Skip to main content

Advertisement

Springer Nature Link
Log in

Compressed \(\text {k}\mathsf {^d}\text {-tree}\) for temporal graphs

  • Regular Paper
  • Published:
Knowledge and Information Systems Aims and scope Submit manuscript

Abstract

Temporal graphs represent vertices and binary relations that change along time. The work in this paper proposes to represent temporal graphs as cells in a 4D binary matrix: two dimensions to represent extreme vertices of an edge and two dimensions to represent the temporal interval when the edge exists. This strategy generalizes the idea of the adjacency matrix for storing static graphs. The proposed structure called Compressed \(\text {k}\mathsf {^d}\text {-tree}\) (\(\text {ck}\mathsf {^d}\text {-tree}\)) is capable of dealing with unclustered data with a good use of space. The \(\text {ck}\mathsf {^d}\text {-tree}\) uses asymptotically the same space than the (worst case) lower bound for storing cells in a 4D binary matrix, without considering any regularity. Techniques that group leaves into buckets and compress nodes with few children show to improve the performance in time and space. An experimental evaluation compares the \(\text {ck}\mathsf {^d}\text {-tree}\) with \(\text {k}\mathsf {^d}\text {-tree}\) (the d-dimensional extension of the \(\text {k}\mathsf {^2}\text {-tree}\)) and with other up-to-date compressed data structures based on inverted indexes and \(\mathsf {Wavelet}\text { Tree}\)s, showing the potential use of the \(\text {ck}\mathsf {^d}\text {-tree}\) for different types of temporal graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Notes

  1. We use the term time instant and time point indistinctly.

  2. Holme and Saramäki defined this as a contact sequence, but we renamed the concept to point-contact temporal contact.

  3. We assume that the boundary is closed at the upper-left corner and open at the lower-right corner.

  4. Other orders have been proposed, but they do not make any improvement on the space or the navigation time [18].

  5. When the input is small, the sorting method is faster than creating a hash table to remove duplicated items.

  6. Available at http://socialnetworks.mpi-sws.org/data-www2009.html.

  7. Downloaded from http://dumps.wikimedia.org/enwiki/.

  8. Downloaded from http://konect.uni-koblenz.de/.

  9. Available at https://github.com/simongog/sdsl-lite.

  10. Available at https://github.com/fclaude/libcds.

  11. The structures were implemented by Susana Ladra (\(\text {k}\mathsf {^2}\text {-tree}\)), Guillermo de Bernardo and Sandra Álvarez (\(\text {ik}\mathsf {^2}\text {-tree}\)).

  12. In this section, we will use B with a suffix to denote the size of the bucket in the \(\text {bck}\mathsf {^d}\text {-tree}\) and b (without a suffix) to denote the block size in bitmaps.

  13. The program used to create the structures failed when the lifetime of the graph is greater than 10,000 instants.

References

  1. Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97

    Article  MathSciNet  MATH  Google Scholar 

  2. Apostolico A, Drovandi G (2009) Graph compression by BFS. Algorithms 2(3):1031–1044

    Article  MathSciNet  Google Scholar 

  3. Álvarez-García S, Brisaboa NR, Fernández JD, Martínez-Prieto MA (2011) Compressed k2-triples for full-in-memory rdf engines. In: Proceedings of the Americas conference on information systems (AMCIS). Association for Information Systems

  4. Aluru S, Sevilgen FE (1999) Dynamic compressed hypertoctrees with application to the N-body problem. In: Proceedings of the 19th conference on foundations of software technology and theoretical computer science. Springer, Berlin

  5. Brisaboa NR, Caro D, Fariña A, Rodríguez A (2014) A compressed suffix-array strategy for temporal-graph indexing. In: Moura E, Crochemore M (eds) String processing and information retrieval. Lecture notes in computer science. Springer International Publishing, pp 77–88

  6. Brisaboa NR, de Bernardo G, Navarro G (2012) Compressed dynamic binary relations. In: Data compression conference (DCC). IEEE Computer Society, pp 52–61

  7. Benoit D, Demaine ED, Ian Munro J, Raman R, Raman V, Srinivasa Rao S (2005) Representing trees of higher degree. Algorithmica 43(4):275–292

    Article  MathSciNet  MATH  Google Scholar 

  8. Brisaboa NR, Ladra S, Navarro G (2009) k2-trees for compact web graph representation. In: International symposium on string processing and information retrieval (SPIRE), vol 5721 of lecture notes in computer science. Springer, Berlin, pp 18–30

