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Measuring the problem-relevant information in input

Published online by Cambridge University Press:  04 April 2009

Stefan Dobrev ,
Rastislav Královič  and
Dana Pardubská
Stefan Dobrev
Affiliation:
Institute of Mathematics, Slovak Academy of Sciences, Slovakia; Stefan.Dobrev@savba.sk
Rastislav Královič
Affiliation:
Department of Computer Science, Comenius University, Bratislava, Slovakia; kralovic@dcs.fmph.uniba.sk;pardubska@dcs.fmph.uniba.sk
Dana Pardubská
Affiliation:
Department of Computer Science, Comenius University, Bratislava, Slovakia; kralovic@dcs.fmph.uniba.sk;pardubska@dcs.fmph.uniba.sk
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Abstract

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of the output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated by the algorithm to the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver upper and lower bounds in both communication modes; in the case of DiffServ problem in helper mode the bounds are tight.

Keywords

Online algorithmscommunication complexityadvice complexitypaging.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 3 , July 2009 , pp. 585 - 613
Copyright
© EDP Sciences, 2009

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References

Achlioptas, D., Chrobak, M. and Noga, J., Competitive analysis of randomized paging algorithms. Theoret. Comput. Sci. 234 (2000) 203–218. CrossRef
Albers, S., On the influence of lookahead in competitive paging algorithms. Algorithmica 18 (1997) 283–305. CrossRef
Albers, S., Online algorithms: A survey. Math. Prog. 97 (2003) 3–26. CrossRef
Belady, L.A., A study of replacement algorithms for virtual storage computers. IBM Systems Journal 5 (1966) 78–101. CrossRef
Ben-David, S. and Borodin, A., A new measure for the study of on-line algorithms. Algorithmica 11 (1994) 73–91. CrossRef
A. Borodin and R. El-Yaniv, Online Computation and Competitive Analysis. Cambridge University Press (1998).
A. Borodin, S. Irani, P. Raghavan and B. Schieber, Competitive paging with locality of reference. In Proc. 23rd Annual ACM Symposium on Theory of Computing (1991) 249–259.
J. Boyar, M.R. Ehmsen and K.S. Larsen, Theoretical Evidence for the superiority of LRU-2 over LRU for the paging problem. In Fourth Workshop on Approximation on Online Algorithms. Lecture Notes Comput. Sci. 4368 (2006) 95–107.
Boyar, J., Larsen, K.S. and Nielsen, M.N., The accommodating function: a generalization of the competitive ratio. SIAM J. Comput. 31 (2001) 233–258. CrossRef
J. Boyar and L.M. Favrholdt, The relative worst order ratio for online algorithms, Algorithms and Complexity, 5th Italian Conference, CIAC 2003, Rome, Italy. Lect. Notes Comput. Sci. 2653 (2003) 58–69.
M. Englert and M. Westermann, lower and upper bounds on FIFO buffer management in QoS switches, In Proc. ESA 2006. Lect. Notes Comput. Sci. 4168 (2006) 352–363.
Fiat, A., Karp, R.M., Luby, M., McGeoch, L.A., Sleator, D.D. and Young, N.E., Competitive paging algorithms. J. Algorithms 12 (1991) 685–699. CrossRef
P. Fraigniaud, C. Gavoille, D. Ilcinkas and A. Pelc, Distributed computing with advice: information sensitivity of graph coloring. In Proc. 34th International Colloquium on Automata, Languages and Programming (ICALP 2007) (2007).
P. Fraigniaud, D. Ilcinkas and A. Pelc, Tree exploration with an oracle. In Proc. 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006). Lect. Notes Comput. Sci. 4162 (2006) 24–37.
P. Fraigniaud, D. Ilcinkas and A. Pelc, Oracle size: a new measure of difficulty for communication problems. In Proc. 25th Ann. ACM Symposium on Principles of Distributed Computing (PODC 2006) (2006) 179–187.
Graham, R.L., Bounds for certain multiprocessing anomalies. Bell Systems Technical Journal 45 (1966) 1563–1581. CrossRef
S. Irany and A.R. Karlin, Online computation. In Approximation Algorithms for NP-Hard Problems, D.S. Hochbaum, Ed. PWS Publishing Company (1997) 521–564.
S. Irani, A.R. Karlin and S. Phillips, Strongly competitive algorithms for paging with locality of reference. In Proc. 3rd Annual ACM-SIAM Symposium on Discrete Algorithms (1992) 228–236.
B. Kalyanasundaram and K. Pruhs, Speed is as Powerful as Clairvoyance. IEEE Symposium on Foundations of Computer Science (1995) 214–221.
Karlin, A.R., Manasse, M.S., Rudolph, L. and Sleator, D.D., Competitive Snoopy Caching. Algorithmica 3 (1988) 79–119. CrossRef
Karp, R., On-line algorithms versus off-line algorithms: how much is it worth to know the future? Proc. IFIP 12th World Computer Congress 1 (1992) 416–429.
E. Koutsoupias and C.H. Papadimitriou, Beyond competitive analysis. In Proc. 34th Annual Symposium on Foundations of Computer Science (1994) 394–400.
Lotker, Z. and Patt-Shamir, B., Nearly optimal FIFO buffer management for DiffServ. PODC 2002 (2002) 134–143.
M.M. Manasse, L.A. McGeoch and D.D. Sleator, Competitive Algorithms for Online Problems. In Proc. 20th Annual Symposium on the Theory of Computing (1988) 322–333.
C.A. Philips, C. Stein, E. Torng and J. Wein, Optimal time-critical scheduling via resource augmentation. In Proc. 29th Annual ACM Symposium on the Theory of Computing (1997) 140–149.
U.M. O'Reilly and N. Santoro, The expressiveness of silence: tight bounds for synchronous communication of information using bits and silence. In Proc. 18th International Workshop on Graph-Theoretic Concepts in Computer Science (1992) 321–332.
P. Raghavan, A statistical adversary for on-line algorithms. In On-Line Algorithms, DIMACS Series in Discrete Mathematics and Theoretical Computer Science (1991) 79–83.
Robbins, H., Remark, A of Stirling's Formula. Amer. Math. Month. 62 (1955) 26–29. CrossRef
Sleator, D.D. and Tarjan, R.E., Amortized efficiency of update and paging rules. Commun. ACM 28 (1985) 202–208. CrossRef
Torng, E., Unified Analysis, A of Paging and Caching. Algorithmica 20 (1998) 175–200. CrossRef
Young, N., On-line paging against adversially biased random inputs. J. Algorithms 37 (2000) 218–235. CrossRef
Young, N., The k-server dual and loose competitiveness for paging. Algorithmica 11 (1994) 525–541. CrossRef