Measuring the problem-relevant information in input
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 585-613.

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.

DOI : https://doi.org/10.1051/ita/2009012
Classification : 68Q25,  68W40,  68Q10,  68Q17
Mots clés : online algorithms, communication complexity, advice complexity, paging
@article{ITA_2009__43_3_585_0,
author = {Dobrev, Stefan and Kr\'alovi\v{c}, Rastislav and Pardubsk\'a, Dana},
title = {Measuring the problem-relevant information in input},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {585--613},
publisher = {EDP-Sciences},
volume = {43},
number = {3},
year = {2009},
doi = {10.1051/ita/2009012},
zbl = {1176.68089},
mrnumber = {2541132},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita/2009012/}
}
TY  - JOUR
AU  - Dobrev, Stefan
AU  - Královič, Rastislav
AU  - Pardubská, Dana
TI  - Measuring the problem-relevant information in input
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2009
DA  - 2009///
SP  - 585
EP  - 613
VL  - 43
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2009012/
UR  - https://zbmath.org/?q=an%3A1176.68089
UR  - https://www.ams.org/mathscinet-getitem?mr=2541132
UR  - https://doi.org/10.1051/ita/2009012
DO  - 10.1051/ita/2009012
LA  - en
ID  - ITA_2009__43_3_585_0
ER  - 
Dobrev, Stefan; Královič, Rastislav; Pardubská, Dana. Measuring the problem-relevant information in input. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 585-613. doi : 10.1051/ita/2009012. http://www.numdam.org/articles/10.1051/ita/2009012/

[1] D. Achlioptas, M. Chrobak and J. Noga, Competitive analysis of randomized paging algorithms. Theoret. Comput. Sci. 234 (2000) 203-218. | MR 1745075 | Zbl 0944.68194

[2] S. Albers, On the influence of lookahead in competitive paging algorithms. Algorithmica 18 (1997) 283-305. | MR 1464707 | Zbl 0879.68046

[3] S. Albers, Online algorithms: A survey. Math. Prog. 97 (2003) 3-26. | MR 2004390 | Zbl 1035.68136

[4] L.A. Belady, A study of replacement algorithms for virtual storage computers. IBM Systems Journal 5 (1966) 78-101.

[5] S. Ben-David and A. Borodin, A new measure for the study of on-line algorithms. Algorithmica 11 (1994) 73-91. | MR 1247989 | Zbl 0782.68052

[6] A. Borodin and R. El-Yaniv, Online Computation and Competitive Analysis. Cambridge University Press (1998). | MR 1617778 | Zbl 0931.68015

[7] 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. | Zbl 0800.68484

[8] 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. | MR 2387702 | Zbl 1129.68432

[9] J. Boyar, K.S. Larsen and M.N. Nielsen, The accommodating function: a generalization of the competitive ratio. SIAM J. Comput. 31 (2001) 233-258. | MR 1857398 | Zbl 0992.68069

[10] 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. | MR 2050620 | Zbl 1032.68988

[11] 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. | MR 2347156 | Zbl 1131.68317

[12] A. Fiat, R.M. Karp, M. Luby, L.A. Mcgeoch, D.D. Sleator and N.E. Young, Competitive paging algorithms. J. Algorithms 12 (1991) 685-699. | Zbl 0753.68018

[13] 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). | MR 2424686 | Zbl 1171.68859

[14] 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. | MR 2298164 | Zbl 1132.68507

[15] 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.

[16] R.L. Graham, Bounds for certain multiprocessing anomalies. Bell Systems Technical Journal 45 (1966) 1563-1581. | Zbl 0168.40703

[17] 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.

[18] 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. | MR 1173897 | Zbl 0834.68049

[19] B. Kalyanasundaram and K. Pruhs, Speed is as Powerful as Clairvoyance. IEEE Symposium on Foundations of Computer Science (1995) 214-221. | Zbl 0938.68545

[20] A.R. Karlin, M.S. Manasse, L. Rudolph and D.D. Sleator, Competitive Snoopy Caching. Algorithmica 3 (1988) 79-119. | MR 925479 | Zbl 0645.68034

[21] R. Karp, 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.

[22] E. Koutsoupias and C.H. Papadimitriou, Beyond competitive analysis. In Proc. 34th Annual Symposium on Foundations of Computer Science (1994) 394-400.

[23] Z. Lotker and B. Patt-Shamir, Nearly optimal FIFO buffer management for DiffServ. PODC 2002 (2002) 134-143.

[24] 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. | Zbl 0796.68042

[25] 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. | Zbl 0962.68010

[26] 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. | MR 1244147 | Zbl 0806.90041

[27] 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. | Zbl 0800.68483

[28] H. Robbins, A Remark of Stirling's Formula. Amer. Math. Month. 62 (1955) 26-29. | MR 69328 | Zbl 0068.05404

[29] D.D. Sleator and R.E. Tarjan, Amortized efficiency of update and paging rules. Commun. ACM 28 (1985) 202-208. | MR 777385

[30] E. Torng, A Unified Analysis of Paging and Caching. Algorithmica 20 (1998) 175-200. | MR 1484535 | Zbl 0895.68058

[31] N. Young, On-line paging against adversially biased random inputs. J. Algorithms 37 (2000) 218-235. | MR 1783254 | Zbl 0971.68073

[32] N. Young, The $k$-server dual and loose competitiveness for paging. Algorithmica 11 (1994) 525-541. | MR 1272524

Cité par Sources :