  9. Brisaboa NR, Ladra S, Navarro G (2013) DACs: bringing direct access to variable-length codes. Inf Process Manag 49(1):392–404

    Article  Google Scholar 

  10. Brisaboa NR, Ladra S, Navarro G (2014) Compact representation of web graphs with extended functionality. Inf Syst 39:152–174

    Article  Google Scholar 

  11. Brodnik A, Ian Munro J (1999) Membership in constant time and almost-minimum space. SIAM J Comput 28(5):1627–1640

    Article  MathSciNet  MATH  Google Scholar 

  12. Caro D, Rodríguez MA, Brisaboa NR (2015) Data structures for temporal graphs based on compact sequence representations. Inf Syst 51:1–26

    Article  Google Scholar 

  13. Clarkson KL (1983) Fast algorithms for the all nearest neighbors problem. In: Proceedings of the 24th annual symposium on foundations of computer science (sfcs 1983). IEEE, pp 226–232

  14. Cha M, Mislove A, Gummadi PK (2009) A measurement-driven analysis of information propagation in the flickr social network. In: International world wide web conference (WWW). ACM, pp 721–730

  15. Claude F, Navarro G (2008) Practical rank/select queries over arbitrary sequences. In: International symposium on string processing and information retrieval (SPIRE), vol 5280 of lecture notes in computer science. Springer, pp 176–187

  16. de Bernardo G, Álvarez-García S, Brisaboa NR, Navarro G, Pedreira O (2013) Compact querieable representations of raster data. In: International symposium on string processing and information retrieval (SPIRE), vol 8214 of lecture notes in computer science. Springer, pp 96–108

  17. de Bernardo G, Brisaboa NR, Caro D, Rodríguez MA (2013) Compact data structures for temporal graphs. In: Data compression conference (DCC). IEEE, p 477

  18. de Bernardo Roca G (2014) New data structures and algorithms for the efficient managementof large spatial datasets. PhD thesis, Universidade da Coruña

  19. Demetrescu C, Eppstein D, Galil Z, Italiano GF (2010) Algorithms and theory of computation handbook, chapter dynamic graph algorithms. Chapman & Hall/CRC, pp 9-1–9-27

  20. Eppstein D, Goodrich MT, Sun JZ (2005) The skip quadtree: a simple dynamic data structure for multidimensional data. In: SCG ’05: proceedings of the twenty-first annual symposium on computational geometry. ACM Request Permissions

  21. Fariña A, Brisaboa N, Navarro G, Claude F, Places A, Rodríguez E (2012) Word-based self-indexes for natural language text. ACM TOIS 30(1):1

    Article  Google Scholar 

  22. Ferreira A, Viennot L (2002) A note on models, algorithms, and data structures for dynamic communication networks. Technical Report RR-4403, INRIA

  23. Gargantini I (1982) An effective way to represent quadtrees. In: Communications of the ACM, pp 1–6

  24. Garcia SA (2014) Compact and Efficient Representations of Graphs. PhD thesis, Universidade da Coruña

  25. Garcia SA, Brisaboa NR, de Bernardo G, Navarro G (2014) Interleaved K2-tree: indexing and navigating ternary relations. In: 2014 data compression conference (DCC). IEEE, pp 342–351

  26. Gog S, Beller T, Moffat A, Petri M (2014) From theory to practice: plug and play with succinct data structures. In: Proceedings of the 13th international symposium on experimental algorithms, (SEA 2014), pp 326–337

  27. Grossi R, Gupta A, Vitter JS (2003) High-order entropy-compressed text indexes. In: Proceedings of the annual ACM-SIAM symposium on discrete algorithms (SODA). ACM/SIAM, pp 841–850

  28. Holme P, Saramäki J (2012) Temporal networks. Phys Rep 519(3):97–125

    Article  Google Scholar 

  29. Hudson B (2009) Succinct representation of well-spaced point clouds. Technical Report. arXiv:0909.3137

  30. Jacobson G (1989) Space-efficient static trees and graphs. In: Proceedings of the 30th annual symposium on foundations of computer science (FOCS). IEEE Computer Society, pp 549–554

  31. Khurana U, Deshpande A (2013) Efficient snapshot retrieval over historical graph data. In: International conference on data engineering (ICDE). IEEE Computer Society, pp 997–1008

