Distance browsing in spatial databases

We compare two different techniques for browsing through a collection of spatial objects stored in an R-tree spatial data structure on the basis of their distances from an arbitrary spatial query object. The conventional approach is one that makes use of a <italic>k</italic>-nearest neighbor algorithm where <italic>k</italic> is known prior to the invocation of the algorithm. Thus if <italic>m</italic> < <italic>k</italic> neighbors are needed, the <italic>k</italic>-nearest neighbor algorithm has to be reinvoked for <italic>m</italic> neighbors, thereby possibly performing some redundant computations. The second approach is incremental in the sense that having obtained the <italic>k</italic> nearest neighbors, the <italic>k</italic> + 1<italic><supscrpt>st</supscrpt></italic> neighbor can be obtained without having to calculate the <italic>k</italic> + 1 nearest neighbors from scratch. The incremental approach is useful when processing complex queries where one of the conditions involves spatial proximity (e.g., the nearest city to Chicago with population greater than a million), in which case a query engine can make use of a pipelined strategy. We present a general incremental nearest neighbor algorithm that is applicable to a large class of hierarchical spatial data structures. This algorithm is adapted to the R-tree and its performance is compared to an existing <italic>k</italic>-nearest neighbor algorithm for R-trees [Rousseopoulos et al. 1995]. Experiments show that the incremental nearest neighbor algorithm significantly outperforms the <italic>k</italic>-nearest neighbor algorithm for distance browsing queries in a spatial database that uses the R-tree as a spatial index. Moreover, the incremental nearest neighbor algorithm usually outperforms the <italic>k</italic>-nearest neighber algorithm when applied to the <italic>k</italic>-nearest neighbor problem for the R-tree, although the improvement is not nearly as large as for distance browsing queries. In fact, we prove informally that at any step in its execution the incremental nearest neighbor algorithm is optimal with respect to the spatial data structure that is employed. Furthermore, based on some simplifying assumptions, we prove that in two dimensions the number of distance computations and leaf nodes accesses made by the algorithm for finding <italic>k</italic> neighbors is <italic>O</italic>(<italic>k</italic> + <italic>k</italic>).