Separating sublinear time computations by approximate diameter

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 randomi...

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Vydané v:	Journal of combinatorial optimization Ročník 18; číslo 4; s. 393 - 416
Hlavní autori:	Fu, Bin, Zhao, Zhiyu
Médium:	Journal Article Konferenčný príspevok..
Jazyk:	English
Vydavateľské údaje:	Boston Springer US 01.11.2009 Springer
Predmet:	Applied sciences Combinatorics Convex and Discrete Geometry Exact sciences and technology Flows in networks. Combinatorial problems Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operational research and scientific management Operational research. Management science Operations Research/Decision Theory Optimization Theory of Computation Randomization Separation of complexity classes Sublinear time algorithm Diameter Complexity class Metric space Query Error bound Randomized algorithm Combinatorial optimization Modeling Computation time Randomized design
ISSN:	1382-6905, 1573-2886
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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.
AbstractList	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.
Author	Zhao, Zhiyu Fu, Bin
Author_xml	– sequence: 1 givenname: Bin surname: Fu fullname: Fu, Bin email: binfu@cs.panam.edu organization: Department of Computer Science, University of Texas–Pan American – sequence: 2 givenname: Zhiyu surname: Zhao fullname: Zhao, Zhiyu organization: Department of Computer Science, University of New Orleans
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Keywords	Randomization Separation of complexity classes Sublinear time algorithm Diameter Complexity class Metric space Query Error bound Randomized algorithm Combinatorial optimization Modeling Computation time Randomized design
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References_xml	– reference: FischerEThe art of uninformed decision: a primer to property testingBull EATCS200175971261024.68045 – reference: 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 – reference: FeigeUOn sums of independent random variables with unbounded variance and estimating the average degree in a graphSIAM J Comput2006359649841098.6002610.1137/S00975397044473042203734 – reference: FuBChenZSublinear time width-bounded separators and their applications to the protein side-chain packing problemJ Comb Optim20081543874070528470210.1007/s10878-007-9092-22386524 – reference: 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/ – reference: ChazelleBLiuDMagenASublinear geometric algorithmsSIAM J Comput20053562764610.1137/S009753970444572X2201450 – reference: KumarRRubinfeldRSublinear time algorithmsSIGACT News200334576710.1145/954092.954103 – reference: Chen L, Fu B (2007) Linear and sublinear time algorithms for the basis of Abelian groups. Electron Colloq Comput Complex TR07-052 – reference: CzumajAErgunFFortnowLMagenINARubinfeldRSohlerCSublinear approximation of euclidean minimum spanning treeSIAM J Comput200535911091086.6814410.1137/S00975397034352972178799 – reference: Goldreich O (2002) Property testing in massive graphs. In: Abello J, Pardalos PM, Resende M (eds) Handbook of massive data sets, pp 123–147 – reference: TrenchWFAdvanced calculus1978New YorkHarper & Row – reference: 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/ – reference: 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 – reference: RonDHandbook of randomized algorithmBull EATCS2001II5976491966913 – reference: Zhao Z, Fu B (2007) A flexible algorithm for pairwise protein structure alignment. In: Proceedings international conference on bioinformatics and computational biology – reference: 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 – reference: 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 – reference: GoldreichSGORonDProperty testing and its connection to learning and approximationJ ACM1998456537501065.6857510.1145/285055.2850601675099 – reference: ChazelleBRubfinfeldRTrevisanLApproximating the minimum spanning tree weight in sublinear timeSIAM J Comput200534137013791081.6812010.1137/S00975397024032442165745 – reference: Zimand M (2007) On derandomizing probabilistic sublinear-time algorithms. In: Proceedings of the 22nd IEEE conference on computational complexity, pp 1–9 – ident: 9248_CR11 doi: 10.1090/dimacs/043/04 – ident: 9248_CR1 doi: 10.1007/11523468_70 – volume: 35 start-page: 627 year: 2005 ident: 9248_CR2 publication-title: SIAM J Comput doi: 10.1137/S009753970444572X – ident: 9248_CR20 doi: 10.1109/CCC.2007.19 – volume-title: Advanced calculus year: 1978 ident: 9248_CR18 – volume: 75 start-page: 97 year: 2001 ident: 9248_CR9 publication-title: Bull EATCS – volume: 35 start-page: 964 year: 2006 ident: 9248_CR8 publication-title: SIAM J Comput doi: 10.1137/S0097539704447304 – volume: 34 start-page: 57 year: 2003 ident: 9248_CR15 publication-title: SIGACT News doi: 10.1145/954092.954103 – ident: 9248_CR7 doi: 10.1109/SFCS.2001.959921 – volume: 15 start-page: 387 issue: 4 year: 2008 ident: 9248_CR10 publication-title: J Comb Optim doi: 10.1007/s10878-007-9092-2 – ident: 9248_CR13 – ident: 9248_CR14 – volume: 35 start-page: 91 year: 2005 ident: 9248_CR6 publication-title: SIAM J Comput doi: 10.1137/S0097539703435297 – volume: 34 start-page: 1370 year: 2005 ident: 9248_CR3 publication-title: SIAM J Comput doi: 10.1137/S0097539702403244 – ident: 9248_CR19 doi: 10.1142/9781848162648_0010 – ident: 9248_CR12 doi: 10.1007/978-1-4615-0005-6_5 – volume: II start-page: 597 year: 2001 ident: 9248_CR17 publication-title: Bull EATCS – ident: 9248_CR4 – ident: 9248_CR5 doi: 10.1145/1007352.1007386 – volume: 45 start-page: 653 year: 1998 ident: 9248_CR16 publication-title: J ACM doi: 10.1145/285055.285060
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SubjectTerms	Applied sciences Combinatorics Convex and Discrete Geometry Exact sciences and technology Flows in networks. Combinatorial problems Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operational research and scientific management Operational research. Management science Operations Research/Decision Theory Optimization Theory of Computation
Title	Separating sublinear time computations by approximate diameter
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