Friedrich, Tobias; Gairing, Martin; Sauerwald, Thomas Quasirandom Load Balancing. Symposium on Discrete Algorithms (SODA) 2010: 1620-1629
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 randomized 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, the randomized rounding approach of Friedrich and Sauerwald [Proceedings of the 41st Annual ACM Symposium on Theory of Computing, 2009, pp. 121-130] can deviate up to \(\Omega(\text{polylog(n))\), and the deterministic algorithm of Rabani, Sinclair, and Wanka [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, 1998, pp. 694-705] 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 on the hypercube as well, our algorithm has a smaller deviation from the idealized process than the previous algorithms. To prove these results, we derive several combinatorial and probabilistic results that we believe to be of independent interest. In particular, we show that first-passage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions.
Bradonjic, Milan; Elsässer, Robert; Friedrich, Tobias; Sauerwald, Thomas; Stauffer, Alexandre Efficient Broadcast on Random Geometric Graphs. Symposium on Discrete Algorithms (SODA) 2010: 1412-1421
A Random Geometric Graph (RGG) in two dimensions is constructed by distributing \(n\) nodes independently and uniformly at random in \([0, \sqrt n]^2\) and creating edges between every pair of nodes having Euclidean distance at most \(r\), for some prescribed \(r\). We analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that with probability \(1 - O(n^{-1})\) this algorithm informs every node in the largest connected component of an RGG within \(O(\sqrt n / r + \log n)\) rounds. This holds for any value of \(r\) larger than the critical value for the emergence of a connected component with \(\Omega(n)\) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in \([0, \sqrt n]^2\), the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is \(\Theta(\sqrt n / r\)), which surprisingly has been an open problem so far.