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All-Distances Sketches

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 388 Accesses

Years and Authors of Summarized Original Work

  • 1997, 2014; Cohen

  • 2002; Palmer, Gibbons, Faloutsos

  • 2004, 2007; Cohen, Kaplan

Problem Definition

All-distances sketches (The term least element lists was used in [3]; the terms MV/D lists and Neighborhood summaries were used in [6].) are randomized summary structures of the distance relations of nodes in a graph. The graph can be directed or undirected, and edges can have uniform or general nonnegative weights.

Preprocessing Cost

A set of sketches, \(\mathop{\mathrm{ADS}}(v)\) for each node v, can be computed efficiently, using a near-linear number of edge traversals. The sketch sizes are well concentrated, with logarithmic dependence on the total number of nodes.

Supported Queries

The sketches support approximate distance-based queries, which include:

  • Distance distribution: The query specifies a node v and value d ≥ 0 and returns the cardinality | N d (v) | of the d-neighborhood of vN d (v) = { u∣d vu  ≤ d}, where d uv...

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Recommended Reading

  1. Boldi P, Vigna S (2014) Axioms for centrality. Internet Math 10:222–262

    Article  MathSciNet  Google Scholar 

  2. Boldi P, Rosa M, Vigna S (2011) HyperANF: approximating the neighbourhood function of very large graphs on a budget. In: WWW, Hyderabad

    Google Scholar 

  3. Cohen E (1997) Size-estimation framework with applications to transitive closure and reachability. J Comput Syst Sci 55:441–453

    Article  MathSciNet  MATH  Google Scholar 

  4. Cohen E (2014) All-distances sketches, revisited: HIP estimators for massive graphs analysis. In: PODS. ACM. http://arxiv.org/abs/1306.3284

  5. Cohen E (2014) Estimation for monotone sampling: competitiveness and customization. In: PODC. ACM. http://arxiv.org/abs/1212.0243, full version http://arxiv.org/abs/1212.0243

  6. Cohen E, Kaplan H (2007) Spatially-decaying aggregation over a network: model and algorithms. J Comput Syst Sci 73:265–288. Full version of a SIGMOD 2004 paper

    Google Scholar 

  7. Cohen E, Kaplan H (2007) Summarizing data using bottom-k sketches. In: PODC, Portland. ACM

    Book  MATH  Google Scholar 

  8. Cohen E, Delling D, Fuchs F, Goldberg A, Goldszmidt M, Werneck R (2013) Scalable similarity estimation in social networks: closeness, node labels, and random edge lengths. In: COSN, Boston. ACM

    Book  Google Scholar 

  9. Cohen E, Delling D, Pajor T, Werneck RF (2014) Timed influence: computation and maximization. Manuscript. ArXiv 1410.6976

    Google Scholar 

  10. Dangalchev C (2006) Residual closeness in networks. Phisica A 365:556–564

    Article  Google Scholar 

  11. Flajolet P, Martin GN (1985) Probabilistic counting algorithms for data base applications. J Comput Syst Sci 31:182–209

    Article  MathSciNet  MATH  Google Scholar 

  12. Flajolet P, Fusy E, Gandouet O, Meunier F (2007) Hyperloglog: the analysis of a near-optimal cardinality estimation algorithm. In: Analysis of algorithms (AOFA), Juan des Pins

    Google Scholar 

  13. Guibas LJ, Knuth DE, Sharir M (1992) Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7:381–413

    Article  MathSciNet  MATH  Google Scholar 

  14. Opsahl T, Agneessens F, Skvoretz J (2010) Node centrality in weighted networks: generalizing degree and shortest paths. Soc Netw 32. http://toreopsahl.com/2010/03/20/

  15. Palmer CR, Gibbons PB, Faloutsos C (2002) ANF: a fast and scalable tool for data mining in massive graphs. In: KDD, Edmonton

    Book  Google Scholar 

  16. Thorup M, Zwick U (2001) Approximate distance oracles. In: Proceedings of the 33th annual ACM symposium on theory of computing, Crete, pp 183–192

    Google Scholar 

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Author information

Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Edith Cohen

  2. Stanford University, Stanford, CA, USA

    Edith Cohen

Authors
  1. Edith Cohen
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Corresponding author

Correspondence to Edith Cohen .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Cohen, E. (2016). All-Distances Sketches. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_574

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