pp 1–35 | Cite as

Stochastic Dominance and the Bijective Ratio of Online Algorithms

  • Spyros Angelopoulos
  • Marc P. RenaultEmail author
  • Pascal Schweitzer


Stochastic dominance is a technique for evaluating the performance of online algorithms that provides an intuitive, yet powerful stochastic order between the compared algorithms. When there is a uniform distribution over the request sequences, this technique reduces to bijective analysis. These methods have been applied in problems such as paging, list update, bin colouring, routing in array mesh networks, and in connection with Bloom filters, and have often provided a clear separation between algorithms whose performance varies significantly in practice. Despite their appealing properties, the above techniques are quite stringent, in that a relation between online algorithms may be either too difficult to establish analytically, or worse, may not even exist. In this paper, we propose remedies to these shortcomings. Our objective is to make all online algorithms amenable to the techniques of stochastic dominance and bijective analysis. First, we establish sufficient conditions that allow us to prove the bijective optimality of a certain class of algorithms for a wide range of problems; we demonstrate this approach in the context of well-studied online problems such as weighted paging, reordering buffer management, and 2-server on the circle. Second, to account for situations in which two algorithms are incomparable or there is no clear optimum, we introduce the bijective ratio as a natural extension of (exact) bijective analysis. Our definition readily generalizes to stochastic dominance. This makes it possible to compare two arbitrary online algorithms for an arbitrary online problem. In addition, the bijective ratio is a generalization of the Max/Max ratio (due to Ben-David and Borodin), and allows for the incorporation of other useful techniques such as amortized analysis. We demonstrate the applicability of the bijective ratio to one of the fundamental online problems, namely the continuous k-server problem on metrics such as the line, the circle, and the star. Among other results, we show that the greedy algorithm attains bijective ratios of O(k) across these metrics.


Online algorithms k-server problem Performance measures for online algorithms Bijective analysis 



