1. Christoph Lenzen, STOC 2011

2. What is Load Balancing? work sharing low-congestion routing optimizing storage utilization hashing

3. Christoph Lenzen, STOC 2011 An Example of Parallel Load Balancing fully connected network small & equal bandwidths (one message/round) n nodes need to send/receive up to n messages minimize number of rounds

4. Christoph Lenzen, STOC 2011 An Example of Parallel Load Balancing fully connected network small & equal bandwidths (one message/round) n nodes need to send/receive up to n messages minimize number of rounds

5. Christoph Lenzen, STOC 2011 An Example of Parallel Load Balancing fully connected network small & equal bandwidths (one message/round) n nodes need to send/receive up to n messages minimize number of rounds ? ? ? but...

6. Christoph Lenzen, STOC 2011 distribute n balls into n bins replace knowledge by randomization! n instances (one for each receiver) Abstraction: Parallel Balls-into-Bins = =

7. Christoph Lenzen, STOC 2011 Naive Approach: Fire-and-Forget throw balls uniformly independently at random (u.i.r.) max. load with high probability (w.h.p.)

8. Christoph Lenzen, STOC 2011 The Power of Two Choices (e.g. Azar et al., SIAM J. of Comp.‘99) inspect two bins and decide take least loaded max. load w.h.p. d choices: w.h.p. possible...but not parallel! Vöcking, JACM‘92

9. Christoph Lenzen, STOC 2011 The Parallel Power of Two Choices strongest upper bound: max. load in r rounds tight for constant r...and certain algorithms: –non-adaptive (fix bins to contact in advance) –symmetric (all choices uniform) Stemann, SPAA‘96 Adler et al., STOC‘95

10. Christoph Lenzen, STOC 2011 An Adaptive Algorithm contact one bin every bin accepts one ball ≈n/e balls remain (w.h.p.) contact 2<e bins <n/e 2 balls remain (w.h.p.) contact k<e 2 bins, and so on

11. Christoph Lenzen, STOC 2011 The Power of the Tower termination in rounds cap # contacted bins at total messages w.h.p. in exp. (for each ball and bin) can enforce max. load 2 tolerates message loss & faulty bins log* xx 01 12 24 316 465,536 5≈10 20,000

12. Christoph Lenzen, STOC 2011 Optimality? for symmetric algorithms: –many balls in symmetric trees for rounds –balls cannot contact many bins w/o incurring messages –if balls in such a tree terminate root gets expected load –many such trees => max. load w.h.p. root

13. Christoph Lenzen, STOC 2011 No Faster Solution Possible, unless... bin loads of are accepted, bins have "identities" known to all balls messages are used place s balls at once

14. Christoph Lenzen, STOC 2011 Exploiting Asymmetry/Bin ID‘s subsets of bins contact random “leader” bin “leader” bins distribute balls in their subset can place balls right away continue with previous algorithm max. load 3 in rounds ? ? ? #1 #3 #2

15. Christoph Lenzen, STOC 2011 How to Use Messages balls “coordinate” constant fraction of bins each ball contacts bins balls find coordinated bins coordinators assign balls to “their” bins proceed with symmetric algorithm

16. Christoph Lenzen, STOC 2011 Summary optimal symmetric solution –max. load 2, mess., rounds constant-time if: –global enumeration of bins – messages – max. load

17. Christoph Lenzen, STOC 2011...hey, What Happened to the Original Problem?! can be solved in rounds can be used to sort keys in rounds Patt-Shamir and Teplitsky, PODC‘11

18. Christoph Lenzen, STOC 2011