Let M be a triangulated, orientable 2-manifold of genus g without boundary, and let h be a height function over M that is linear within each triangle. We present a kinetic data structure (KDS) for maintaining the Reeb graph R of h as the heights of M's vertices vary continuously with time. Assuming the heights of two vertices of M become equal only O(1) times, the KDS processes O((k + g) n \polylog n) events; n is the number of vertices in M, and k is the number of external events which change the combinatorial structure of R. Each event is processed in O(\log^2 n) time, and the total size of our KDS is O(gn). The KDS can be extended to maintain an augmented Reeb graph as well.
@InProceedings{agarwal_et_al:LIPIcs.FSTTCS.2017.8, author = {Agarwal, Pankaj K. and Fox, Kyle and Nath, Abhinandan}, title = {{Maintaining Reeb Graphs of Triangulated 2-Manifolds}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.8}, URN = {urn:nbn:de:0030-drops-84043}, doi = {10.4230/LIPIcs.FSTTCS.2017.8}, annote = {Keywords: Reeb graphs, 2-manifolds, topological graph theory} }
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