Toward spatial joins for polygons

Efficient evaluation of spatial join is an important issue in spatial databases. The traditional evaluation strategy is to perform a join of "minimum bounding rectangles" (MBR) of the spatial objects (MBR-filter) and evaluate the actual join of the objects using the results of the join on approximations. Improvements to add additional filtering using more accurate approximations were also considered. In the present paper, we develop efficient algorithms for evaluating joins of "trapezoids" without using MBR'S. For the case where there are no intersecting non-horizontal boundaries of trapezoids in the same set, a spatial join of two sets of N trapezoids can be evaluated in O(N logb N+k) I/Os, where b is the page size and k the number of trapezoid intersections. For the general case without any assumptions, a join can be done in O((N+l+k) logb N) I/Os, where l is the total number of intersections of non-horizontal boundaries within the same set, and N, k, b are the same as above. The new algorithms can be used to evaluate spatial joins for polygons. One possibility is to decompose polygons into trapezoids and apply a trapezoid join algorithm. In particular, this approach is efficient for "I/O bounded polygons" (each of which can be retrieved in a constant number of I/Os). Given two sets of N "I/O bounded polygons, we show that in the case where there are no boundary intersections among polygons of the same set, the join of the two sets can be computed in O(N log/sub b/ N+k) I/Os, and in the case where there is no such assumption, the join takes O((N+l+k) log/sub b/ N) I/Os, where b is the page size, k the number of pairs of intersecting polygons, and l the number of boundary intersections within the same polygon set. Another possibility is to approximate objects by I/O bounded polygons (e.g., 5-corner convex polygons) which are finer than rectangles and use the new algorithms as a filter.