tight bounds for parallel randomized load balancing christoph lenzen and roger wattenhofer
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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?
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