Multidimensional Binary Search Trees Used for Associative Searching

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Multidimensional Binary Search Trees Used for
Associative Searching

• Jon Louis Bentley
• Stanford University
• Communications of the ACM, September 1975, vol.
  18, no. 9
• 1975 ACM Student Award
• Paper Second Place

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Notations

• Let P be a set of n points in the plane.
• A 2-dimensional rectangle range query on P asks
  for the points from P lying inside a query
  rectangle x xy y.
• A point p(px,py) lies inside this rectangle if
  and only if x?px?x and y?py?y.

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Construction

• Algorithm BUILDKDTREE(P,depth)
• //Input A set of points P and the current depth.
  
• //Output The root of a kd-tree storing P.
• 1. if P contains only one point then
• 2. return a leaf storing this point
• 3. else if depth is even then
• 4. Split P into two subsets with a
  vertical line l through the median x-coordinate
  of the points
• in P. Let P1 be the set of points to the left and
  P2 be the set of points to the right. Points
  exactly on
• the line belong to P1.
• 5. else
• 6. Split P into two subsets with a
  horizontal line l through the median
  y-coordinate of the points
• in P. Let P1 be the set of points above l and P2
  be the points below l. Points exactly on the line
  belong
• to P1.
• 7. Vright BUILDKDTREE(P1,depth1)

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Query
Algorithm SEARCHKDTREE(v,R) Input. The root of
a (subtree of a) kd-tree and a range R. Output.
All points at leave below v that lie in the
range. 01 if v is a leaf 02 Report the point
stored at v if it lies in R. 03 else 04 if
region(lc(v)) is fully contained in R 05
ReportSubtree(lc(v)) 06 else 07 if
region(lc(v)) intersects R 08
SEARCHKDTREE(lc(v),R) 09 if region(rc(v)) is
fully contain in R 10 ReportSubtree(rc(v)) 11
else 12 if region(rc(v)) intersects R 13
SEARCHKDTREE(rc(v),R)
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Insertion
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Demo

• http//www.rolemaker.dk/nonRoleMaker/uni/algogem/k
  dtree.htm

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Discussion

• Priority search trees v.s. Kd-trees