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15
Rumour spreading and graph conductance
 IN PROCEEDINGS OF THE 21ST ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2010
"... We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws ..."
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Cited by 35 (2 self)
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We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws a connection between rumour spreading and the spectral sparsification procedure of Spielman and Teng [23].
The Cover Time of Deterministic Random Walks
, 2010
"... The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this “deterministic random walk ” covers all vertices (or all edges). We present general techniques to der ..."
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Cited by 11 (2 self)
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The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this “deterministic random walk ” covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.
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
Distributed Selfish Load Balancing on Networks
"... We study distributed load balancing in networks with selfish agents. In the simplest model considered here, there are n identical machines represented by vertices in a network and m ≫ n selfish agents that unilaterally decide to move from one vetex to another if this improves their experienced load. ..."
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Cited by 6 (2 self)
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We study distributed load balancing in networks with selfish agents. In the simplest model considered here, there are n identical machines represented by vertices in a network and m ≫ n selfish agents that unilaterally decide to move from one vetex to another if this improves their experienced load. We present several protocols for concurrent migration that satisfy desirable properties such as being based only on local information and computation and the absence of global coordination or cooperation of agents. Our main contribution is to show rapid convergence of the resulting migration process to states that satisfy different stability or balance criteria. In particular, the convergence time to a Nash equilibrium is only logarithmic in m and polynomial in n, where the polynomial depends on the graph structure. Using a slight modification with neutral moves, a perfectly balanced state can be reached after additional time polynomial in n. Inaddition, we show reduced convergence times to approximate Nash equilibria. Finally, we extend our results to networks of machines with different speeds or to agents that have different weights and show similar results for convergence to approximate and exact Nash equilibria. 1
Randomized Diffusion for Indivisible Loads
"... We present a new randomized diffusionbased algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and ..."
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Cited by 4 (4 self)
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We present a new randomized diffusionbased algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement). In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constantdegree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds compared to the algorithm of Rabani, Sinclair, and Wanka [14]. For hypercubes we obtain a polynomial improvement. In contrast to previous papers, our algorithm is vertexbased and not edgebased. This means excess tokens are assigned to vertices instead to edges, and the vertex reallocates all of its excess tokens by itself. This approach avoids nodes having “negative loads ” (like in [8, 10]), but causes additional dependencies for the analysis.
Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies
, 1201
"... We consider the problem of balancing load items (tokens) on networks. Starting with an arbitrary load distribution, we allow in each round nodes to exchange tokens with their neighbors. Thegoalisto achieveadistribution whereall nodeshavenearlythe samenumber of tokens. For the continuous case where t ..."
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Cited by 3 (1 self)
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We consider the problem of balancing load items (tokens) on networks. Starting with an arbitrary load distribution, we allow in each round nodes to exchange tokens with their neighbors. Thegoalisto achieveadistribution whereall nodeshavenearlythe samenumber of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains whose convergence is rather wellunderstood in terms of their spectral gap. However, since for many applications load items cannot be divided arbitrarily, we focus on the discrete case where the load is composed of indivisible tokens. Unfortunately, this discretization entails a nonlinear behavior due to its rounding errors, which makes the analysis much harder than in the continuous case. Therefore, it has been a major open problem to understand the limitations of discrete load balancing and its relation to the continuous case. We investigate several randomized protocols for different communication models in the discrete case. Ourresults demonstratethat there is almost no deviationbetween the discrete
Smoothed Analysis of Balancing Networks
, 2009
"... In a load balancing network each processor has an initial collection of unitsize jobs, tokens, and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule cruc ..."
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Cited by 2 (1 self)
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In a load balancing network each processor has an initial collection of unitsize jobs, tokens, and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially effects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformlyrandom assignment rule, and the arbitrary one (in the spirit of smoothedanalysis) by starting from an arbitrary assignment of balancer directions, then flipping each assignment with probability α independently. For a large class of balancing networks our result implies that after O(log n) rounds the discrepancy is whp O((1/2−α) log n+log log n). This matches and generalizes the known bounds for α = 0 and α = 1/2.
Tight Bounds for Randomized Load Balancing . . .
, 2012
"... We consider the problem of balancing load items (tokens) on networks. Starting with an arbitrary load distribution, we allow in each round nodes to exchange tokens with their neighbors. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous ca ..."
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We consider the problem of balancing load items (tokens) on networks. Starting with an arbitrary load distribution, we allow in each round nodes to exchange tokens with their neighbors. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains whose convergence is fairly wellunderstood in terms of their spectral gap. However, in many applications load items cannot be divided arbitrarily and we need to deal with the discrete case where the load is composed of indivisible tokens. This discretization entails a nonlinear behavior due to its rounding errors, which makes the analysis much harder than in the continuous case. Therefore, it has been a major open problem to understand the limitations of discrete load balancing and its relation to the continuous case. We investigate several randomized protocols for different communication models in the discrete case. Our results demonstrate that there is almost no difference between the discrete
Load Balancing in Distributed Systems using Diffusion Technique
"... The purpose of load balancing algorithm is to distribute the excess load from heavily loaded nodes to underloaded nodes. A new dynamic load balancing algorithm is proposed based on diffusion approach (DDD) for homogeneous systems where the processing capacities of all nodes in the system are equal. ..."
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The purpose of load balancing algorithm is to distribute the excess load from heavily loaded nodes to underloaded nodes. A new dynamic load balancing algorithm is proposed based on diffusion approach (DDD) for homogeneous systems where the processing capacities of all nodes in the system are equal. The proposed algorithm works iteratively to balance the load among the nodes in a system. The dynamic distributed diffusion algorithm has been developed for coarse and large granularity applications, where the load shall be treated as an Integer quantity. The functioning of the proposed algorithm is demonstrated by using a random graph & simulation has shown the proposed algorithm performs better in terms of time taken to balance the load, minimizing the load variance among the nodes and maximizing the throughput.