Abstract
We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs.
We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses \(O(n (\log n \log \log n)^2)\) space, supports queries in worst case \(O(\log ^3 n)\) time, and updates in \(O(\log ^7 n)\) amortized time. This leads to an \(O(n\alpha (n)\log ^7 n)\) time algorithm (where \(\alpha \) is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward \(O(n^2 \log n)\) time algorithm.
The Netherlands Organisation for Scientific Research (NWO) is supporting T. Castermans (project number 314.99.117), B. Speckmann (project number 639.023.208), F. Staals (project number 612.001.651), and K. Verbeek (project number 639.021.541).
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- 1.
http://glammap.net/glamdev/maps/1, best viewed in Chrome. GlamMap currently does not implement the algorithm described in this article.
- 2.
For example, the catalogue of WorldCat contains more than 321 million library records at hundreds of thousands of distinct locations.
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Castermans, T., Speckmann, B., Staals, F., Verbeek, K. (2018). Agglomerative Clustering of Growing Squares. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_20
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DOI: https://doi.org/10.1007/978-3-319-77404-6_20
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