Skip to main content

Eulerian tour algorithms for data visualization and the PairViz package


PairViz is an R package that produces orderings of statistical objects for visualization purposes. We abstract the ordering problem to one of constructing edge-traversals of (possibly weighted) graphs. PairViz implements various edge traversal algorithms which are based on Eulerian tours and Hamiltonian decompositions. We describe these algorithms, their PairViz implementation and discuss their properties and performance. We illustrate their application to two visualization problems, that of assessing rater agreement, and model comparison in regression.

This is a preview of subscription content, access via your institution.


  • Allison T, Cicchetti D (1976) Sleep in mammals: ecological and constitutional correlates. Science 194: 732–734

    Article  Google Scholar 

  • Alspach B, Bermond JC, Sotteau D (1990) Decomposition into cycles I: Hamilton decompositions. In: Hahn G, Sabidussi G, Woodrow RE (eds) Cycles and rays. Kluwer, Boston, pp 9–18

    Google Scholar 

  • Ankerst M, Berchtold S, Keim DA (1998) Similarity clustering of dimensions for an enhanced visualization of multidimensional data. IEEE Symp Inf Vis 1998: 52–60

    Google Scholar 

  • Bendix F, Kosara R, Hauser H (2005) Parallel sets: visual analysis of categorical data. IEEE Symp Inf Vis 2005: 1–18

    Google Scholar 

  • Cleveland WS (1995) Visualizing data. Hobart Press, Summit, NJ

    Google Scholar 

  • Csardi G, Nepusz T (2006) The igraph software package for complex network research. InterJournal, Complex Systems 1695

  • Edmonds J, Johnson EL (1973) Matching, euler tours, and the Chinese postman. Math Program 5: 88–124

    Article  MathSciNet  MATH  Google Scholar 

  • Fleiss JL (1971) Measuring nominal scale agreement among many raters. Psychol Bull 76: 378–382

    Article  Google Scholar 

  • Friendly M, Kwan E (2003) Effect ordering for data displays. Comput Stat Data Anal 43: 509–539

    Article  MathSciNet  MATH  Google Scholar 

  • Gamer M, Lemon J, Fellows I (2009) irr: various coefficients of interrater reliability and agreement. R package version 0.82

  • Gentleman R, Whalen E, Huber W, Falcon S (2010) graph: a package to handle graph data structures. R package version 1.26.0

  • Gross, JL, Yellen, J (eds) (2004) Handbook of graph theory. CRC Press, London

    MATH  Google Scholar 

  • Hierholzer C (1873) Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math Annalen VI:30–32

    Article  MathSciNet  Google Scholar 

  • Hofmann H (2006) Multivariate categorical data–mosaic plots. In: Unwin A, Theus M, Hofmann H (eds) Graphics of large datasets, visualizing a million. Springer, New York, pp 105–124

    Chapter  Google Scholar 

  • Hurley CB (2004) Clustering visualizations of multidimensional data. J Comput Graphical Stat 13: 788–806

    Article  MathSciNet  Google Scholar 

  • Hurley CB, Oldford RW (2009) PairViz: visualization using Eulerian tours and Hamiltonian decompositions, R package version 1.1

  • Hurley CB, Oldford RW (2010) Pairwise display of high dimensional information via Eulerian tours and Hamiltonian decompositions. J Comput Graphical Stat 19(4): 861–886

    Article  Google Scholar 

  • Hurley CB, Oldford RW (2011) Graphs as navigational infrastructure for high dimensional data spaces. Comput Stat. doi:10.1007/s00180-011-0228-6

  • Lucas DE (1892) Recréations Mathématiques, Vol. II. Gauthier Villars, Paris

    Google Scholar 

  • Theus M (2002) Interactive data visualization using mondrian. J Stat Softw 7(11): 1–9

    Google Scholar 

  • Unwin A, Volinsky C, Winkler S (2003) Parallel coordinates for exploratory modelling analysis. Comput Stat Data Anal 43: 553–564

    Article  MathSciNet  MATH  Google Scholar 

  • Wegman EJ (1990) Hyperdimensional data analysis using parallel coordinates. J Am Stat Assoc 85: 664–675

    Article  Google Scholar 

  • Wilkinson L, Anand A, Grossman R (2005) Graph-theoretic scagnostics. IEEE Symp Inf Vis 2005: 157–164

    Google Scholar 

  • Wills G (2000) A good, simple axis. Stat Comput Stat Graphics Newsl 11: 20–25

    Google Scholar 

  • Yang J, Peng W, Ward MO, Rundenmeister EA (2003) Interactive hierarchical dimension ordering, spacing and filtering for exploration of high dimensional datasets. IEEE Symp Inf Vis 2003: 105–112

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to C. B. Hurley.

Additional information

C. B. Hurley’s Research was supported in part by a Research Frontiers Grant from Science Foundation Ireland. R. W. Oldford’s Research was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hurley, C.B., Oldford, R.W. Eulerian tour algorithms for data visualization and the PairViz package. Comput Stat 26, 613–633 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Hamiltonian
  • Eulerian tour
  • Seriation
  • Visualization
  • Parallel coordinates