegel, H., Schneider, R., and
  Seeger, B. 1990. The R-tree an efficient and
  robust access method for points and rectangles.
  In Proceedings of the 1990 ACM SIGMOD
  international Conference on Management of Data
  (Atlantic City, New Jersey, United States, May 23
  - 26, 1990). SIGMOD '90. ACM, New York, NY,
  322-331.