A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs
We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms.
In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.
- 7.Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proceedings of 21st Symposium on Computational Geometry, pp. 263–271 (2005)Google Scholar
- 8.de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic 2-centers in the black-box model. In: Proceedings of 29th Symposium on Computational Geometry, pp. 145–154 (2013)Google Scholar
- 13.Guibas, L.J.: Kinetic data structures. In: Mehta, D.P., Sahni, S. (eds.) Handbook of Data Structures and Applications, pp. 23.1–23.18. Chapman and Hall/CRC, Boca Raton (2004)Google Scholar
- 15.Katoh, N., Tokuyama, T., Iwano, K.: On minimum and maximum spanning trees of linearly moving points. In: Proceedings of 33rd Symposium on Foundations of Computer Science, pp. 396–405 (1992)Google Scholar