The document discusses how asymmetry helps load balancing by presenting the Always-Go-Left algorithm. It summarizes that: 1) The Always-Go-Left algorithm improves upon the uniform greedy algorithm by partitioning bins into groups and placing balls into bins with the fewest balls using an asymmetric tie-breaking rule. 2) The maximum load achieved by Always-Go-Left is related to Fibonacci numbers and is significantly less than the uniform greedy algorithm. 3) The combination of partitioning bins and an unfair tie-breaking rule is crucial to reducing maximum load, as partitioning alone or an asymmetric rule alone does not provide the same benefits.

1. How Asymmetry Helps Load
Balancing
Berthold Voecking
Speaker: Wenkai Dai
Tutor: Dr.Tobias Friedrich &
Dr.Thomas Sauerwald

2. Introduction

3. 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

4. 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

5. 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
•

6. 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

7. 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].

8. 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

9. Always-Go-Left Algorithm Example
Always-Go-Left when there are several locations with the same max load

10. 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

11. Analysis of Always-Go-Left Algorithm
• Max load is related to the Fibonacci numbers
• Define , is golden ration
•
• In general, and
•

12. 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

13. 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

14. 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

15. 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

16. 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.

17. 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.

18. 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

19. 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.

20. 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

21. 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.

22. The following proof is on blackboard

23. Experiment Result

24. 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

25. Mensa Time and Thanks

26. Questions