Document Open Access Logo

Amortised Analysis of Dynamic Data Structures (Invited Talk)

Author Eva Rotenberg



PDF
Thumbnail PDF

File

LIPIcs.STACS.2023.2.pdf
  • Filesize: 401 kB
  • 2 pages

Document Identifiers

Author Details

Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark

Cite AsGet BibTex

Eva Rotenberg. Amortised Analysis of Dynamic Data Structures (Invited Talk). In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 2:1-2:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.2

Abstract

In dynamic data structures, one is interested in efficiently facilitating queries to a data set, while being able to efficiently perform updates as the data set undergoes changes. Often, relaxing the efficiency measure to the amortised setting allows for simpler algorithms. A well-known example of a data structure with amortised guarantees is the splay tree by Sleator and Tarjan [Daniel D. Sleator and Robert E. Tarjan, 1985]. Similarly, in data structures for dynamic graphs, one is interested in efficiently maintaining some information about the graph, or facilitating queries, as the graph undergoes changes in the form of insertion and deletion of edges. Examples of such information include connectivity, planarity, and approximate sparsity of the graph: is the graph presently connected? Is it planar? Has its arboricity grossly exceeded some specified number α̃? The related queries could be: is a connected to b? Are the edges uv and uw consecutive in the ordering around u in its current planar embedding? Or, report the O(α) out-edges of vertex x. In this talk, we will see Brodal and Fagerberg’s amortised algorithm for orienting sparse graphs (i.e. of arboricity ≤ α), so that each vertex has O(α) out-edges [Gerth Stølting Brodal and Rolf Fagerberg, 1999]. The algorithm itself is extremely simple, and uses an elegant amortised argument in its analysis. Then, we will visit the problem of dynamic planarity testing: is the graph presently planar? Here, we will see an elegant amortised reduction to the seemingly easier problem, where planarity-violating edges may be detected and rejected [Eppstein et al., 1996]. We will see a sketch of how the current state-of-the-art algorithm for efficient planarity testing [Jacob Holm and Eva Rotenberg, 2020] uses ideas similar to those in [Gerth Stølting Brodal and Rolf Fagerberg, 1999] to analyse the behaviour of a greedy algorithm via a possibly inefficient algorithm with provably low recourse [Jacob Holm and Eva Rotenberg, 2020]. If time permits, we will touch upon a recent simple amortised data structure for maintaining information in dynamic forests [Jacob Holm et al., 2023], which builds on ideas from splay trees. The talk concludes with some open questions in the area.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Amortised analysis
  • splaying
  • dynamic graphs
  • planarity testing

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Stephen Alstrup, Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Maintaining information in fully dynamic trees with top trees. ACM Trans. Alg., 1(2):243-264, 2005. Google Scholar
  2. Gerth Stølting Brodal and Rolf Fagerberg. Dynamic representation of sparse graphs. In WADS '99, volume 1663 of Lecture Notes in Computer Science, pages 342-351. Springer, 1999. URL: https://doi.org/10.1007/3-540-48447-7_34.
  3. Richard Cole. On the dynamic finger conjecture for splay trees. part II: the proof. SIAM J. Comput., 30(1):44-85, 2000. URL: https://doi.org/10.1137/S009753979732699X.
  4. Richard Cole, Bud Mishra, Jeanette P. Schmidt, and Alan Siegel. On the dynamic finger conjecture for splay trees. part I: splay sorting log n-block sequences. SIAM J. Comput., 30(1):1-43, 2000. URL: https://doi.org/10.1137/S0097539797326988.
  5. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification: I. planarity testing and minimum spanning trees. Journal of Computer and Systems Sciences, 52(1):3-27, February 1996. URL: https://doi.org/10.1006/jcss.1996.0002.
  6. Zvi Galil, Giuseppe F. Italiano, and Neil Sarnak. Fully dynamic planarity testing with applications. Journal of the ACM, 46(1):28-91, 1999. URL: https://doi.org/10.1145/300515.300517.
  7. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, 2001. URL: https://doi.org/10.1145/502090.502095.
  8. Jacob Holm and Eva Rotenberg. Fully-Dynamic Planarity Testing in Polylogarithmic Time. In STOC 2020, pages 167-180. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384249.
  9. Jacob Holm and Eva Rotenberg. Worst-case polylog incremental spqr-trees: Embeddings, planarity, and triconnectivity. In SODA 2020, pages 2378-2397. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.146.
  10. Jacob Holm, Eva Rotenberg, and Alice Ryhl. Splay Top Trees. In SOSA '23. SIAM, 2023. Google Scholar
  11. John E. Hopcroft and Robert Endre Tarjan. Efficient planarity testing. Journal of the ACM, 21(4):549-568, 1974. URL: https://doi.org/10.1145/321850.321852.
  12. Johannes A. La Poutré. Alpha-algorithms for incremental planarity testing (preliminary version). In STOC '94, pages 706-715, 1994. URL: https://doi.org/10.1145/195058.195439.
  13. Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362-391, 1983. URL: https://doi.org/10.1016/0022-0000(83)90006-5.
  14. Daniel D. Sleator and Robert E. Tarjan. Self-Adjusting Binary Search Trees. J. ACM, 32(3):652-686, 1985. URL: https://doi.org/10.1145/3828.3835.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail