A Simple Greedy Algorithm for Dynamic Graph Orientation

Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With...

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Published in:	Algorithmica Vol. 82; no. 2; pp. 245 - 259
Main Authors:	Berglin, Edvin, Brodal, Gerth Stølting
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2020 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graphs Greedy algorithms Mathematics of Computing Special Issue: Algorithms and Computation Theory of Computation Graph arboricity Edge orientations Dynamic graph algorithms
ISSN:	0178-4617, 1432-0541
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Abstract	Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O log n flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O 1 and O log n flips nearly matching out-degree bounds to their respective amortized solutions.
AbstractList	Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in Ologn flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O1 and Ologn flips nearly matching out-degree bounds to their respective amortized solutions. Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O log n flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O 1 and O log n flips nearly matching out-degree bounds to their respective amortized solutions.
Author	Berglin, Edvin Brodal, Gerth Stølting
Author_xml	– sequence: 1 givenname: Edvin surname: Berglin fullname: Berglin, Edvin email: berglin@cs.au.dk organization: Aarhus University – sequence: 2 givenname: Gerth Stølting surname: Brodal fullname: Brodal, Gerth Stølting organization: Aarhus University
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References	Brodal, G.S., Fagerberg, R.: Dynamic representation of sparse graphs. In: Proceedings 6th International Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science, vol. 1663, pp. 342–351. Springer (1999) He, M., Tang, G., Zeh, N.: Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In: Proceedings 25th International Symposium on Algorithms and Computation (ISAAC), Lecture Notes in Computer Science, vol. 8889, pp. 128–140. Springer (2014) Kopelowitz, T., Krauthgamer, R., Porat, E., Solomon, S.: Orienting fully dynamic graphs with worst-case time bounds. In: Proceedings 41st International Colloquium Automata, Languages, and Programming (ICALP), Part II, Lecture Notes in Computer Science, vol. 8573, pp. 532–543. Springer (2014) LevcopoulosCOvermarsMHA balanced search tree with O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1)$$\end{document} worst-case update timeActa Inform.198826326927797533310.1007/BF00299635 Peleg, D., Solomon, S.: Dynamic (1+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+ \varepsilon $$\end{document})-approximate matchings: a density-sensitive approach. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 712–729. Society for Industrial and Applied Mathematics (2016) Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 692–711. Society for Industrial and Applied Mathematics (2016) KannanSNaorMRudichSImplicit representation of graphsSIAM J. Discrete Math.199254596603118682710.1137/0405049 KowalikŁAdjacency queries in dynamic sparse graphsInf. Process. Lett.20071025191195231345810.1016/j.ipl.2006.12.006 Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: International Colloquium on Automata, Languages, and Programming, pp. 167–179. Springer (2015) Dietz, P., Sleator, D.: Two algorithms for maintaining order in a list. In: Proceedings 19th Annual ACM Symposium on Theory of Computing (STOC), pp. 365–372. ACM (1987) Berglin, E.: Geometric covers, graph orientations, counter games. Ph.D. thesis, Aarhus University (2017) NeimanOSolomonSSimple deterministic algorithms for fully dynamic maximal matchingACM Trans. Algorithms2016121734628431400.05241 Ł Kowalik (528_CR9) 2007; 102 528_CR2 528_CR1 O Neiman (528_CR11) 2016; 12 528_CR8 528_CR12 528_CR4 528_CR3 528_CR6 S Kannan (528_CR7) 1992; 5 528_CR5 C Levcopoulos (528_CR10) 1988; 26
References_xml	– reference: Kopelowitz, T., Krauthgamer, R., Porat, E., Solomon, S.: Orienting fully dynamic graphs with worst-case time bounds. In: Proceedings 41st International Colloquium Automata, Languages, and Programming (ICALP), Part II, Lecture Notes in Computer Science, vol. 8573, pp. 532–543. Springer (2014) – reference: Peleg, D., Solomon, S.: Dynamic (1+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+ \varepsilon $$\end{document})-approximate matchings: a density-sensitive approach. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 712–729. Society for Industrial and Applied Mathematics (2016) – reference: Brodal, G.S., Fagerberg, R.: Dynamic representation of sparse graphs. In: Proceedings 6th International Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science, vol. 1663, pp. 342–351. Springer (1999) – reference: Berglin, E.: Geometric covers, graph orientations, counter games. Ph.D. thesis, Aarhus University (2017) – reference: KowalikŁAdjacency queries in dynamic sparse graphsInf. Process. Lett.20071025191195231345810.1016/j.ipl.2006.12.006 – reference: Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 692–711. Society for Industrial and Applied Mathematics (2016) – reference: Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: International Colloquium on Automata, Languages, and Programming, pp. 167–179. Springer (2015) – reference: Dietz, P., Sleator, D.: Two algorithms for maintaining order in a list. In: Proceedings 19th Annual ACM Symposium on Theory of Computing (STOC), pp. 365–372. ACM (1987) – reference: He, M., Tang, G., Zeh, N.: Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In: Proceedings 25th International Symposium on Algorithms and Computation (ISAAC), Lecture Notes in Computer Science, vol. 8889, pp. 128–140. Springer (2014) – reference: KannanSNaorMRudichSImplicit representation of graphsSIAM J. Discrete Math.199254596603118682710.1137/0405049 – reference: NeimanOSolomonSSimple deterministic algorithms for fully dynamic maximal matchingACM Trans. Algorithms2016121734628431400.05241 – reference: LevcopoulosCOvermarsMHA balanced search tree with O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1)$$\end{document} worst-case update timeActa Inform.198826326927797533310.1007/BF00299635 – ident: 528_CR2 doi: 10.1007/978-3-662-47672-7_14 – ident: 528_CR8 doi: 10.1007/978-3-662-43951-7_45 – volume: 102 start-page: 191 issue: 5 year: 2007 ident: 528_CR9 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2006.12.006 – volume: 5 start-page: 596 issue: 4 year: 1992 ident: 528_CR7 publication-title: SIAM J. Discrete Math. doi: 10.1137/0405049 – volume: 12 start-page: 7 issue: 1 year: 2016 ident: 528_CR11 publication-title: ACM Trans. Algorithms doi: 10.1145/2700206 – ident: 528_CR6 doi: 10.1007/978-3-319-13075-0_11 – ident: 528_CR12 doi: 10.1137/1.9781611974331.ch51 – ident: 528_CR4 doi: 10.1007/3-540-48447-7_34 – volume: 26 start-page: 269 issue: 3 year: 1988 ident: 528_CR10 publication-title: Acta Inform. doi: 10.1007/BF00299635 – ident: 528_CR3 doi: 10.1137/1.9781611974331.ch50 – ident: 528_CR5 doi: 10.1145/28395.28434 – ident: 528_CR1
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Snippet	Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one... Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graphs Greedy algorithms Mathematics of Computing Special Issue: Algorithms and Computation Theory of Computation
Title	A Simple Greedy Algorithm for Dynamic Graph Orientation
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