# Online Search with Time-Varying Price Bounds

- 163 Downloads
- 18 Citations

## Abstract

Online search is a basic online problem. The fact that its optimal deterministic/randomized solutions are given by simple formulas (however with difficult analysis) makes the problem attractive as a target to which other practical online problems can be transformed to find optimal solutions. However, since the upper/lower bounds of prices in available models are constant, natural online problems in which these bounds vary with time do not fit in the available models.

We present two new models where the bounds of prices are not constant but vary with time in certain ways. The first model, where the upper and lower bounds of (logarithmic) prices have decay speed, arises from a problem in concurrent data structures, namely to maximize the (appropriately defined) freshness of data in concurrent objects. For this model we present an optimal deterministic algorithm with competitive ratio
\(\sqrt{D}\)
, where *D* is the known duration of the game, and a nearly-optimal randomized algorithm with competitive ratio
\(\frac{\ln D}{1+\ln2-\frac{2}{D}}\)
. We also prove that the lower bound of competitive ratios of randomized algorithms is asymptotically
\(\frac{\ln D}{4}\)
.

The second model is inspired by the fact that some applications do not utilize the decay speed of the lower bound of prices in the first model. In the second model, only the upper bound decreases *arbitrarily* with time and the lower bound is constant. Clearly, the lower bound of competitive ratios proved for the first model holds also against the stronger adversary in the second model. For the second model, we present an optimal randomized algorithm. Our numerical experiments on the freshness problem show that this new algorithm achieves much better/smaller competitive ratios than previous algorithms do, for instance 2.25 versus 3.77 for *D*=128.

## Keywords

Search algorithms Online algorithms Competitive analysis Game theory## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
- 2.Chen, G.H., Kao, M.Y., Lyuu, Y.D., Wong, H.K.: Optimal buy-and-hold strategies for financial markets with bounded daily returns. In: Proc. of ACM Symp. on Theory of Computing (STOC), pp. 119–128 (1999) Google Scholar
- 3.Cho, J., Garcia-Molina, H.: Synchronizing a database to improve freshness. In: SIGMOD ’00: Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, pp. 117–128 (2000) Google Scholar
- 4.Damaschke, P., Ha, P.H., Tsigas, P.: Competitive freshness algorithms for wait-free objects. In: Proceedings of the European Conference on Parallel Computing (Euro-Par). Lecture Notes in Computer Science, vol. 4128, pp. 811–820. Springer, Berlin (2006) Google Scholar
- 5.El-Yaniv, R.: Competitive solutions for online financial problems. ACM Comput. Surv.
**30**(1), 28–69 (1998) CrossRefMathSciNetGoogle Scholar - 6.El-Yaniv, R., Fiat, A., Karp, R., Turpin, G.: Competitive analysis of financial games. In: Proc. of the 33rd Symp. on Foundations of Computer Science, pp. 327–333 (1992) Google Scholar
- 7.El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Optimal search and one-way trading online algorithms. Algorithmica
**30**(1), 101–139 (2001) MATHCrossRefMathSciNetGoogle Scholar - 8.Ha, P.H., Papatriantafilou, M., Tsigas, P.: Efficient self-tuning spin-locks using competitive analysis. J. Syst. Softw.
**80**(7), 1077–1090 (2007) CrossRefGoogle Scholar - 9.Ha, P.H., Papatriantafilou, M., Tsigas, P.: Self-tuning reactive diffracting trees. J. Parallel Distrib. Comput.
**67**(6), 674–694 (2007) MATHCrossRefGoogle Scholar - 10.Ha, P.H., Tsigas, P.: Reactive multi-word synchronization for multiprocessors. J. Instr. Lev. Parallelism
**6**(Special issue) (2004). http://www.jilp.org/vol6/v6paper3.pdf - 11.Herlihy, M.: Wait-free synchronization. ACM Trans. Program. Syst.
**11**(1), 124–149 (1991) CrossRefGoogle Scholar - 12.Herlihy, M., Wing, J.: Linearizability: a correctness condition for concurrent objects. ACM Trans. Program. Lang. Syst.
**12**(3), 463–492 (1990) CrossRefGoogle Scholar - 13.Kang, K.D., Son, S.H., Stankovic, J.A.: Managing deadline miss ratio and sensor data freshness in real-time databases. IEEE Trans. Knowl. Data Eng.
**16**(10), 1200–1216 (2004) CrossRefGoogle Scholar - 14.Labrinidis, A., Roussopoulos, N.: Exploring the tradeoff between performance and data freshness in database-driven web servers. VLDB J.
**13**(3), 240–255 (2004) CrossRefGoogle Scholar - 15.Li, W.S., Po, O., Hsiung, W.P., Candan, K.S., Agrawal, D.: Engineering and hosting adaptive freshness-sensitive web applications on data centers. In: WWW ’03: Proceedings of the 12th International Conference on World Wide Web, pp. 587–598 (2003) Google Scholar
- 16.Ling, Y., Chen, W.: Measuring cache freshness by additive age. SIGOPS Oper. Syst. Rev.
**38**(3), 12–17 (2004) CrossRefGoogle Scholar - 17.Pacitti, E., Simon, E.: Update propagation strategies to improve freshness in lazy master replicated databases. VLDB J.
**8**(3–4), 305–318 (2000) Google Scholar