Nowadays there are a large number of applications, such as digital library information retrieval, business data analysis, CAD/CAM, multimedia applications with images and sound, real-time process control and scientific computation, with data sets about gigabytes, terabytes or even petabytes. Because data distributions are too large to be stored accurately, maintaining compact and accurate summarized information about underlying data is of crucial important.The summarizing problem for Level 1 (disjoint and non-disjoint) topological relationship has been well studied for the past few years. However the spatial database users are often interested in a much richer set of spatial relations such as contains. Little work has been done on summarization for Level 2 topological relationship which includes contains, contained, overlap, equal and disjoint relations.We study the problem of effective summatization to represent the underlying data distribution to answer window queries for Level 2 topologicalrelationship. Cell-density based approach has been demonstrated as an effective way to this problem. But the challenges are the accuracy of the results and the storage space required which should be linearly proportional to the number of cells to be practical.In this thesis, we present several novel techniques to effectively construct cell density based spatial histograms. Based on the framework proposed, exact results could be obtained in constant time for aligned window queries. To minimize the storage space of the framework, an approximate algorithm with the approximate ratio 19/12 is presented, while the problem is shown NP-hard generally. Because the framework requires only a storage space linearly proportional to the number of cells, it is practical for many popularreal datasets. To conform to a limited storage space, effective histogram construction and query algorithms are proposed which can provide approximate resultsbut with high accuracy. The problem for non-aligned window queries is also investigated and techniques of un-even partitioned space are developed to support non-aligned window queries. Finally, we extend our techniques to 3D space. Our extensive experiments against both synthetic and real world datasets demonstrate the efficiency of the algorithms developed in this thesis.
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