Triangle Counting in Dynamic Graph Streams
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a fe...
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| Vydané v: | Algorithmica Ročník 76; číslo 1; s. 259 - 278 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
Springer US
01.09.2016
Springer Verlag |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered
insert-only
streams. We present a new algorithm estimating the number of triangles in
dynamic
graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous solutions for various graph classes, for example sparse graphs with a relatively small number of triangles. Also, for graphs with constant transitivity coefficient, a common situation in real graphs, this is the first algorithm achieving constant processing time per edge. The result is achieved by a novel approach combining sampling of vertex triples and sparsification of the input graph. In the course of the analysis of the algorithm we present a lower bound on the number of pairwise independent 2-paths in general graphs which might be of independent interest. At the end of the paper we discuss lower bounds on the space complexity of triangle counting algorithms that make no assumptions on the structure of the graph. |
|---|---|
| AbstractList | Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered
insert-only
streams. We present a new algorithm estimating the number of triangles in
dynamic
graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous solutions for various graph classes, for example sparse graphs with a relatively small number of triangles. Also, for graphs with constant transitivity coefficient, a common situation in real graphs, this is the first algorithm achieving constant processing time per edge. The result is achieved by a novel approach combining sampling of vertex triples and sparsification of the input graph. In the course of the analysis of the algorithm we present a lower bound on the number of pairwise independent 2-paths in general graphs which might be of independent interest. At the end of the paper we discuss lower bounds on the space complexity of triangle counting algorithms that make no assumptions on the structure of the graph. Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered insert-only streams. We present a new algorithm estimating the number of triangles in dynamic graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous solutions for various graph classes, for example sparse graphs with a relatively small number of triangles. Also, for graphs with constant transitivity coefficient, a common situation in real graphs, this is the first algorithm achieving constant processing time per edge. The result is achieved by a novel approach combining sampling of vertex triples and sparsification of the input graph. In the course of the analysis of the algorithm we present a lower bound on the number of pairwise independent 2-paths in general graphs which might be of independent interest. At the end of the paper we discuss lower bounds on the space complexity of triangle counting algorithms that make no assumptions on the structure of the graph. |
| Author | Bulteau, Laurent Froese, Vincent Kutzkov, Konstantin Pagh, Rasmus |
| Author_xml | – sequence: 1 givenname: Laurent surname: Bulteau fullname: Bulteau, Laurent organization: Technische Universität Berlin, Inria, LBBE – sequence: 2 givenname: Vincent surname: Froese fullname: Froese, Vincent organization: Technische Universität Berlin – sequence: 3 givenname: Konstantin surname: Kutzkov fullname: Kutzkov, Konstantin email: kutzkov@gmail.com organization: NEC Laboratories Europe – sequence: 4 givenname: Rasmus surname: Pagh fullname: Pagh, Rasmus organization: IT University of Copenhagen |
