On MBR Approximation of Histories for Historical Queries: Expectations and Limitations

Traditional approaches for efficiently processing historical queries, where a history is a multidimensional time-series, employ a two step filter-and-refine scheme. In the filter step, an approximation of each history often as a set of minimum bounding hyper-rectangles (MBRs) is organized using a spatial index structure such as R-tree. The index is used to prune redundant disk accesses and to reduce the number of pairwise comparisons required in the refine step. To improve the efficiency of the filtering step, a heuristic is used to decrease the expected number of MBRs that overlap with a query, by reducing the volume of empty space indexed by the index. The heuristic selects, among all possible splitting schemes of a history, the one which results to a set of MBRs with minimum total volume. Although this heuristic is expected to improve the performance of spatial and history based queries with small temporal and spatial extents, in many real settings, the performance of historical queries depends on the extent of the query. Moreover, the optimal approximation of a history is not always the one with minimum total volume. In this paper, we present the limitations of using volume as a criteria for approximating histories, specially in high dimensional cases, where it is not feasible to index MBRs by traditional spatial index structures.