Tight Bounds for PowerPoint Presentation, PPT - DocSlides

Tight Bounds for

Parallel Randomized Load BalancingChristoph Lenzen and Roger Wattenhofer

What is Load Balancing?

work sharing

low-congestion

routing

optimizing

storage

utilization

hashing

An Example of Parallel Load Balancing

fully connected network small & equal bandwidths (one message/round)n nodes need to send/receive up to n messagesminimize number of rounds

An Example of Parallel Load Balancing

fully connected network small & equal bandwidths (one message/round)n nodes need to send/receive up to n messagesminimize number of rounds

An Example of Parallel Load Balancing

fully connected network small & equal bandwidths (one message/round)n nodes need to send/receive up to n messagesminimize number of rounds

?

?

?

but...

distribute n balls into n binsreplace knowledge by randomization!n instances (one for each receiver)

Abstraction: Parallel Balls-into-Bins

=

=

Naive Approach: Fire-and-Forget

throw balls uniformly independently at random (u.i.r.)max. load with high probability (w.h.p.)

The Power of Two Choices (e.g. Azar et al., SIAM J. of Comp.‘99)

inspect two bins and decide take least loadedmax. load w.h.p.d choices: w.h.p. possible...but not parallel!

Vöcking,

JACM‘92

The Parallel Power of Two Choices

strongest upper bound: max. load in r roundstight 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

An Adaptive Algorithm

contact one bin every bin accepts one ball≈ n/e balls remain (w.h.p.)contact 2<e bins< n/e2 balls remain (w.h.p.)contact k<e2 bins, and so on

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 2tolerates message loss & faulty bins

log*

x

x

011224316465,5365≈1020,000

Optimality?

for symmetric algorithms:many balls in symmetric trees for roundsballs cannot contact many bins w/o incurring messagesif balls in such a tree terminate root gets expected load many such trees => max. load w.h.p.

root

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

Exploiting Asymmetry/Bin ID‘s

subsets of binscontact random “leader” bin“leader” bins distribute balls in their subsetcan place balls right awaycontinue with previous algorithmmax. load 3 in rounds

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#1

#3

#2

How to Use Messages

balls “coordinate” constant fraction of binseach ball contacts bins balls find coordinated binscoordinators assign balls to “their” binsproceed with symmetric algorithm

Summary

optimal symmetric solutionmax. load 2, mess., roundsconstant-time if:global enumeration of bins messages max. load

...hey, What Happened to the Original Problem?!

can be solved in roundscan be used to sort keys in rounds

Patt-Shamir and Teplitsky, PODC‘11

Thank You!

Questions orComments?