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External segment trees

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Abstract

The segment tree is a well-known internal data structure with numerous applications in computational geometry. It allows the dynamical maintenance of a set of intervals such that the intervals enclosing a query point can be found efficiently (point enclosure search).

In this paper we transfer the underlying principle of the segment tree in a nontrivial way to secondary storage and arrive at the EST-an external file structure with the same functionality and the following properties: (1) Point enclosure searches are very efficient—only very few pages are accessed that are not filled to more than 50% with result intervals. (2) A page filling of 50% is guaranteed—on the average it will be around 70%. Although the segment tree represents, in the worst case, each interval by a logarithmic number offragments, in practical cases fragmentation remains low and the storage requirements about linear. (3) The EST is balanced and the update algorithms are efficient. (4) Unlike many other file structures for spatial objects the EST has no problems with an arbitrarydensity, that is, an arbitrarily large number of intervals covering any point of the line.

Furthermore, the EST can be used as a file structureconstructor in the following sense: Let there be a file structureX supporting searches for objects with propertyx and suppose it is necessary to maintain a collection of objects with associated (e.g., time) intervals. Then an EST-X structure that supports searches for objects with propertyx present at timet can be built. This suggests using the EST as a building block in the implementation of temporal database systems. Other applications include the one-dimensional indexing of collections of spatial objects in two or more dimensions.

More generally, this paper shows techniques for mapping internal tree structures with node lists (other examples: range tree, interval tree) to secondary memory. In this context an intriguing theoretical problem, thecover-balancing problem, is solved: Given a tree whose nodes have associatedweights partitioned into subtrees whose weights must lie in a certain range, maintain this partition under weight changes at arbitrary nodes. This is in contrast to classical balancing problems where updates occur only at the leaves.

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  1. G. Blankenagel

    Present address: Kommunalverband Ruhrgebiet, Kronprinzenstrasse 35, D-45128, Essen, Germany

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  1. Praktische Informatik IV, FernUniversität Hagen, D-58084, Hagen, Germany

    G. Blankenagel & R. H. Güting

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Communicated by Franco P. Preparata.

This work was supported by the DFG (Deutsche Forschungsgemeinschaft) under Grant Cr 65/2-5.

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Blankenagel, G., Güting, R.H. External segment trees. Algorithmica 12, 498–532 (1994). https://doi.org/10.1007/BF01188717

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