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 Subramanian October 9, 2018 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Balls in Bins Model 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 Example: Random Allocation 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 log n (1 + o (1)) log log n with high probability as n → ∞ . Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Example: Power of d choices 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 Example: Power of d choices Theorem Let m = n. The maximum load under power-of-d-choices allocation policy is log log n + Θ(1) with high probability as n → ∞ . log d Much better than random allocation! Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Motivation 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 Motivation 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 log 6 n …… Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Motivation 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 Alternative ways of sampling 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¨ oking (2003): d bins are sampled from d disjoint groups of 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 = log Θ(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 Alternative ways of sampling 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¨ oking (2003): d bins are sampled from d disjoint groups of 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 = log Θ(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.’s Model 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 Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model 2 7 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Alon et al.’s Model Theorem (Alon et al.) If the graph G is an expander graph whose girth is greater than 10 log k − 1 log n, then the maximum load after inserting n balls into log n n bins is (1 + o (1)) 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 Motivation Alon et al. raised the following open question ... Let W 1 and W 2 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 of W 1 and W 2 , and pick the least loaded one. Does the log n maximal load decrease from Θ( 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 Main Results We analyzed two models for allocating n balls into n bins. For both models: Bins are associated with vertices of a k -regular graph G n . G n is fixed throughout the process. W 1 [ j ] , W 2 [ j ] , · · · , W d [ j ] are candidate bins for the j -th ball. The j -th ball is allocated to the least loaded bin of W 1 [ j ] , W 2 [ j ] , · · · , W d [ 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 I Model 1 W 1 [ j ] , W 2 [ j ] , · · · , W d [ j ] are independent non-backtracking random walks. Except when either the random walkers’ paths “intersect”. or a reset has not occurred for T steps Then W 1 [ j ] , W 2 [ j ] , · · · , W d [ j ] are reset to independent uniform random positions . Assumption 1 The graph G n 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 Model I n = 10 bins 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Model I n = 10 bins 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Model I n = 10 bins 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Model I n = 10 bins 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Model I n = 10 bins 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
Introduction Related Work Main Results Discussion References Model I n = 10 bins intersected! 2 d = 2 walkers/choices 7 G = 3-regular graph T = 4 1 3 8 6 9 10 5 4 9 10 1 2 3 4 5 6 7 8 Dengwang Tang Balanced Allocation with Random Walk
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