CMPUT 498 Orthogonal Range Searching

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CMPUT 498Orthogonal Range Searching

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Database queries
a geometric view of database queries
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1-Dimensional range searching

• Given a set of real numbers, report the points
  inside an interval
• Solution using arrays is not dynamic and does not
  generalise to higher dimensions
• Store in a balanced binary tree
• leaves given numbers
• interior splitting value largest value in left
  subtree

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1-Dimensional range searching

• keys
• light grey search paths
• dark grey points in search interval
• Split node is the node at which the search paths
  bifurcate

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Discussion

• Can this algorithm output a point not in the
  interval?
• Can a point in the interval not be reported?
• Time and space?
• Notion of output sensitivity
• n number of points
• k number of points in interval
• time is in O(k log n)

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2-Dimensional range searching

• report points in a rectangle
• assume points are in general position

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kd-tree

• leaf nodes input points
• interior nodes splitting lines

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line l
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Correspondencebetween data structure and point
set
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Query rectangle
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Searching in a k-d tree
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Recurrence
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Data structures seen so far

• Convex hull arrays
• Segment intersection
• priority queue
• balanced binary tree
• (DCEL)
• Voronoi diagram
• ibid.
• Delaunay triangulation
• Udirected graph
• Directed Acyclic Graph (DAG)
• Binary Space Partitioning
• Binary tree (OO classes?)
• k-d trees
• Balanced binary search tree

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Single vs. multi-level data structures
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Storage space

• What is the total storage cost?
• How many times is a single point stored?

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• Query time?

• O(log2 n k)

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Reference
Chap. 5