Randomized diffusion for indivisible loads

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe present a new randomized diffusion-based 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 constant-degree 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 vertex-based and not edge-based. 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.

Parallel rotor walks on finite graphs and applications in discrete load balancing

“…This can lead to negative load on some of the nodes. In contrast, the algorithm of [3] avoids this problem. It calculates a the total number of extra tokens (difference between the total continuous flow forwarded over all edges minus the number of tokens forwarded by the discrete algorithm after rounding down) a nodes should forward to its neighbours and sends these tokens to randomly chosen neighbours (without replacement).…”

“…This yields an imbalance bound of O(n 1/2d ) for tori, and O(d log n) for expanders. For hypercubes, the same technique with a tighter analysis gives a bound of Θ(log 2 n) [3]. In fact, due to the technique used, these results are not only imbalance bounds, but also discrepancy bounds.…”

“…Many papers suggest randomized rounding processes [2,3,11] to convert the continuous load that is transferred over an edge into discrete load. The algorithms in [2,11] round the continuous load on each edge independently from all other edges.…”

“…This gives us discrepancy bounds of O(log 3/2 n) and O(1) for hypercube and r-dimensional torus with r = O(1), respectively, which improve over the best existing bounds of O(log 2 n) and O(n 1/r ). Also, as a result of switching to the load balancing view, we observe that the existing load balancing results can be translated to rotor walk discrepancy bounds not previously noticed in the rotor walk literature.We also use the idea of rotor walks to propose and analyze a randomized rounding discrete load balancing process that achieves the same balancing quality as similar protocols [11,3], but uses fewer number of random bits compared to [3], and avoids the negative load problem of [11]. …”

“…We also use the idea of rotor walks to propose and analyze a randomized rounding discrete load balancing process that achieves the same balancing quality as similar protocols [11,3], but uses fewer number of random bits compared to [3], and avoids the negative load problem of [11].…”