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Local Divergence of Markov Chains and the Analysis of Iterative LoadBalancing Schemes
 IN PROCEEDINGS OF THE 39TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS ’98
, 1998
"... We develop a general technique for the quantitative analysis of iterative distributed load balancing schemes. We illustrate the technique by studying two simple, intuitively appealing models that are prevalent in the literature: the diffusive paradigm, and periodic balancing circuits (or the dimensi ..."
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Cited by 61 (2 self)
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We develop a general technique for the quantitative analysis of iterative distributed load balancing schemes. We illustrate the technique by studying two simple, intuitively appealing models that are prevalent in the literature: the diffusive paradigm, and periodic balancing circuits (or the dimension exchange paradigm). It is well known that such load balancing schemes can be roughly modeled by Markov chains, but also that this approximation can be quite inaccurate. Our main contribution is an effective way of characterizing the deviation between the actual loads and the distribution generated by a related Markov chain, in terms of a natural quantity which we call the local divergence. We apply this technique to obtain bounds on the number of rounds required to achieve coarse balancing in general networks, cycles and meshes in these models. For balancing circuits, we also present bounds for the stronger requirement of perfect balancing, or counting.
Quantum simulations of classical random walks and undirected graph connectivity
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Efficiency and nash equilibria in a scrip system for p2p networks
 IN ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2006
"... A model of providing service in a P2P network is analyzed. It is shown that by adding a scrip system, a mechanism that admits a reasonable Nash equilibrium that reduces free riding can be obtained. The effect of varying the total amount of money (scrip) in the system on efficiency (i.e., social welf ..."
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Cited by 25 (5 self)
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A model of providing service in a P2P network is analyzed. It is shown that by adding a scrip system, a mechanism that admits a reasonable Nash equilibrium that reduces free riding can be obtained. The effect of varying the total amount of money (scrip) in the system on efficiency (i.e., social welfare) is analyzed, and it is shown that by maintaining the appropriate ratio between the total amount of money and the number of agents, efficiency is maximized. The work has implications for many online systems, not only P2P networks but also a wide variety of online forums for which scrip systems are popular, but formal analyses have been lacking.
Mixing Times
 AMS DIMACS SERIES
, 1998
"... The critical issue in the complexity of Markov chain sampling techniques has been "mixing time", the number of steps of the chain needed to reach its stationary distribution. It turns out that there are many ways to define mixing timemore than a dozen are considered herebut they fal ..."
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Cited by 25 (1 self)
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The critical issue in the complexity of Markov chain sampling techniques has been "mixing time", the number of steps of the chain needed to reach its stationary distribution. It turns out that there are many ways to define mixing timemore than a dozen are considered herebut they fall into a small number of classes. The parameters in each class lie within constant multiples of one another, independent of the chain. Furthermore, there are interesting connections between these classes related to time reversal. This work is more in the nature of a long research paper than a survey, with many new results and proofs. Some of the results have appeared in recent articles or were known previously for the important special case of reversible chains.
Nonbacktracking random walks mix faster
, 2006
"... We compute the mixing rate of a nonbacktracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the ..."
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Cited by 24 (6 self)
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We compute the mixing rate of a nonbacktracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a highgirth regular expander on n vertices, then a typical nonbacktracking random walk of length n on G does not visit a vertex more than log n (1 + o(1)) log log n times, and this result is tight. In this sense, the multiset of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times. 1
Nearperfect load balancing by randomized rounding
 In 41st Annual ACM Symposium on Theory of Computing (STOC’09
, 2009
"... We consider and analyze a new algorithm for balancing indivisible loads on a distributed network with n processors. The aim is minimizing the discrepancy between the maximum and minimum load. In every timestep paired processors balance their load as evenly as possible. The direction of the excess t ..."
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Cited by 15 (9 self)
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We consider and analyze a new algorithm for balancing indivisible loads on a distributed network with n processors. The aim is minimizing the discrepancy between the maximum and minimum load. In every timestep paired processors balance their load as evenly as possible. The direction of the excess token is chosen according to a randomized rounding of the participating loads. We prove that in comparison to the corresponding model of Rabani, Sinclair, and Wanka (1998) with arbitrary roundings, the randomization yields an improvement of roughly a square root of the achieved discrepancy in the same number of timesteps on all graphs. For the important case of expanders we can even achieve a constant discrepancy in O(log n(log log n) 3) rounds. This is optimal up to log log nfactors while the best previous algorithms in this setting either require Ω(log 2 n) time or can only achieve a logarithmic discrepancy. This result also demonstrates that with randomized rounding the difference between discrete and continuous load balancing vanishes almost completely.
The Evolution of the Mixing Rate of a Simple Random Walk on the Giant Component of a Random Graph
, 2008
"... In this article we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most O ( √ ln n), proving that the mixin ..."
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Cited by 13 (0 self)
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In this article we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most O ( √ ln n), proving that the mixing time in this case is �((ln n/d) 2) asymptotically almost surely. As the average degree grows these become negligible and it is the diameter of the largest component that takes over, yielding mixing time �(ln n / ln d) a.a.s.. We proved these results during the 2003–04 academic year. Similar results but for constant d were later proved independently by Benjamini et al. in [3].
Quasirandom Load Balancing
"... We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm approximates the idealized p ..."
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Cited by 10 (6 self)
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We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm approximates the idealized process (where the tokens are divisible) on important network topologies surprisingly closely. On ddimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant. In contrast to that, the randomized rounding approach of Friedrich and Sauerwald [8] can deviate up to Ω(polylogn) and the deterministic algorithm of Rabani, Sinclair and Wanka [23] has a deviation of Ω(n 1/d). This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness. We further show that also on the hypercubeour algorithm has a smaller deviation from the idealized process than the previous algorithms. To prove these results, we derive several combinatorial andprobabilistic results thatwe believe to beof independent interest. In particular, we show that firstpassage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions. 1
Reversal of Markov Chains and the Forget Time
, 1996
"... We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to "forget" where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A sec ..."
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Cited by 8 (2 self)
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We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to "forget" where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state. The third is the reset time, the minimum expected time between independent samples from the stationary distribution. Our main results state that the mixing time of a chain is equal to the mixing time of the timereversed chain, while the forget time of a chain is equal to the reset time of the reverse chain. In particular, the forget time and the reset time of a timereversible chain are equal. Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset time. We also derive an explicit formula for the forget time, in terms o...
Quantitative common carotid artery blood flow: prediction of internal carotid artery stenosis
 Magn. Reson. Imaging 4 37–42 Brands P J, Hoeks A P, Hofstra L and Reneman R S
, 1986
"... We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences — α, the total influence on a site, as studied by Dobrushin; α ′ , the total influence of ..."
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Cited by 7 (1 self)
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We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences — α, the total influence on a site, as studied by Dobrushin; α ′ , the total influence of a site, as studied by Dobrushin and Shlosman; and α ′ ′ , the total influence of a site in any given context, which is related to the pathcoupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then randomupdate Glauber dynamics (in which a randomlychosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematicscan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α ′ and α ′ ′ , and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrixbalancing to show that the DobrushinShlosman condition α ′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α ′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heatbath Glauber dynamics for proper qcolourings of a degree ∆ graph G when q ≥ 2∆. 1