Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Lower Bounds for Dynamic Connectivity

  • Mihai Pătraşcu
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_214

Years and Authors of Summarized Original Work

  • 2004; Pătraşcu, Demaine

Problem Definition

The dynamic connectivity problem requests maintenance of a graph G subject to the following operations:

insert( u, v):

insert an undirected edge (u, v) into the graph.

delete( u, v):

delete the edge (u, v) from the graph.

connected( u, v):

test whether u and v lie in the same connected component.

Let m be an upper bound on the number of edges in the graph. This entry discusses cell-probe lower bounds for this problem. Let tu be the complexity of insert and delete and tq the complexity of query.

The Partial-Sums Problem

Lower bounds for dynamic connectivity are intimately related to lower bounds for another classic problem: maintaining partial sums. Formally, the problem asks one to maintain an array \( { A[1 .. n] } \)

Keywords

Dynamic trees 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Alstrup S, Husfeldt T, Rauhe T (1998) Marked ancestor problems. In: Proceedings of the 39th IEEE symposium on foundations of computer science (FOCS), pp 534–543Google Scholar
  2. 2.
    Eppstein D, Italiano GF, Tamassia R, Tarjan RE, Westbrook JR, Yung M (1992) Maintenance of a minimum spanning forest in a dynamic planar graph. J Algorithm 13:33–54. See also SODA'90MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Fredman ML, Saks ME (1989) The cell probe complexity of dynamic data structures. In: Proceedings of the 21st ACM symposium on theory of computing (STOC), pp 345–354Google Scholar
  4. 4.
    Husfeldt T, Rauhe T (2003) New lower bound techniques for dynamic partial sums and related problems. SIAM J Comput 32:736–753. See also ICALP'98MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Miltersen PB (1999) Cell probe complexity – a survey. In: 19th conference on the foundations of software technology and theoretical computer science (FSTTCS) (Advances in Data Structures Workshop)Google Scholar
  6. 6.
    Pătraşcu M, Demaine ED (2006) Logarithmic lower bounds in the cell-probe model. SIAM J Comput 35:932–963. See also SODA'04 and STOC'04MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Pătraşcu M, Tarniţă C (2007) On dynamic bit-probe complexity. Theor Comput Sci 380:127–142. See also ICALP'05MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Sleator DD, Tarjan RE (1983) A data structure for dynamic trees. J Comput Syst Sci 26:362–391, See also STOC'81MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Thorup M (2000) Near-optimal fully-dynamic graph connectivity. In: Proceedings of the 32nd ACM symposium on theory of computing (STOC), pp 343–350Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mihai Pătraşcu
    • 1
  1. 1.Computer Science and Artificial Intelligence Laboratory (CSAIL)Massachusetts Institute of Technology (MIT)CambridgeUSA