Efficient Load-Balancing through Distributed Token Dropping

Abstract

We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for stable orientations and more generally for locally optimal semi-matchings. The prior work by Czygrinow et al. (DISC 2012) finds a stable orientation in (Δ 5 ) rounds in graphs of maximum degree Δ, while we improve it to (Δ 4 ) and also prove a lower bound of Ω(Δ). For the more general problem of locally optimal semi-matchings, the prior upper bound is ( 5 ) and our new algorithm runs in ( • 4 ) rounds, which is an improvement for = ( ); here and are the maximum degrees of customers and servers, respectively.

Distributed Edge Coloring in Time Polylogarithmic in $Δ$

“…Our goal is to color each edge either red or blue such that every red edge has at most (1+ε)∆ adjacent red edges and every blue edge has at most (1+ε)∆ adjacent blue edges. To achieve this, we generalize an idea that was presented in [14]. There, the authors show how to efficiently solve a problem known as locally optimal semi-matching [19] and in particular a special case of this problem, a so-called stable edge orientation of a graph.…”

“…In [14], the authors introduce a tool that they call the token dropping game and which is used in particular to compute stable edge orientations. The token dropping game works as follows.…”

“…Every time a token passes through an edge, the edge is deleted. In [14], it is shown that the token dropping game can be solved in time O(L • ∆ 2 ) if the directed graph of the game has no cycles and the longest directed path is of length L. It is shown that this algorithm can be used to compute a stable edge orientation and thus a perfect defective 2-coloring in ∆-regular 2-colored bipartite graphs in O(∆ 4 ) rounds.…”

On the Complexity of Load Balancing in Dynamic Networks

“…A different line of work studies load balancing in client-server networks: here, servers communicate with the clients, in order to assign clients to servers in a balanced way. While the scenario in our work is modeled by an arbitrary communication network, load balancing in client-server networks is modeled using bipartite graphs, and admits very different algorithmic results, both in the centralized setting [9,19,22,25,28] and in the distributed setting [1,8,12,24].…”

“…Consider a matching-based algorithm, and an adversary that works as follows: after a pair connects, the adversary orders the pair in ascending order, swapping the nodes if necessary. For example, if two adjacent node had weights (3,8) and the algorithm chose to re-balance them to (7,4), then the adversary will swap the nodes to get (4,7). In terms of the corresponding loads arrays, we will have a t [i] = 3, a t [i + 1] = 8, and after the balance and swap, a t+1 [i] = 4, a t+1 [i + 1] = 7.…”