Tight Bounds for Parallel Randomized Load Balancing Christoph Lenzen and Roger Wattenhofer
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Christoph Lenzen, STOC 2011 Tight Bounds for Parallel Randomized Load Balancing Christoph Lenzen and Roger Wattenhofer
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Tight Bounds for Parallel Randomized Load Balancing Christoph Lenzen and Roger Wattenhofer. What is Load Balancing?. optimizing storage utilization. work sharing. hashing. low-congestion routing. An Example of Parallel Load Balancing. fully connected network - PowerPoint PPT Presentation
Transcript of Tight Bounds for Parallel Randomized Load Balancing Christoph Lenzen and Roger Wattenhofer
Christoph Lenzen, STOC 2011
Tight Bounds forParallel Randomized Load Balancing
Christoph Lenzen and Roger Wattenhofer
Christoph Lenzen, STOC 2011
What is Load Balancing?
work sharing
low-congestionrouting
optimizingstorage
utilization
hashing
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
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
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...
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
=
=
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.))lnln/(ln nn
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!
)ln(ln n
)/ln(ln dn Vöcking,JACM‘92
)ln/ln(ln dn
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)
rnn
/1lnln/ln
Stemann, SPAA‘96
Adler et al., STOC‘95
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/e2 balls remain (w.h.p.)• contact k<e2 bins, and so on
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
)1(log* On 1)(log*)2(log* xx
log* x x0 1
1 2
2 4
3 16
4 65,536
5 ≈1020,000
nlog)(nO)1(O
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.
)(log*))1(1( no)(n
)1()1(
root
Christoph Lenzen, STOC 2011
No Faster Solution Possible, unless...
• bin loads of are accepted,
• bins have "identities" known to all balls
• messages are used)(n
)1(
place sballs at once
no ))1(1(
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(
no ))1(1(
? ? ?
#1 #3#2
)1(O
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
)1(no ))1(1(
)(n
X XXX)1( )1(
)1(O
Christoph Lenzen, STOC 2011
Summary
• optimal symmetric solution
– max. load 2, mess., rounds
• constant-time if:
– global enumeration of bins
– messages
– max. load
)(nO )1(log* On
)(n
)1(
Christoph Lenzen, STOC 2011
...hey, What Happened to the Original Problem?!
• can be solved in rounds
• can be used to sort keys in rounds
)1(O
2n )1(O
Patt-Shamir and Teplitsky, PODC‘11
Christoph Lenzen, STOC 2011
Thank You!
Questions orComments?
