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 surprisingly closely approximates the idealized process (where the tokens are divisible) on important network topologies. On d-dimensional 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 (2009) can deviate up to Omega(polylog(n)) and the deterministic algorithm of Rabani, Sinclair and Wanka (1998) has a deviation of Omega(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 hypercube our algorithm has a smaller deviation from the idealized process than the previous algorithms.Comment: 25 page

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Discrete Load Balancing in Heterogeneous Networks with a Focus on Second-Order Diffusion

In this paper we consider a wide class of discrete diffusion load balancing algorithms. The problem is defined as follows. We are given an interconnection network and a number of load items, which are arbitrarily distributed among the nodes of the network. The goal is to redistribute the load in iterative discrete steps such that at the end each node has (almost) the same number of items. In diffusion load balancing nodes are only allowed to balance their load with their direct neighbors. We show three main results. Firstly, we present a general framework for randomly rounding the flow generated by continuous diffusion schemes over the edges of a graph in order to obtain corresponding discrete schemes. Compared to the results of Rabani, Sinclair, and Wanka, FOCS'98, which are only valid w.r.t. the class of homogeneous first order schemes, our framework can be used to analyze a larger class of diffusion algorithms, such as algorithms for heterogeneous networks and second order schemes. Secondly, we bound the deviation between randomized second order schemes and their continuous counterparts. Finally, we provide a bound for the minimum initial load in a network that is sufficient to prevent the occurrence of negative load at a node during the execution of second order diffusion schemes. Our theoretical results are complemented with extensive simulations on different graph classes. We show empirically that second order schemes, which are usually much faster than first order schemes, will not balance the load completely on a number of networks within reasonable time. However, the maximum load difference at the end seems to be bounded by a constant value, which can be further decreased if first order scheme is applied once this value is achieved by second order scheme.Comment: Full version of paper submitted to ICDCS 201

Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies

We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. 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 well-understood 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 non-linear behavior due to its rounding errors, which makes this analysis much harder than in the continuous case. We investigate several randomized protocols for different communication models in the discrete case. As our main result, we prove that for any regular network in the matching model, all nodes have the same load up to an additive constant in (asymptotically) the same number of rounds as required in the continuous case. This generalizes and tightens the previous best result, which only holds for expander graphs, and demonstrates that there is almost no difference between the discrete and continuous cases. Our results also provide a positive answer to the question of how well discrete load balancing can be approximated by (continuous) Markov chains, which has been posed by many researchers.Comment: 74 pages, 4 figure

Conflict-free star-access in parallel memory systems

We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates. i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs-tori and hypercubes-that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing. (C) 2006 Elsevier Inc. All rights reserved

The Impact of Randomisation in Load Balancing and Random Walks

The real world is full of uncertainties. Classical analyses usually favour deterministic cases, which in practice can be too restricted. Hence it motivates us to add in randomness to make models similar to practical situations. In this thesis, we mainly study two network problems taken from the distributed computing world: iterative load balancing and random walks. An interesting observation is that the problems we study, though not quite related regarding their real world applications, can be linked by the same mathematical toolkit: Markov chain theory. These problems have been heavily studied in the literature. However, their assumptions are mostly \emph{deterministic}, which causes less flexibility and generality to the real world settings. The novelty of this thesis is that we add randomness in these problems in order to observe worst cases vs. average cases (load balancing) and static cases vs. dynamic cases (random walks). For iterative load balancing, the randomness is added on the number of tasks over the entire network. Previous works often assumed worst case initial loads, which may be wasteful sometimes. Hence we relax this condition and assume the loads are drawn from different probability distributions. In particular, we no longer assume the initial loads are chosen by an adversary. Instead, we assume the initial loads on each processor are sampled from independent and identically distributed (i.i.d.) probability distributions. We then study the same problems as in classical settings, i.e., the time needed for the load balancing process to reach a sufficiently small discrepancy. Our main result implies that under such a regime, the time required to balance a network can be much faster. An insightful observation is that the load discrepancy is proportional to the term $t^{-1/4}$ where $t$ is the time used to run the protocol. This implies two main improvements compared with previous works: first, when the initial discrepancy is the same, our regime can reach small discrepancy faster; second, we have established a connection between the time and the discrepancy while previous analyses do not have. For random walks, the randomness is added on the network topologies. This means at each time step (considering discrete times), the underlying network can change randomly. In particular, we want the graph ``evolves'' instead of changing arbitrarily. To model the graph changing process, we adopt a model commonly used in the literature, i.e., the edge-Markovian model. If an edge does not exist between the two nodes, then it will appear in the next step with probability $p$, and if it does then in the next step it will disappear with probability $q$. This model can simulate real world scenarios such as adding friends with each other in social networks or a disruption between two remotely connected computers. Our main contributions regarding random walks include the following results. First, we divided the edge-Markovian graph model into different regimes in a parameterised way. This provides an intuitive path to similar analyses of dynamic graph models. Dynamic models are often hard to analyse in the field because of its complicated nature. We present a possible strategy to reach some feasible solutions by using parameters ($p, q$ above) to control the process. Second, we again analyse the random walk behaviours on such models. We have found that under certain regimes, the random walk still shows similar behaviours especially its mixing nature as in static settings. For the other regimes, we also show either weaker mixing or no mixing results

Dynamic Averaging Load Balancing on Arbitrary Graphs

In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing