Abstract
We break the long standing cubic time bound of \(O(n^3)\) for the Minimum Weight Polygon Triangulation problem by showing that the well known dynamic programming algorithm, reported independently by Gilbert and Klincsek, can be optimized with a faster algorithm for the \((min,+)\)-product using look-up tables. In doing so, we also show that the well known Floyd-Warshall algorithm can be optimized in a similar manner to achieve a sub-cubic time bound for the All Pairs Shortest Paths problem without having to resort to recursion in the semi-ring theory.
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Notes
- 1.
Named after the Belgian mathematician Eugene Charles Catalan (1814–1894).
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Bae, S.E., Shinn, TW., Takaoka, T. (2016). Minimum Weight Polygon Triangulation Problem in Sub-Cubic Time Bound. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_24
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