On k - d Range

We present two new k-dimensional data structures, called the PKD-tree and the PKD-tree, respectively. They are explored for indexing combined text and point data, and evaluated experimentally for orthogonal range search (for 2 ≤ k ≤ 100 and n up to 1,000,000) using synthetic data and real data. We compared the range search performance of the PKD-tree with the PKD-tree, the k-d tree, the Pyramid technique, the R∗-tree and naive search. The experimental results show that the PKDtree and the PKD-tree have very good performance for large k, and they always outperform the Pyramid technique, and are better than k-d tree and the R∗-tree when k ≥ log2 n. For a PKD-tree built from n uniform and random data points, an orthogonal range search with a query square W of side length ∆ visits O(k log n + n(1− (1− 2∆))) nodes for ∆ ≤ 0.5.