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| Cites_doi | 10.1007/978-3-642-23719-5_57 10.1016/0022-0000(79)90044-8 10.1007/978-3-642-31585-5_53 10.1103/PhysRevE.83.056119 10.1145/2220357.2220361 10.1007/s000370050018 10.1137/060658400 10.1145/2213977.2214056 10.1145/1989284.1989289 10.1145/2487575.2487678 10.1145/1142351.1142388 10.1145/1557019.1557111 10.1137/100800774 10.1007/11533719_72 10.1561/0400000002 10.1007/BF02523189 10.1103/RevModPhys.74.47 10.1145/2380718.2380723 10.1137/1.9781611972832.2 10.7155/jgaa.00245 10.1145/1064092.1064116 10.1016/j.ipl.2011.12.007 10.1145/2213556.2213560 |
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| Keywords | Streaming algorithms Triangle counting Randomized approximation algorithms |
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| References | Jowhari, H., Ghodsi, M.: New Streaming Algorithms for Counting Triangles in Graphs. In: Computing and Combinatorics, 11th Annual International Conference, COCOON 2005, Kunming, China, August 16–29, 710–716 (2005) PaghAPaghRUniform hashing in constant time and optimal spaceSIAM J. Comput.20083818596239952010.1137/0606584001165.68027 Jha, M., Seshadhri, C., Pinar, A.: A space efficient streaming algorithm for triangle counting using the birthday paradox. In: The 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, Chicago, IL, USA, August 11–14, 589–597 (2013) Aiello, W., Chung, F.R.K., Lu, L.: A Random Gmodel for Massive Graphs. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21–23, 2000, Portland, OR, USA (STOC), 171–180 (2000) MuthukrishnanSData streams: algorithms and applicationsFound. Trends Theor. Comput. Sci.2005121237950710.1561/04000000021128.68025 PaghRRasmusTCharalmposEColorful triangle counting and a MapReduce implementationInf. Process. Lett.20121127277281287916510.1016/j.ipl.2011.12.0071237.68245 AlonNYusterRZwickUFinding and counting given length cyclesAlgorithmica1997173209223142573410.1007/BF025231890865.68093 PavanATangwongsanKTirthapuraSWuKLCounting and sampling triangles from a graph streamPVLDB201361418701881 PǎtraşcuMThorupMThe power of simple tabulation hashingJ. ACM20125931429462181281.68089 AlbertRBarabasiALStatistical mechanics of complex networksRev. Mod. Phys.2002744797189509610.1103/RevModPhys.74.471205.82086 Arbitman, Y., Naor, M., Segev, G.: Backyard Cuckoo Hashing: Constant Worst-Case Operations with a Succinct Representation. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23–26, 2010, Las Vegas, Nevada, USA, 787–796 (2010) TsourakakisCEKolountzakisMNGaryGLTriangle sparsifiersJ. Graph Algorithms Appl.2011156703726284497810.7155/jgaa.002451276.05120 ThorupMZhangYTabulation-based 5-independent hashing with applications to linear probing and second moment estimationSIAM J. Comput.2012412293331291432910.1137/1008007741246.68107 Backstrom, L., Boldi, P., Rosa, M., Ugander, J., Vigna, S.: Four degrees of separation. Web Sci. 2012, WebScience ’12, Evanston, IL, USA, June 22–24, 33–42 (2012) Tsourakakis, C.E., Kang, U., Miller, G.L., Faloutsos, C.: DOULION: Counting Triangles in Massive Graphs with a Coin. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Paris, France, June 28–July 1, 837–846 (2009) Williams, V.V.: Multiplying Matrices Faster than Coppersmith-Winograd. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 887–898 (2012) BerryJWHendricksonBLaVioletteRAPhillipsCATolerating the community detection resolution limit with edge weightingPhys. Rev. E20118355611910.1103/PhysRevE.83.056119 Buriol, L.S., Frahling, G., Leonardi, S., Marchetti-Spaccamela, A., Sohler, C.: Counting triangles in data streams. In: Proceedings of the Twenty-Fifth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 26–28, Chicago, Illinois, USA, 253–262 (2006) Ahn, K.J., Guha, S., McGregor, A.: Graph Sketches: Sparsification, Spanners, and Subgraphs. In: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 