Balanced Allocation with Random Walk Based Sampling Dengwang Tang - - PowerPoint PPT Presentation
Introduction Related Work Main Results Discussion References Balanced Allocation with Random Walk Based Sampling Dengwang Tang Electrical and Computer Engineering Department University of Michigan, Ann Arbor Joint work with Vijay
Introduction Related Work Main Results Discussion References
Dengwang Tang
Electrical and Computer Engineering Department University of Michigan, Ann Arbor
Joint work with Vijay Subramanian
October 9, 2018
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
m balls are placed into n bins sequentially according to some policy The dispatching policy is usually random or partially random The maximum load (i.e. number of balls in the fullest bin) is usually the quantity of interest Applications
Resource Allocation Distributed Hash Tables Load Balancing in Cloud Computing
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Random Allocation Policy:
Each ball is inserted into a bin uniformly at random The choice of bin for each ball is independent
Theorem Let m = n. The maximum load under random allocation policy is (1 + o(1))
log n log log n with high probability as n → ∞.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Power of d choice policy [Azar et al 1999]:
For each ball, sample d bins uniformly and independently Place the ball into the least loaded bin among d sampled bins Ties are broken arbitrarily
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Theorem Let m = n. The maximum load under power-of-d-choices allocation policy is log log n
log d
+ Θ(1) with high probability as n → ∞. Much better than random allocation!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
At each time, uniform random bins are chosen Are there any other process which returns a uniform random bin? Random walk on k-regular graphs.
Stationary distribution: Uniform on all vertices.
What about sampling bins using d independent random walks?
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Randomness (or Entropy) is an precious resource in computer. How many times per ball do you need to throw dice in order to implement power-of-d-choices with n bins? Answer: d log6 n
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
In Computer Science Literature, random walks on certain graphs are utilized to derandomize a randomized algorithm Can we “derandomize” power-of-d-choices policy?
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Alon et al. (2007): d = 1 bin is sampled through a non-backtracking random walk of length n on a high girth expander graph. V¨
bins respectively. Kenthapadi and Panigraphy (2006): d = 2 bins are sampled from a random edge of a high degree regular graph. Pourmiri (2016): d bins are sampled by a random walk starting from a random vertex each time. Godfrey (2008): A random set of d = logΘ(1) n bins is chosen each time, where the random set satisfy some “balancedness” property.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Alon et al. (2007): d = 1 bin is sampled through a non-backtracking random walk of length n on a high girth expander graph. V¨
bins respectively. Kenthapadi and Panigraphy (2006): d = 2 bins are sampled from a random edge of a high degree regular graph. Pourmiri (2016): d bins are sampled by a random walk starting from a random vertex each time. Godfrey (2008): A random set of d = logΘ(1) n bins is chosen each time, where the random set satisfy some “balancedness” property.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Alon et al. (2007) investigated the the maximum load after inserting n balls into n bins based on the location of a non-backtracking random walker
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Theorem (Alon et al.) If the graph G is an expander graph whose girth is greater than 10 logk−1 log n, then the maximum load after inserting n balls into n bins is (1 + o(1))
log n log log n w.h.p. as n → ∞.
Same performance as random allocation But randomness used is reduced!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Alon et al. raised the following open question ... Let W1 and W2 denote two non-backtracking random walks on an expander of high girth, and suppose that in each step we are given a choice between the two current locations
maximal load decrease from Θ(
log n log log n) to Θ(log log n) in this
setting as well? ... We answered this question!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
We analyzed two models for allocating n balls into n bins. For both models:
Bins are associated with vertices of a k-regular graph Gn. Gn is fixed throughout the process. W1[j], W2[j], · · · , Wd[j] are candidate bins for the j-th ball. The j-th ball is allocated to the least loaded bin of W1[j], W2[j], · · · , Wd[j].
We consider the case where k, d are fixed and n → ∞.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Model 1 W1[j], W2[j], · · · , Wd[j] are independent non-backtracking random
either the random walkers’ paths “intersect”.
