Skip to main content
SpringerLink
Log in

Separating sublinear time computations by approximate diameter

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

We study the problem of separating sublinear time computations via approximating the diameter for a sequence S=p 1 p 2 ⋅⋅⋅ p n of points in a metric space, in which any two consecutive points have the same distance. The computation is considered respectively under deterministic, zero error randomized, and bounded error randomized models. We obtain a class of separations using various versions of the approximate diameter problem based on restrictions on input data. We derive tight sublinear time separations for each of the three computation models via proving that computation with O(n r) time is strictly more powerful than that with O(n r−ε) time, where r and ε are arbitrary parameters in (0,1) and (0,r) respectively. We show that, for any parameter r∈(0,1), the bounded error randomized sublinear time computation in time O(n r) cannot be simulated by any zero error randomized sublinear time algorithm in o(n) time or queries; and the same is true for zero error randomized computation versus deterministic computation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Badoiu M, Czumaj A, Indyk P, Sohler C (2005) Facility location in sublinear time. In: Proceedings of 32nd annual international colloquium on automata, languages and programming, pp 866–877

  • Chazelle B, Liu D, Magen A (2005a) Sublinear geometric algorithms. SIAM J Comput 35:627–646

    Article  MathSciNet  Google Scholar 

  • Chazelle B, Rubfinfeld R, Trevisan L (2005b) Approximating the minimum spanning tree weight in sublinear time. SIAM J Comput 34:1370–1379

    Article  MATH  MathSciNet  Google Scholar 

  • Chen L, Fu B (2007) Linear and sublinear time algorithms for the basis of Abelian groups. Electron Colloq Comput Complex TR07-052

  • Czumaj A, Sohler C (2004) Estimating the weight of metric minimum spanning trees in sublinear-time. In: Proceedings of the 36th annual. ACM, Symposium on theory of computing, pp 175–183

  • Czumaj A, Ergun F, Fortnow L, Magen INA, Rubinfeld R, Sohler C (2005) Sublinear approximation of euclidean minimum spanning tree. SIAM J Comput 35:91–109

    Article  MATH  MathSciNet  Google Scholar 

  • Drineas P, Kannan R (2001) Fast Monte-Carlo algorithms for approximate matrix multiplication. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer. Science, pp 452–459

  • Feige U (2006) On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM J Comput 35:964–984

    Article  MATH  MathSciNet  Google Scholar 

  • Fischer E (2001) The art of uninformed decision: a primer to property testing. Bull EATCS 75:97–126

    MATH  Google Scholar 

  • Fu B, Chen Z (2008) Sublinear time width-bounded separators and their applications to the protein side-chain packing problem. J Comb Optim 15(4):387–407

    Article  MATH  MathSciNet  Google Scholar 

  • Goldreich O (1997) Combinatorial property testing (a survey). In: Pardalos P, Rajasekaran S, Rolim J (eds) Proceesdings of the DIMACS workshop on radnomziation methods in algorithm design. DIMACS series in discrete mathematics and theoretical computer science, vol 43, pp 45–59

  • Goldreich O (2002) Property testing in massive graphs. In: Abello J, Pardalos PM, Resende M (eds) Handbook of massive data sets, pp 123–147

  • Goldreich O, Ron D (2000) On testing expansion in bounded-degree graphs. Technical Report 00-20, Electronic Colloquium on Computational Complexity. http://www.eccc.uni-trier.de/eccc/

  • Goldreich O, Ron D (2005) Approximating average parameters of graphs. Technical Report 05-73, Electronic Colloquium on Computational Complexity. http://www.eccc.uni-trier.de/eccc/

  • Kumar R, Rubinfeld R (2003) Sublinear time algorithms. SIGACT News 34:57–67

    Article  Google Scholar 

  • Goldreich SGO, Ron D (1998) Property testing and its connection to learning and approximation. J ACM 45:653–750

    Article  MATH  MathSciNet  Google Scholar 

  • Ron D (2001) Handbook of randomized algorithm. Bull EATCS II:597–649

    MathSciNet  Google Scholar 

  • Trench WF (1978) Advanced calculus. Harper & Row, New York

    Google Scholar 

  • Zhao Z, Fu B (2007) A flexible algorithm for pairwise protein structure alignment. In: Proceedings international conference on bioinformatics and computational biology

  • Zimand M (2007) On derandomizing probabilistic sublinear-time algorithms. In: Proceedings of the 22nd IEEE conference on computational complexity, pp 1–9

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas–Pan American, Edinburg, TX, 78539, USA

    Bin Fu

  2. Department of Computer Science, University of New Orleans, New Orleans, LA, 70148, USA

    Zhiyu Zhao

Authors
  1. Bin Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiyu Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Fu.

Additional information

This research is supported in part by the National Science Foundation Early Career Award 0845376.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fu, B., Zhao, Z. Separating sublinear time computations by approximate diameter. J Comb Optim 18, 393–416 (2009). https://doi.org/10.1007/s10878-009-9248-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-009-9248-3

Keywords

Search

Navigation