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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Tight Bounds for
Parallel Randomized Load BalancingChristoph Lenzen and Roger Wattenhofer
Slide2What is Load Balancing?
work sharing
low-congestion
routing
optimizing
storage
utilization
hashing
Slide3An 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
Slide4An 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
Slide5An 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...
Slide6distribute n balls into n binsreplace knowledge by randomization!n instances (one for each receiver)
Abstraction: Parallel Balls-into-Bins
=
=
Slide7Naive Approach: Fire-and-Forget
throw balls uniformly independently at random (u.i.r.)max. load with high probability (w.h.p.)
Slide8The 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
Slide9The 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
Slide10An 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
Slide11The 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
Slide12Optimality?
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
Slide13No Faster Solution Possible, unless...
bin loads of are accepted,bins have "identities" known to all balls messages are used
place
s
balls at once
Slide14Exploiting 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
Slide15How to Use Messages
balls “coordinate” constant fraction of binseach ball contacts bins balls find coordinated binscoordinators assign balls to “their” binsproceed with symmetric algorithm
Slide16Summary
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
Slide18Thank You!
Questions orComments?
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