Advertisement

Chains-into-Bins Processes

  • Tuğkan Batu
  • Petra Berenbrink
  • Colin Cooper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

Abstract

The study of balls-into-bins processes or occupancy problems has a long history. These processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins process is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper, we analyze the maximum load for the chains-into-bins problem, which is defined as follows. There are n bins, and m objects to be allocated. Each object consists of balls connected into a chain of length ℓ, so that there are m ℓ balls in total. We assume the chains cannot be broken, and that the balls in one chain have to be allocated to ℓ consecutive bins. We allow each chain d independent and uniformly random bin choices for its starting position. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for d ≥ 2 and m·ℓ= O(n), the maximum load is ((ln ln m)/ln d) + O(1) with probability \(1-\tilde O(1/m^{d-1})\).

Keywords

Balls-into-bins processes chains-into-bins processes random processes offline assignment 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced Allocations. SIAM J. Computing 29, 180–200 (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berenbrink, P., Czumaj, A., Steger, A., Vöcking, B.: Balanced Allocations: The Heavily Loaded Case. SIAM J. Computing 35, 1350–1385 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Englert, M.: Chains of Length Two into Bins. Manuscript, University of AachenGoogle Scholar
  4. 4.
    Fekete, S., Köhler, E., Teich, J.: Optimal FPGA module placement with temporal precedence constraints. In: Proc. of the Conference on Design, Automation and Test in Europe (DATE 2001), pp. 658–667 (2001)Google Scholar
  5. 5.
    Kenthapadi, K., Panigrahy, R.: Balanced allocation on graphs. In: Proc. of 17th Annual Symposium on Discrete Algorithms (SODA 2006), pp. 434–443 (2006)Google Scholar
  6. 6.
    Knuth, D.E.: Sorting and Searching, 2nd edn. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998)Google Scholar
  7. 7.
    Mitzenmacher, M., Richa, A.W., Sitaraman, R.: The Power of Two Random Choices: A Survey of Techniques and Results. In: Handbook of Randomized Computing (2000)Google Scholar
  8. 8.
    Mitzenmacher, M., Prabhakar, B., Shah, D.: Load Balancing with Memory. In: Proc. of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pp. 799–808 (2002)Google Scholar
  9. 9.
    Patterson, D.A., Gibson, G.A., Katz, R.H.: A Case for Redundant Arrays of Inexpensive Disks (RAID). In: Proc. of SIGMOD International Conference on Management of Data, pp. 109–116 (1988)Google Scholar
  10. 10.
    Sanders, P., Vöcking, B.: Tail Bounds And Expectations For Random Arc Allocation And Applications. Combinatorics, Probability & Computing 12(3) (2003)Google Scholar
  11. 11.
    Steiger, C., Walder, H., Platzner, M.: Operating Systems for Reconfigurable Embedded Platforms: Online Scheduling of Real-Time Tasks. IEEE Trans. Computers 53(11), 1393–1407 (2004)CrossRefGoogle Scholar
  12. 12.
    Talwar, K., Wieder, U.: Balanced allocations: the weighted case. In: Proc. of the 39th Symposium on Theory of Computing (STOC), pp. 256–265 (2007)Google Scholar
  13. 13.
    Vöcking, B.: How Asymmetry Helps Load Balancing. J. ACM 50(4), 568–589 (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tuğkan Batu
    • 1
  • Petra Berenbrink
    • 2
  • Colin Cooper
    • 3
  1. 1.Department of MathematicsLondon School of EconomicsLondonUK
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  3. 3.Department of Computer ScienceKing’s College LondonLondonUK

Personalised recommendations