Download - Seminar load balance
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How Asymmetry Helps Load Balancing
Berthold Voecking
Speaker: Wenkai Dai
Tutor: Dr.Tobias Friedrich & Dr.Thomas Sauerwald
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Introduction
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Formal task
• Given n balls and n bins
• Randomly placing sequentially n balls into n bins
• Goal: minimize the maximum number of balls in the same bin
• balls are indistinguishable and global knowledge of previously assigned balls is not available
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Basic idea
• Each ball is placed into a bin chosen independently and uniformly at random from the set of all bins
• The expected maximum load of
• Give the proof later
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Better Idea
• Azar et al.[1994, 1999] suggest uniform greedy algorithm
• For each ball, choose d>= 2 locations independently and uniformly at random from the set of bins
• Place the ball into the bin with the fewest ball
• Maximum load of only
• d=2 yields great improvement over one choice, larger d just a smaller factor better than 2 choice
•
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Three types of selection
– (1) uniform and independent
– (2) nonuniform and independent
• A nonuniform algorithm may choose the first location from the bins 0 to n/2-1 and the second location from the bins n/2 to n -1
– (3) nonuniform and dependent
• The second choice may depend on the first choice, if the first location is i then the second location is
–Goal is to improve the uniform greedy algorithm
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Three Classes of Algorithm• [n]={0…n-1} denote the set of bins, 3 classes of algorithms
depending on how the sample location is chosen from the probability space [n]^d
• Class 1: Uniform and independent. Each of the d locations of a ball is chosen uniformly and independently at random from [n].
• Class 2: (Possibly) nonuniform and independent. For 1<=i<=d, the ith location of a ball is chosen independently at random from [n] as defined by a probability distribution Di : [n][0,1].
• Class 3: (Possibly) nonuniform and (possibly) dependent. The d locations of a ball are chosen at random from the set [n]^d as defined by a probability distribution D : [n]^d[0,1].
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Always-Go-Left Algorithm
• Introduce a multi-choice algorithm of class2 giving smaller maximum load than uniform greedy algorithm in class1
• Partitions the bins into d groups of almost equal size
• For each ball, choose one location from each group
• The i-th location of each ball is chose uniformly and independently at random from the i-th group
• Insert ball into a bin with minimum load among d locations
• Tie-breaking by asymmetric Always-Go-Left rule
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Always-Go-Left Algorithm Example
Always-Go-Left when there are several locations with the same max load
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d-ary Fibonacci numbers
• d-ary Fibonacci numbers
• For k<= 0, Fd(k)=0, Fd(1) =1, and for k>=1,
• if d =2, it’s standard fibonacci sequence
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Analysis of Always-Go-Left Algorithm
• Max load is related to the Fibonacci numbers
• Define , is golden ration
•
• In general, and
•
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Analysis of Always-Go-Left Algorithm
• We have so that
• Even for d =2, there is a significant improvement
• AGL yields maximum load of instead of
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Analysis of Always-Go-Left Algorithm
• Uniform greedy scheme achieves the best load balancing among all algorithms of class 1
• This results holds regardless of the used tie-breaking mechanism, showing that the tie-breaking mechanism is irrelevant in the uniform case
• Partitioning the bins and using a fair tie-breaking doesn’t reduce the number of balls in the fullest bin below
• The combination of partitioning and unfair tie breaking is very crucial for our result
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Is The Further Improvement Possible ?
• Whether other kind of choices for the d locations or other schemes for deciding which of these locations receives the ball can improve the result
• Negative answer by this theorem
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Conclude
• By Theorems 1 and 2, apart from some additive constants, the AGL algorithm achieves the best possible maximum load among all the sequential multi-choice algorithms, namely
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Generalization
• Interesting to assuming more balls than bins or even an infinite sequence of insertions and deletions
• An oblivious adversary specifies a possibly infinite sequence of insertions and deletions of balls
• All the requests on-line, the sequence of insertions and deletions is presented one by one without knowing future requests
• Time t denote time at which request is presented but not yet served. A ball is said to exist at time t if it is stored in one of the bins at this time.
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On-line Model conclusion
• The uniform greedy get a maximum load of
• Always-Go-Left get
• Multiple choice processes are fundamentally different from the single-choice variant because the multiple-choice does not increase with the number of balls but depending only on n and d.
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Proof of the upper bounds
• Use a witness tree to upper-bound the probability for the event a bin contains too many balls
• This witness tree is a rooted tree the nodes of which represent balls whose randomly chosen locations are arranged in a bad fashion
• Simplifying assumptions
• All the events are stochastically independent
• At most n balls exist at any time, h=1
• Finally it will remove all these assumption
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Witness Tree
• A bad event when the maximum load exceeds some threshold value, implies the “ activation of a witness tree”
• the probability for the existence of an activated witness tree upper-bounds the probability that a bad event occurs.
• It will show the activation of a witness tree is unlikely. Consequently, the bad event that is witnessed by this structure is unlikely as well.
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Symmetric Witness Tree
• A symmetric witness tree of order L is a complete d-ary tree of height L with d^L leaf nodes
• Each node v represents a ball, but some ball may be represented by several nodes
• Not every assignment of balls to nodes gives a witness tree
• Each non-root node v with parent node u has to exist at the insertion time of u’s ball
• Each node of the witness tree describes an event that may occur or not depending on the random choices for the locations of the balls
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Symmetric Witness Tree
• The edge event are defined in terms of the alternative locations of balls instead of their final resting place
• If all of edges and all its leaf node are activated then we say this tree is activated.
• existence of a bin with more than L+3 balls implies the existence of an activated witness tree of order L.
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The following proof is on blackboard
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Experiment Result
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Summary
• Always-Go-Left process yields a smaller maximum load than the uniform greedy process
• The both of asymmetry and the partitioning of the set of bins are crucial
• Using an asymmetric tie-breaking mechanism without partitioning doesn’t help
• Also using partitioning but with fair tie-breaking doesn’t help either
• Multiple choice is totally different from the single choice
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Mensa Time and Thanks
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Questions