Approximate Orthogonal Range Search using Patricia Tries

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We use Patricia tries to answer E-approximate orthogonal range search on a set of n random points and rectangles in k-dimensional space. Given n k-dimensional random points or rectangles and a k-dimensional query rectangle, E-approximate orthogonal range query counts (or reports) the points in the query rectangle or the rectangles intersecting the query rectangle, allowing errors near the boundary of the query rectangle. Points within a distance of a function of E the boundary of the query rectangle might be misclassified. The approximate orthogonal range search time using Patricia tries is determined theoretically to be O(k log n/E[superscript k-1]) for cubical range queries. Patricia tries are evaluated experimentally for E-approximate orthogonal range counting and reporting queries (for 2 < k < 10 and n up to 1,000,000) using uniformly distributed random points and rectangles, and we compared the performance of the Patricia trie for k-d points with the k-d tree and the adaptive k-d tree. The experimental results show that allowing small errors can significantly improve the query execution time of the approximate range counting. For epsilon = 0.05, an average of 50% fewer nodes are visited for the Patricia trie (compared to the exact range search).