A tighter bound for Nearest Neighbor Query

Nearest Neighbor query is one of the most important spatial queries. Using R-tree index structure and bounds on the distances it can be processed very efficiently. In this work we tried a modification of R-tree nodes and defined a new upper bound. We extended the R-tree nodes by 4 bits per dimension and experimentally compared the performance of our bound with MinMaxDistance. Our bound is good on sparse database where overlapping of the objects and MBRs are rare. Introduction: Nearest Neighbor Query is an important type of query wherever a good sense of distance is available. In spatial database nearest neighbor is defined as the one with lowest euclidian distance from the query object. Finding the nearest neighbor from a large spatial database involves accessing the objects and measuring the distances. Both of these tasks can be made easy with the indexing technique called R-tree. With an R-tree built on a spatial relation a lower bound (admissible heuristic) (4) called MinDist and an upper bound called MinMaxDist are defined in (1). These bounds help the search of the Nearest Neighbor in a great deal. Defining a tighter bound than these two using R-tree is impossible to us. These bounds are the tightest with the representation of R-tree nodes. We tried some simple modification to R-tree node structure and used one of them to define a new upper bound which we expected to be better than MinMaxDist bounds in some cases at the least. Next section defines the extension of the R-tree and the bound. In following sections we describe the measures we used in the experiments and the results. In all the sections we describe the two dimensional case and the idea is easily extendable to higher dimensions. Modification of the R-tree: If we are given a Minimum Bounding Rectangle (MBR) of an object MinMaxDist is the tightest upper bound on the smallest distance from a point to that object. To make this bound tighter we need to increase the information content in the object representation which is MBR in case of R-tree. We opted for a very small increase. For each possible edge of the MBR we store the orientation or position of the spatial object. An object may touch an edge of an MBR at several points or at a single point. The part of the edge that contains the touched region of the object can be completely contained in the upper half of the edge or completely contained in the lower half of the edge or may span over both upper and lower portion of the edge. These three situations can be represented by only two bits. Each pair of bit can also be represented as a base-4 number. For each dimension we have two parallel faces. Thus if we have n dimensions we would need 4n bits to represent the whole shape of the object within the MBR. Alternatively we can represent the whole shape with a string of 2n base-4 numbers where first n numbers represent the lower faces parallel to axes and the last n numbers represent the higher faces. The sequences of the axes in both of the halves are preserved. This string of base-4 numbers are represented by B. In Figure 1 two examples of the bit representation of the shapes are shown.