conference paper published yes Gramoz Goranci author Monika H Henzinger author 540c9bbd-f2de-11ec-812d-d04a5be856300000-0002-5008-6530 Pan Peng author ESA: Annual European Symposium on Algorithms We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest. Schloss Dagstuhl - Leibniz-Zentrum für Informatik2017Vienna, Austria eng 1868-8969 978-3-95977-049-1 1712.0647310.4230/LIPICS.ESA.2017.45 87 yes G. Goranci, M. Henzinger, P. Peng, in:, 25th Annual European Symposium on Algorithms, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Goranci G, Henzinger M, Peng P. The power of vertex sparsifiers in dynamic graph algorithms. In: <i>25th Annual European Symposium on Algorithms</i>. Vol 87. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2017. doi:<a href="https://doi.org/10.4230/LIPICS.ESA.2017.45">10.4230/LIPICS.ESA.2017.45</a> Goranci, Gramoz, Monika Henzinger, and Pan Peng. “The Power of Vertex Sparsifiers in Dynamic Graph Algorithms.” In <i>25th Annual European Symposium on Algorithms</i>, Vol. 87. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. <a href="https://doi.org/10.4230/LIPICS.ESA.2017.45">https://doi.org/10.4230/LIPICS.ESA.2017.45</a>. Goranci, Gramoz, et al. “The Power of Vertex Sparsifiers in Dynamic Graph Algorithms.” <i>25th Annual European Symposium on Algorithms</i>, vol. 87, 45, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, doi:<a href="https://doi.org/10.4230/LIPICS.ESA.2017.45">10.4230/LIPICS.ESA.2017.45</a>. Goranci G, Henzinger M, Peng P. 2017. The power of vertex sparsifiers in dynamic graph algorithms. 25th Annual European Symposium on Algorithms. ESA: Annual European Symposium on Algorithms, LIPIcs, vol. 87, 45. Goranci, G., Henzinger, M., &#38; Peng, P. (2017). The power of vertex sparsifiers in dynamic graph algorithms. In <i>25th Annual European Symposium on Algorithms</i> (Vol. 87). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPICS.ESA.2017.45">https://doi.org/10.4230/LIPICS.ESA.2017.45</a> G. Goranci, M. Henzinger, and P. Peng, “The power of vertex sparsifiers in dynamic graph algorithms,” in <i>25th Annual European Symposium on Algorithms</i>, Vienna, Austria, 2017, vol. 87. 118332022-08-12T10:46:26Z2024-11-06T11:58:29Z