Tight Bounds for PowerPoint Presentation

Tight Bounds for PowerPoint Presentation

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Parallel Randomized Load Balancing. Christoph Lenzen and Roger Wattenhofer. What is Load Balancing?. work sharing. low-congestion. routing. optimizing. storage. utilization. hashing. An Example of Parallel Load Balancing. ID: 589098

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Slide1

Tight Bounds for

Parallel Randomized Load BalancingChristoph Lenzen and Roger Wattenhofer

Slide2

What is Load Balancing?

work sharing

low-congestion

routing

optimizing

storage

utilization

hashing

Slide3

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

Slide4

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

Slide5

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...

Slide6

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

Abstraction: Parallel Balls-into-Bins

=

=

Slide7

Naive Approach: Fire-and-Forget

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

Slide8

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

Slide9

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

Slide10

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

Slide11

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

Slide12

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

Slide13

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

Slide14

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

?

?

?

#1

#3

#2

Slide15

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

Slide16

Summary

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

Slide17

...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

Slide18

Thank You!

Questions orComments?


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