On k-d Range Search with Patricia Tries

Patricia tries are explored for indexing combined text and spatial data. A combined text and spatial data range search algorithm is presented for reporting all data from a set of size $n$ intersecting a query hyperrectangle. We also use Patricia tries to answer $\epsilon$-approximate orthogonal range search on a set of $n$ random points and hyperrectangles in $k$-dimensional data space. $\epsilon$-approximate orthogonal range counting queries can be answered in $O(k {\rm log} n \epsilon^{k-1})$ time, and the number of nodes visited for orthogonal range counting queries is shown to be $O({\rm log} n + k(1+2n^{1/k}\Delta)^{k-1})$ for cubical range of side length $\Delta$. Patricia tries are evaluated experimentally for both orthogonal range search and $\epsilon$-approximate orthogonal range search (for $2 \leq k \leq 14$ and $n$ up to 1,000,000) using uniformly distributed random data. The expected range search time is determined theoretically and found to agree with experimental results.