How Much Information about the Future Is Needed?

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of 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 between the algorithm and 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 tight bounds in both communication modes.

[1]  Cynthia A. Phillips,et al.  Optimal Time-Critical Scheduling via Resource Augmentation (Extended Abstract) , 1997, Symposium on the Theory of Computing.

[2]  Yossi Azar,et al.  Algorithms - ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11-13, 2006, Proceedings , 2006, ESA.

[3]  Richard M. Karp,et al.  On-Line Algorithms Versus Off-Line Algorithms: How Much is it Worth to Know the Future? , 1992, IFIP Congress.

[4]  Prabhakar Raghavan,et al.  A Statistical Adversary for On-line Algorithms , 1991, On-Line Algorithms.

[5]  Sandy Irani,et al.  Strongly competitive algorithms for paging with locality of reference , 1992, SODA '92.

[6]  Sandy Irani,et al.  Online computation , 1996 .

[7]  Susanne Albers,et al.  Online algorithms: a survey , 2003, Math. Program..

[8]  Anna R. Karlin,et al.  Competitive snoopy caching , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[9]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .

[10]  Joan Boyar,et al.  The relative worst order ratio for online algorithms , 2007, TALG.

[11]  Neal E. Young,et al.  Thek-server dual and loose competitiveness for paging , 1994, Algorithmica.

[12]  Christos H. Papadimitriou,et al.  Beyond competitive analysis [on-line algorithms] , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[13]  Herbert S. Wilf,et al.  Algorithms and Complexity , 2010, Lecture Notes in Computer Science.

[14]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[15]  Robert E. Tarjan,et al.  Amortized efficiency of list update and paging rules , 1985, CACM.

[16]  Allan Borodin,et al.  A new measure for the study of on-line algorithms , 2005, Algorithmica.

[17]  Bala Kalyanasundaram,et al.  Speed is as powerful as clairvoyance , 2000, JACM.

[18]  Andrzej Pelc,et al.  Oracle size: a new measure of difficulty for communication tasks , 2006, PODC '06.

[19]  Una-May O'Reilly,et al.  The Expressiveness of Silence: Tight Bounds for Synchronous Communication of Information Using Bits and Silence , 1992, WG.

[20]  Joan Boyar,et al.  The Relative Worst Order Ratio for On-Line Algorithms , 2003, CIAC.

[21]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[22]  Boaz Patt-Shamir,et al.  Nearly optimal FIFO buffer management for DiffServ , 2002, PODC '02.

[23]  Amos Fiat,et al.  Competitive Paging Algorithms , 1991, J. Algorithms.

[24]  Cynthia A. Phillips,et al.  Optimal Time-Critical Scheduling via Resource Augmentation , 1997, STOC '97.

[25]  Matthias Englert,et al.  Lower and Upper Bounds on FIFO Buffer Management in QoS Switches , 2006, ESA.

[26]  Allan Borodin,et al.  Competitive paging with locality of reference , 1991, STOC '91.

[27]  Eric Torng.  A Unified Analysis of Paging and Caching , 1998, Algorithmica.

[28]  Lyle A. McGeoch,et al.  Competitive algorithms for on-line problems , 1988, STOC '88.

[29]  Andrzej Pelc,et al.  Distributed Computing with Advice: Information Sensitivity of Graph Coloring , 2007, ICALP.

[30]  Laszlo A. Belady,et al.  A Study of Replacement Algorithms for Virtual-Storage Computer , 1966, IBM Syst. J..

[31]  Bala Kalyanasundaram,et al.  Speed is as powerful as clairvoyance [scheduling problems] , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.