  1. 1.
    Adamaszek, A., Czumaj, A., Englert, M., Räcke, H.: Almost tight bounds for reordering buffer management. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pp. 607–616 (2011)Google Scholar
  2. 2.
    Anagnostopoulos, A., Dombry, C., Guillotin-Plantard, N., Kontoyiannis, I., Upfal, E.: Stochastic analysis of the k-server problem on the circle. In: Proceedings of the 21st International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA), pp. 21–34 (2010)Google Scholar
  3. 3.
    Angelopoulos, S., Dorrigiv, R., López-Ortiz, A.: List update with locality of reference. In: Proceedings of the 8th Latin American Theoretical Informatics Symposium (LATIN), pp. 399–410 (2008)Google Scholar
  4. 4.
    Angelopoulos, S., Dorrigiv, R., López-Ortiz, A.: On the separation and equivalence of paging strategies and other online algorithms. Algorithmica 81(3), 1152–1179 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Angelopoulos, S., Schweitzer, P.: Paging and list update under bijective analysis. J. ACM 60(2), 7:1–7:18 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Avigdor-Elgrabli, N., Rabani, Y.: An optimal randomized online algorithm for reordering buffer management. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 1–10 (2013)Google Scholar
  7. 7.
    Avigdor-Elgrabli, N., Rabani, Y.: An improved competitive algorithm for reordering buffer management. ACM Trans. Algorithms 11(4), 35:1–35:15 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bansal, N., Buchbinder, N., Naor, J.: A primal-dual randomized algorithm for weighted paging. J. ACM 59(4), 19:1–19:24 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bartal, Y., Koutsoupias, E.: On the competitive ratio of the work function algorithm for the k-server problem. Theor. Comput. Sci. 324(2–3), 337–345 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Baumgartner, A., Manger, R., Hocenski, Z.: Work function algorithm with a moving window for solving the on-line k-server problem. J. Comput. Inf. Technol. 15(4), 325–330 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ben-David, S., Borodin, A.: A new measure for the study of on-line algorithms. Algorithmica 11(1), 73–91 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  13. 13.
    Borodin, A., Nielsen, M.N., Rackoff, C.: (Incremental) priority algorithms. Algorithmica 37(4), 295–326 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Boyar, J., Ehmsen, M.R., Larsen, K.S.: A theoretical comparison of LRU and LRU-K. Acta Inform. 47(7–8), 359–374 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Boyar, J., Favrholdt, L., Larsen, K.: Relative worst-order analysis: a survey. In: Böckenhauer, H.J., Komm, D., Unger, W. (eds.) Adventures Between Lower Bounds and Higher Altitudes. LNCS, vol. 11011, pp. 216–230. Springer, Berlin (2018)Google Scholar
  16. 16.
    Boyar, J., Favrholdt, L.M., Larsen, K.S.: The relative worst-order ratio applied to paging. J. Comput. Syst. Sci. 73(5), 818–843 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Boyar, J., Irani, S., Larsen, K.S.: A comparison of performance measures for online algorithms. Algorithmica 72(4), 969–994 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Boyar, J., Larsen, K.S., Nielsen, M.N.: The accommodating function: a generalization of the competitive ratio. SIAM J. Comput. 31(1), 233–258 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Calderbank, A.R., Coffman Jr., E.G., Flatto, L.: Sequencing problems in two-server systems. Math. Oper. Res. 10(4), 585–598 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Calderbank, A.R., Coffman Jr., E.G., Flatto, L.: Sequencing two servers on a sphere. Commun. Stat. Stoch. Models 1(1), 17–28 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chrobak, M., Karloff, H.J., Payne, T.H., Vishwanathan, S.: New results on server problems. SIAM J. Discrete Math. 4(2), 172–181 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chrobak, M., Larmore, L.L.: An optimal on-line algorithm for k-servers on trees. SIAM J. Comput. 20(1), 144–148 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Chrobak, M., Larmore, L.L.: The server problem and on-line games. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 7, 11–64 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Dorrigiv, R., López-Ortiz, A.: A survey of performance measures for on-line algorithms. SIGACT News 36(3), 67–81 (2005)CrossRefGoogle Scholar
  25. 25.
    Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Stochastic analyses for online combinatorial optimization problems. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 942–951 (2008)Google Scholar
  26. 26.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45(9), 1563–1581 (1966)zbMATHCrossRefGoogle Scholar
  27. 27.
    Hiller, B., Vredeveld, T.: Probabilistic analysis of online bin coloring algorithms via stochastic comparison. In: Proceedings of the 16th Annual European Symposium on Algorithms (ESA), pp. 528–539 (2008)Google Scholar
  28. 28.
    Hiller, B., Vredeveld, T.: Simple optimality proofs for least recently used in the presence of locality of reference. Technical report, Maastricht University of Business and Economics (2009)Google Scholar
  29. 29.
    Hiller, B., Vredeveld, T.: Probabilistic alternatives for competitive analysis. Comput. Sci. R&D 27(3), 189–196 (2012)Google Scholar
  30. 30.
    Hofri, M.: Should the two-headed disk be greedy?—Yes, it should. Inf. Process. Lett. 16(2), 83–85 (1983)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Keane, M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kenyon, C.: Best-fit bin-packing with random order. In: Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 359–364 (1996)Google Scholar
  33. 33.
    Koutsoupias, E.: The k-server problem. Comput. Sci. Rev. 3(2), 105–118 (2009)zbMATHCrossRefGoogle Scholar
  34. 34.
    Koutsoupias, E., Papadimitriou, C.: Beyond competitive analysis. SIAM J. Comput. 30, 300–317 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Koutsoupias, E., Papadimitriou, C.H.: On the k-server conjecture. J. ACM 42(5), 971–983 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Lumetta, S.S., Mitzenmacher, M.: Using the power of two choices to improve bloom filters. Internet Math. 4(1), 17–33 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Manasse, M.S., McGeoch, L.A., Sleator, D.D.: Competitive algorithms for on-line problems. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computation (STOC), pp. 322–333 (1988)Google Scholar
  38. 38.
    Mitzenmacher, M.: Bounds on the greedy routing algorithm for array networks. J. Comput. Syst. Sci. 53(3), 317–327 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Hoboken (2002)zbMATHGoogle Scholar
  40. 40.
    Räcke, H., Sohler, C., Westermann, M.: Online scheduling for sorting buffers. In: Proceedings of the 10th Annual European Symposium on Algorithms (ESA), pp. 820–832 (2002)CrossRefGoogle Scholar
  41. 41.
    Raghavan, P., Snir, M.: Memory versus randomization in on-line algorithms. IBM J. Res. Dev. 38(6), 683–708 (1994)CrossRefGoogle Scholar
  42. 42.
    Rudec, T., Baumgartner, A., Manger, R.: Measuring true performance of the work function algorithm for solving the on-line k-server problem. J. Comput. Inf. Technol. 18(4), 361–367 (2010)zbMATHCrossRefGoogle Scholar
  43. 43.
    Rudec, T., Baumgartner, A., Manger, R.: A fast work function algorithm for solving the k-server problem. Cent. Eur. J. Oper. Res. 21(1), 187–205 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Seshadri, S., Rotem, D.: The two headed disk: stochastic dominance of the greedy policy. Inf. Process. Lett. 57(5), 273–277 (1996)zbMATHCrossRefGoogle Scholar
  45. 45.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, Cambridge (1994)zbMATHGoogle Scholar
  46. 46.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28, 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Wolfstetter, E.: Topics in Microeconomics. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  48. 48.
    Young, N.E.: On-line caching as cache size varies. In: Proceedings of the 2nd Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms (SODA), pp. 241–250 (1991)Google Scholar
  49. 49.
    Young, N.E.: The \(k\)-server dual and loose competitiveness for paging. Algorithmica 11(6), 525–541 (1994)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Young, N.E.: Bounding the diffuse adversary. In: Proceedings of the 9th Annual ACM-SIAM symposium on Discrete Algorithms (SODA), pp. 420–425 (1998)Google Scholar
  51. 51.
    Young, N.E.: On-line paging against adversarially biased random inputs. J. Algorithms 37(1), 218–235 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Young, N.E.: On-line file caching. Algorithmica 33(3), 371–383 (2002)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6, LIP6ParisFrance
  2. 2.Computer Sciences DepartmentUniversity of Wisconsin – MadisonMadisonUSA
  3. 3.TU KaiserslauternKaiserslauternGermany

Personalised recommendations