Then W1[j], W2[j], · · · , Wd[j] are reset to independent uniform random positions. Assumption 1 The graph Gn is such that Pr (randomly initialized random walkers intersect before T) < 0.1
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
intersected!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Reset!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
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n = 10 bins d = 2 walkers/choices G = 3-regular graph T = 4
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Theorem (T., Subramanian) In Model 1, for any T = o(√n), if Assumption 1 holds, then the maximum load does not exceed log log n
log d
+ Θ(1) with high probability as n → ∞. Same performance as power-of-d-choices! But less randomness used!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Comparing the randomness used per reset period:
For power-of-d: (length of reset period)·d log6 n. For Model 1: d log6(kn)+(length of reset period - 1)·d log6(k − 1)
If G is a cycle graph and T = ⌊n1/3⌋, then the number of dice throw per ball ≈ dn−1/3 log6(2n) → 0
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Sufficient Exploration of Bins
Resets are frequent
Decorrelating the load vector and random walker position
Load of sampled bin is equal to that load at last reset. The distribution of the random walker positions is always independent uniform. Relate the distribution of the load of a sampled bin to the empirical distribution of load of bins at the last reset time.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Model 2 (Alon et al.’s Model) W1[j], W2[j], · · · , Wd[j] are independent non-backtracking random walks on G. Assumption 2 girth(Gn) = Ω(logk−1 n) (Gn)n is a sequence of expander graphs.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Definition (Alon et al.) Let (Gn)n be a sequence of k-regular graphs with |Gn| = n. Let k = λ1,n ≥ λ2,n ≥ · · · λn,n be the eigenvalues of Gn. Define λn := max{λ2,n, |λn,n|}. (Gn)n is said to be expander graphs if lim sup
n→∞ λn < k
Explicit Construction of graphs that satisfies Assumption 2 exists! e.g. Lubotzky, Phillips, and Sarnak’s Ramanujan Graph (1987)
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Theorem (T., Subramanian) In Model 2, if Assumption 2 holds, then the maximum load does not exceed log log n
log d
+ Θ(1) with high probability as n → ∞.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Sufficient exploration of bins:
Non-backtracking random walks on expanders has O(log n) mixing time [Alon et al 2011]. Mixing effect replaces resets.
Decorrelating queue length vector and random walker locations
In a high girth graph, “intersections” are unlikely to happen within one mixing interval. The number of visits to a vertex within one mixing interval is bounded by Θ(1) w.p.1 .
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
In load balancing applications, is the graph G costing too much memory? No! If G is a graph with structure, we can compute the neighbors rather than storing them.
For example, LPS Graph is a Cayley graph of PSL(2, q). One can find the neighbors of a given vertex through multiplying 2 × 2 matrices!
Model 1 could cost memory through storing the paths of random walkers. But Model 2 eliminated this cost through a stronger assumption on the graph.
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Why non-backtracking random walk? Because simple random walk doesn’t perform well! Theorem (T., Subramanian) For Model 2, if we replace the non-backtracking random walks with simple random walks, then the maximum load is greater than Θ (log n) w.h.p. as n → ∞. Proof idea: short-term revisits are frequent. It’s even worse than random allocation!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
What if we have a weaker girth assumption? Theorem (T., Subramanian) If Gn is a graph such that each vertex is contained in a cycle of length g = g(n), then the maximum load yielded by Model 2 is larger than Θ( log n
g ) almost surely.
Proof idea: short-term revisits are frequent. If g = o(
log n log log n), Model 2 could perform worse than
power-of-d!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Theorem (T., Subramanian) If (Gn)n is a sequence of expander graph with girth girth(Gn) = Ω(
log n log log n), then the maximum load yielded by Model
2 is larger than Θ(log log n) with high probability. Not necessarily the same performance as power-of-d, but close!
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Random walk based sampling can work for balls-in-bins. However we need to...
ensure that short-term revisits are rare. e.g. non-backtracking random walk on high girth graphs, reset at intersections ensure sufficient exploration of bins. e.g. expander graphs, reset to uniform random positions
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Queuing system setting: low traffic and high traffic Lower bound on maximum load for Model 1 and Model 2 Apply random walk based methods to other randomized algorithms
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Harsha Honappa Yang Xiao National Science Foundation
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Balanced Allocations SIAM Journal on Computing, Vol. 29, No. 1, 180 – 200.
Non-backtracking random walks mix faster Communications in Contemporary Mathematics, Vol. 9, No. 04, 585-603
Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References
Dengwang Tang Balanced Allocation with Random Walk