5–14 (2012) Kane, D.M., Mehlhorn, K., Sauerwald, T., Sun, H.: Counting Arbitrary Subgraphs in Data Streams. In: Automata, Languages, and Programming—39th International Colloquium, ICALP: Warwick, UK, July 9–13, Proceedings. Part II 598–609 (2012) Leonardi, S.: List of Open Problems in Sublinear Algorithms: Problem 11. http://sublinear.info/11 Manjunath, M., Mehlhorn, K., Panagiotou, K., Sun, H.: Approximate Counting of Cycles in Streams. In: Algorithms—ESA 2011—19th Annual European Symposium, Saarbrücken, Germany, September 5–9, 677–688 (2011) KolountzakisMNMillerGLPengRRichardTCharalamposEEfficient triangle counting in large graphs via degree-based vertex partitioningInt. Math.201281–216118529004921245.05120 BecchettiLBoldiPCastilloCGionisAEfficient algorithms for large-scale local triangle countingACM Trans. Knowl. Discov. Data20101311328 CarterLJWegmanMNUniversal classes of hash functionsJ. Comput. Syst. Sci.197918214315453217310.1016/0022-0000(79)90044-80412.68090 Jowhari, H., Saglam, M., Tardos, G.: Tight bounds for Lp samplers, finding duplicates in streams, and related problems. In: Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2011, June 12–16, Athens, Greece, 49–58 (2011) Seshadhri, C., Pinar, A., Kolda, T.: Triadic Measures on Graphs: The Power of Wedge Sampling. In: Proceedings of the 13th SIAM International Conference on Data Mining, May 2–4. Austin, Texas, USA, 10–18 (2013) Frahling, G., Indyk, P., Sohler, C.: Sampling in dynamic data streams and applications. In: Symposium on Computational Geometry 142–149 (2005) KremerINisanNRonDOn randomized one-round communication complexityComput. Complex.1999812149170021510.1007/s0003700500180942.68059 LJ Carter (36_CR10) 1979; 18 M Thorup (36_CR26) 2012; 41 R Pagh (36_CR22) 2012; 112 36_CR27 36_CR29 36_CR24 A Pavan (36_CR23) 2013; 6 JW Berry (36_CR8) 2011; 83 R Albert (36_CR3) 2002; 74 MN Kolountzakis (36_CR16) 2012; 8 I Kremer (36_CR17) 1999; 8 CE Tsourakakis (36_CR28) 2011; 15 36_CR7 36_CR9 N Alon (36_CR4) 1997; 17 36_CR5 L Becchetti (36_CR6) 2010; 13 36_CR1 36_CR2 36_CR19 36_CR18 A Pagh (36_CR21) 2008; 38 M Pǎtraşcu (36_CR25) 2012; 59 36_CR13 36_CR12 36_CR15 S Muthukrishnan (36_CR20) 2005; 1 36_CR14 36_CR11 |
| References_xml | – reference: Jha, M., Seshadhri, C., Pinar, A.: A space efficient streaming algorithm for triangle counting using the birthday paradox. In: The 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, Chicago, IL, USA, August 11–14, 589–597 (2013) – reference: AlonNYusterRZwickUFinding and counting given length cyclesAlgorithmica1997173209223142573410.1007/BF025231890865.68093 – reference: Ahn, K.J., Guha, S., McGregor, A.: Graph Sketches: Sparsification, Spanners, and Subgraphs. In: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 5–14 (2012) – reference: Jowhari, H., Ghodsi, M.: New Streaming Algorithms for Counting Triangles in Graphs. In: Computing and Combinatorics, 11th Annual International Conference, COCOON 2005, Kunming, China, August 16–29, 710–716 (2005) – reference: CarterLJWegmanMNUniversal classes of hash functionsJ. Comput. Syst. Sci.197918214315453217310.1016/0022-0000(79)90044-80412.68090 – reference: KolountzakisMNMillerGLPengRRichardTCharalamposEEfficient triangle counting in large graphs via degree-based vertex partitioningInt. Math.201281–216118529004921245.05120 – reference: BerryJWHendricksonBLaVioletteRAPhillipsCATolerating the community detection resolution limit with edge weightingPhys. Rev. E20118355611910.1103/PhysRevE.83.056119 – reference: Manjunath, M., Mehlhorn, K., Panagiotou, K., Sun, H.: Approximate Counting of Cycles in Streams. In: Algorithms—ESA 2011—19th Annual European Symposium, Saarbrücken, Germany, September 5–9, 677–688 (2011) – reference: PavanATangwongsanKTirthapuraSWuKLCounting and sampling triangles from a graph streamPVLDB201361418701881 – reference: Williams, V.V.: Multiplying Matrices Faster than Coppersmith-Winograd. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 887–898 (2012) – reference: AlbertRBarabasiALStatistical mechanics of complex networksRev. Mod. Phys.2002744797189509610.1103/RevModPhys.74.471205.82086 – reference: Frahling, G., Indyk, P., Sohler, C.: Sampling in dynamic data streams and applications. In: Symposium on Computational Geometry 142–149 (2005) – reference: Buriol, L.S., Frahling, G., Leonardi, S., Marchetti-Spaccamela, A., Sohler, C.: Counting triangles in data streams. In: Proceedings of the Twenty-Fifth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 26–28, Chicago, Illinois, USA, 253–262 (2006) – reference: PaghRRasmusTCharalmposEColorful triangle counting and a MapReduce implementationInf. Process. Lett.20121127277281287916510.1016/j.ipl.2011.12.0071237.68245 – reference: PǎtraşcuMThorupMThe power of simple tabulation hashingJ. ACM20125931429462181281.68089 – reference: Kane, D.M., Mehlhorn, K., Sauerwald, T., Sun, H.: Counting Arbitrary Subgraphs in Data Streams. In: Automata, Languages, and Programming—39th International Colloquium, ICALP: Warwick, UK, July 9–13, Proceedings. Part II 598–609 (2012) – reference: Tsourakakis, C.E., Kang, U., Miller, G.L., Faloutsos, C.: DOULION: Counting Triangles in Massive Graphs with a Coin. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Paris, France, June 28–July 1, 837–846 (2009) – reference: TsourakakisCEKolountzakisMNGaryGLTriangle sparsifiersJ. Graph Algorithms Appl.2011156703726284497810.7155/jgaa.002451276.05120 – reference: Leonardi, S.: List of Open Problems in Sublinear Algorithms: Problem 11. http://sublinear.info/11 – reference: Jowhari, H., Saglam, M., Tardos, G.: Tight bounds for Lp samplers, finding duplicates in streams, and related problems. In: Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2011, June 12–16, Athens, Greece, 49–58 (2011) – reference: Seshadhri, C., Pinar, A., Kolda, T.: Triadic Measures on Graphs: The Power of Wedge Sampling. In: Proceedings of the 13th SIAM International Conference on Data Mining, May 2–4. Austin, Texas, USA, 10–18 (2013) – reference: Arbitman, Y., Naor, M., Segev, G.: Backyard Cuckoo Hashing: Constant Worst-Case Operations with a Succinct Representation. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23–26, 2010, Las Vegas, Nevada, USA, 787–796 (2010) – reference: BecchettiLBoldiPCastilloCGionisAEfficient algorithms for large-scale local triangle countingACM Trans. Knowl. Discov. Data20101311328 – reference: Aiello, W., Chung, F.R.K., Lu, L.: A Random Gmodel for Massive Graphs. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21–23, 2000, Portland, OR, USA (STOC), 171–180 (2000) – reference: ThorupMZhangYTabulation-based 5-independent hashing with applications to linear probing and second moment estimationSIAM J. Comput.2012412293331291432910.1137/1008007741246.68107 – reference: MuthukrishnanSData streams: algorithms and applicationsFound. Trends Theor. Comput. Sci.2005121237950710.1561/04000000021128.68025 – reference: KremerINisanNRonDOn randomized one-round communication complexityComput. Complex.1999812149170021510.1007/s0003700500180942.68059 – reference: PaghAPaghRUniform hashing in constant time and optimal spaceSIAM J. Comput.20083818596239952010.1137/0606584001165.68027 – reference: Backstrom, L., Boldi, P., Rosa, M., Ugander, J., Vigna, S.: Four degrees of separation. 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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Bioinformatics Computational Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Mathematics of Computing Theory of Computation |
| Title | Triangle Counting in Dynamic Graph Streams |
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