Dynamic Graph Coloring

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the numbe...

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Published in:	Algorithmica Vol. 81; no. 4; pp. 1319 - 1341
Main Authors:	Barba, Luis, Cardinal, Jean, Korman, Matias, Langerman, Stefan, van Renssen, André, Roeloffzen, Marcel, Verdonschot, Sander
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.04.2019 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Deletion Graph coloring Graph theory Lower bounds Mathematics of Computing Theory of Computation Tradeoffs Dynamic coloring Graphs Data structures Amortized algorithms
ISSN:	0178-4617, 1432-0541
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Abstract	In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d > 0 , the first algorithm maintains a proper O ( C d N 1 / d ) -coloring while recoloring at most O ( d ) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O ( C d ) -coloring with O ( d N 1 / d ) recolorings per update. The two converge when d = log N , maintaining an O ( C log N ) -coloring with O ( log N ) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c -coloring of a 2-colorable graph on N vertices must recolor at least Ω ( N 2 c ( c - 1 ) ) vertices per update, for any constant c ≥ 2 .
AbstractList	In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d > 0 , the first algorithm maintains a proper O ( C d N 1 / d ) -coloring while recoloring at most O ( d ) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O ( C d ) -coloring with O ( d N 1 / d ) recolorings per update. The two converge when d = log N , maintaining an O ( C log N ) -coloring with O ( log N ) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c -coloring of a 2-colorable graph on N vertices must recolor at least Ω ( N 2 c ( c - 1 ) ) vertices per update, for any constant c ≥ 2 . In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d>0, the first algorithm maintains a proper O(CdN1/d)-coloring while recoloring at most O(d) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(Cd)-coloring with O(dN1/d) recolorings per update. The two converge when d=logN, maintaining an O(ClogN)-coloring with O(logN) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on N vertices must recolor at least Ω(N2c(c-1)) vertices per update, for any constant c≥2.
Author	Verdonschot, Sander Cardinal, Jean van Renssen, André Korman, Matias Barba, Luis Roeloffzen, Marcel Langerman, Stefan
Author_xml	– sequence: 1 givenname: Luis surname: Barba fullname: Barba, Luis organization: Department of Computer Science, ETH Zürich – sequence: 2 givenname: Jean surname: Cardinal fullname: Cardinal, Jean organization: Département d’Informatique, Université Libre de Bruxelles – sequence: 3 givenname: Matias surname: Korman fullname: Korman, Matias organization: Tohoku University – sequence: 4 givenname: Stefan surname: Langerman fullname: Langerman, Stefan organization: Département d’Informatique, Université Libre de Bruxelles – sequence: 5 givenname: André surname: van Renssen fullname: van Renssen, André organization: School of Information Technologies, University of Sydney – sequence: 6 givenname: Marcel surname: Roeloffzen fullname: Roeloffzen, Marcel email: m.j.m.roeloffzen@tue.nl organization: TU Eindhoven – sequence: 7 givenname: Sander surname: Verdonschot fullname: Verdonschot, Sander organization: School of Computer Science, Carleton University
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Sci., vol. 10389, pp. 97–108 (2017) – reference: Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: Proc. 45th IEEE Sym. Found. Comp. Sci., pp. 499–508 (2004) – reference: HalldórssonMMParallel and on-line graph coloringJ. Algorithms1997232265280144196810.1006/jagm.1996.08360873.68165 – reference: Saxe, J.B., Bentley, J.L.: Transforming static data structures to dynamic structures (abridged version). In: Proc. 20th Symp. Found. Comp. Sci., pp. 148–168 (1979) – reference: BaswanaSGuptaMSenSFully dynamic maximal matching in O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}(\log n)$$\end{document} update timeSIAM J. Comput.201544188113331356810.1137/1309141401314.05155 – reference: Ouerfelli, L., Bouziri, H.: Greedy algorithms for dynamic graph coloring. In: Proc. Int. Conf. on Comm., Comp. and Control App., pp. 1–5 (2011) – reference: BentleyJLSaxeJBDecomposable searching problems I: static-to-dynamic transformationJ. Algorithms19801430135860486910.1016/0196-6774(80)90015-20461.68065 – reference: Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: Proc. 43rd IEEE Sym. Found. Comp. Sci., pp. 679–688 (2002) – reference: Henzinger, M., Krinninger, S., Nanongkai, D.: A subquadratic-time algorithm for decremental single-source shortest paths. In: Proc. 25th ACM-SIAM Symp. on Discr. Alg., pp. 1053–1072 (2014) – reference: OvermarsMHvan LeeuwenJWorst-case optimal insertion and deletion methods for decomposable searching problemsInf. Process. Lett.198112416817310.1016/0020-0190(81)90093-40459.68026 – reference: ThorupMFully-dynamic min-cutCombinatorica200727191127231078910.1007/s00493-007-0045-21135.68024 – reference: Dutot, A., Guinand, F., Olivier, D., Pigné, Y.: On the decentralized dynamic graph-coloring problem. In: Proc. Worksh. Compl. Sys. and Self-Org. Mod. (2007) – reference: HalldrssonMMSzegedyMLower bounds for on-line graph coloringTheor. Comput. Sci.19941301163174128713710.1016/0304-3975(94)90157-00822.68081 – reference: Kapron, B.M., King, V., Mountjoy, B.: Dynamic graph connectivity in polylogarithmic worst case time. In: Proc. 24th ACM-SIAM Symp. on Discr. Alg., pp. 1131–1142 (2013) – reference: Coudert, O.: Exact coloring of real-life graphs is easy. In: Proceedings of 34th Design Automation Conference. ACM, pp. 121–126 (1997). https://doi.org/10.1145/266021.266047 – reference: Preuveneers, D., Berbers, Y.: ACODYGRA: an agent algorithm for coloring dynamic graphs. In: Symb. Num. Alg. Sci. Comp., pp. 381–390 (2004) – reference: LovászLSaksMETrotterWTAn on-line graph coloring algorithm with sublinear performance ratioDiscrete Math.1989751–331932510014040679.05031 – reference: HolmJDe LichtenbergKThorupMPoly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivityJ. ACM2001484723760214492810.1145/502090.5020951127.68408 – reference: DemetrescuCFinocchiIItalianoPMehtaDSahniSDynamic graphsHandbook on Data Structures and Applications, Computer and Information Science2005Boca RatonCRC Press – reference: VishwanathanSRandomized online graph coloringJ. Algorithms1992134657669118720710.1016/0196-6774(92)90061-G0768.68185 – reference: BaswanaSKhuranaSSarkarSFully dynamic randomized algorithms for graph spannersACM Trans. Algorithms20128435298191310.1145/2344422.23444251295.05221 – reference: BorowieckiPSidorowiczEDynamic coloring of graphsFundam. Inf.2012114210512829618551236.05077 – volume: 23 start-page: 265 issue: 2 year: 1997 ident: 473_CR10 publication-title: J. Algorithms doi: 10.1006/jagm.1996.0836 – ident: 473_CR18 doi: 10.1109/CCCA.2011.6031437 – volume: 27 start-page: 91 issue: 1 year: 2007 ident: 473_CR24 publication-title: Combinatorica doi: 10.1007/s00493-007-0045-2 – volume: 12 start-page: 168 issue: 4 year: 1981 ident: 473_CR19 publication-title: Inf. Process. Lett. doi: 10.1016/0020-0190(81)90093-4 – ident: 473_CR20 – ident: 473_CR15 doi: 10.1007/978-1-4684-2001-2_9 – volume: 75 start-page: 319 issue: 1–3 year: 1989 ident: 473_CR16 publication-title: Discrete Math. doi: 10.1016/0012-365X(89)90096-4 – ident: 473_CR23 doi: 10.1109/SFCS.1979.47 – ident: 473_CR1 doi: 10.1007/978-3-319-62127-2_9 – volume: 8 start-page: 35 issue: 4 year: 2012 ident: 473_CR3 publication-title: ACM Trans. Algorithms doi: 10.1145/2344422.2344425 – ident: 473_CR9 – volume-title: Algorithms and Theory of Computation Handbook year: 2010 ident: 473_CR7 – volume: 3 start-page: 103 year: 2007 ident: 473_CR26 publication-title: Theory Comput. doi: 10.4086/toc.2007.v003a006 – volume: 13 start-page: 657 issue: 4 year: 1992 ident: 473_CR25 publication-title: J. Algorithms doi: 10.1016/0196-6774(92)90061-G – ident: 473_CR14 doi: 10.1137/1.9781611973105.81 – volume: 1 start-page: 301 issue: 4 year: 1980 ident: 473_CR4 publication-title: J. Algorithms doi: 10.1016/0196-6774(80)90015-2 – ident: 473_CR22 doi: 10.1109/FOCS.2004.22 – volume: 114 start-page: 105 issue: 2 year: 2012 ident: 473_CR5 publication-title: Fundam. Inf. doi: 10.3233/FI-2012-620 – ident: 473_CR6 doi: 10.1145/266021.266047 – ident: 473_CR21 doi: 10.1109/SFCS.2002.1181993 – ident: 473_CR12 doi: 10.1137/1.9781611973402.79 – volume: 48 start-page: 723 issue: 4 year: 2001 ident: 473_CR13 publication-title: J. 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Snippet	In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Deletion Graph coloring Graph theory Lower bounds Mathematics of Computing Theory of Computation Tradeoffs
Title	Dynamic Graph Coloring
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