Nearest Neighbour Matching Using Multidimensional Binary Search Trees

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Nearest Neighbour Matching Using Multidimensional
Binary Search Trees

• Henry Stern
• Supervisor Pat KeastReader Mike Shepherd

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Nearest Neighbour Matching

• A fundamental computational geometry problem.
• Given a set of data points and target points in a
  metric space, find the nearest data point to each
  target point.
• Cost increases with both the number of points and
  the dimensionality of the space.

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Process of Nearest Neighbour Matching

• A k-dimensional Euclidean ball with radius r,
  centered around the target point contains all
  neighbours where distance(targetlt-gtpoint) lt r.
• During a nearest neighbour match, the radius of
  this ball is the distance from the target point
  to the best known match.

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Simple Nearest Neighbour Matching Algorithm

• Every point is examined in turn.
• If the point being examined is closer to the
  target than all of the previously known matches,
  it is saved as the candidate for nearest
  neighbour.

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Simple but Expensive

• Simple nearest neighbour matching algorithm runs
  in O(nk) time with space complexity O(nk).
• Cheapest solution for a single query.
• For multiple queries, it does not take advantage
  of any previous queries.

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What Other Options Are There?

• Quad Trees
• Multidimensional Binary Search Trees

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Quad Trees

• Sacrifices space complexity for time complexity.
• O(n) time complexity, expected linear.
• O(n 2k) space complexity.

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Multidimensional Binary Search Tree

• kd-trees.
• Sacrifices time complexity for space complexity.
• O(n) time complexity, expected linear.
• O(n) space complexity.

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Query Optimization for kd-trees

• Region Compression
• Splitting Strategies

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Region Compression

• Tight bounds for each node are calculated.
• O(n) method while tree is being constructed.
• Significantly reduces query costs.

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Splitting Strategies

• The choice of splitting dimension can have a
  significant impact on query performance.
• Success of splitting strategy is highly
  data-dependent.
• Three strategies examined
• Median of the next dimension (Simple).
• Median of the widest dimension.
• Median of the most deviant dimension.

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Splitting Strategy vs. Query Cost
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Query Cost

• Proportional to dimensionality and number of
  points.
• Critical point when k approaches log n.
• After the critical point, query cost slowly
  approaches n.

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Conclusion

• Nearest neighbour matching in sub-linear time is
  a difficult problem.
• Time complexity vs. space complexity.
• Difficult to minimize trade-off.
• The kd-tree is ineffective for high-dimensional
  nearest neighbour matching.