# Internal and external algorithms for the points-in-regions problem — the INSIDE join of geo-relational algebra

• Gabriele Blankenagel
• Ralf Hartmut Güting
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 333)

## Abstract

We consider the problem of collectively locating a set of points within a set of disjoint polygonal regions when neither for points nor for regions preprocessing is allowed. This problem arises in geometric database systems. More specifically it is equivalent to computing the inside join of geo-relational algebra, a conceptual model for geo-data management. We describe efficient algorithms for solving this problem based on plane-sweep and divide-and-conquer, requiring O(n (log n)+t) and O(log2 n)+t) time, respectively, and O(n) space, where n is the total number of points and edges, and t the number of reported (point,region) pairs. Since the algorithms are meant to be practically useful we consider apart from the internal versions — running completely in main memory — also versions that run internally but use much less than linear space and versions that run externally, that is, require only a constant amount of internal memory regardless of the amount of data to be processed. Comparing plane-sweep and divide-and-conquer, it turns out that divide-and-conquer can be expected to perform much better in the external case even though it has a higher internal asymptotic worst-case complexity.

An interesting theoretical by-product is a new general technique for handling arbitrarily large sets of objects clustered on a single x-coordinate within a planar divide-and-conquer algorithm and a proof that the resulting "unbalanced" dividing does not lead to a more than logarithmic height of the tree of recursive calls.

## References

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Blankenagel, G., and R.H. Güting, Internal and External Algorithms for the Points-In-Regions Problem — the INSIDE Join of Geo-Relational Algebra. Universität Dortmund, Abteilung Informatik, Forschungsbericht 228, 1987; to appear in Algorithmica.Google Scholar
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