  32. Kunegis J (2013) Konect: the koblenz network collection. In: Proceedings of the 22nd international conference on world wide web companion, WWW ’13 Companion, pp 1343–1350, Republic and Canton of Geneva, Switzerland, 2013. International World Wide Web Conferences Steering Committee

  33. Labouseur AG, Birnbaum J, Olsen PW, Spillane SR, Vijayan J, Hwang J-H, Han W-S (2014) The G* graph database: efficiently managing large distributed dynamic graphs. Distrib Parallel Databases

  34. Matsuyama T, Hao LV, Nagao M (1984) A file organization for geographic information systems based on spatial proximity. Comput Vis Graph Image Process 26(3):303–318

    Article  Google Scholar 

  35. Nicosia V, Tang J, Mascolo C, Musolesi M, Russo G, Latora V (2013) Graph metrics for temporal networks. In: Temporal networks, understanding complex systems. Springer Berlin Heidelberg, pp 15–40

  36. Pagh R (1999) Low redundancy in static dictionaries with O(1) worst case lookup time. In: ICAL ’99: proceedings of the 26th international colloquium on automata, languages and programming. Springer, Berlin

  37. Ren C, Lo E, Kao B, Zhu X, Cheng R (2011) On querying historical evolving graph sequences. Proc VLDB Endow (PVLDB) 4(11):726–737

    Google Scholar 

  38. Raman R, Raman V, Srinivasa Rao S (2002) Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In: Proceedings SODA’12, pp 233–242

  39. Sadakane K (2003) New text indexing functionalities of the compressed suffix arrays. J Algorithms 48(2):294–313

    Article  MathSciNet  MATH  Google Scholar 

  40. Samet H (2006) Foundations of multidimensional and metric data structures. Morgan Kaufmann, Burlington, MA

    MATH  Google Scholar 

  41. Xuan BB, Ferreira A, Jarry A (2003) Computing shortest, fastest, and foremost journeys in dynamic networks. Int J Found Comput Sci 14(02):267–285

    Article  MathSciNet  MATH  Google Scholar 

  42. Yahoo! Labs (2014) Yahoo! network flows data, version 1.0. http://webscope.sandbox.yahoo.com/catalog.php?datatype=g

  43. Zukowski M, Héman S, Nes N, Boncz PA (2006) Super-scalar ram-cpu cache compression. In: Proceedings ICDE’06, p 59

  44. Zhang J, Long X, Suel T (2008) Performance of compressed inverted list caching in search engines. In: Proceedings WWW’08, pp 387–396

Download references

Acknowledgments

Diego Caro and M. Andrea Rodríguez were partially funded by Fondef D09I1185. Diego Caro was also funded by CONICYT PhD scholarship and M. Andrea Rodríguez by Fondecyt 1140428 and MINECO (PGE and FEDER) Grant TIN2013-46801-C4-3-R. Nieves Brisaboa and Antonio Fariña are funded by MINECO (PGE and FEDER) Grants TIN2013-46238-C4-3-R and TIN2013-47090-C3-3-P; CDTI, AGI and MINECO Grant CDTI-00064563/ITC-20133062; ICT COST Action IC1302; and by Xunta de Galicia (co-founded with FEDER) Grant GRC2013/053. We also thank to Diego Seco for his help in the preliminary discussions of the structures, to Gonzalo Navarro and Simon Gog for their suggestions on the improvement of the data structures and the experimental evaluation, and to Claudio Sanhueza from Yahoo! Labs who helps us with the P-Yahoo-Session dataset.

Author information

Authors and Affiliations

  1. Computer Science Department, University of Concepción, Concepción, Chile

    Diego Caro & M. Andrea Rodríguez

  2. Database Lab, Facultade de Informática, University of A Coruña, A Coruña, Spain

    Nieves R. Brisaboa & Antonio Fariña

Authors
  1. Diego Caro
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. M. Andrea Rodríguez
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Nieves R. Brisaboa
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Antonio Fariña
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Diego Caro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caro, D., Rodríguez, M.A., Brisaboa, N.R. et al. Compressed \(\text {k}\mathsf {^d}\text {-tree}\) for temporal graphs. Knowl Inf Syst 49, 553–595 (2016). https://doi.org/10.1007/s10115-015-0908-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10115-015-0908-6

Keywords