International audienceWe establish a bound for the classic PUSH-PULL rumor spreading protocol on general graphs, in terms of the vertex expansion of the graph. We show that O(log^2 (n) / α) rounds suffice with high probability to spread a rumor from any single node to all n nodes, in any graph with vertex expansion at least α. This bound matches a known lower bound, and settles the natural question on the relationship between rumor spreading and vertex expansion asked by Chierichetti, Lattanzi, and Panconesi (SODA 2010). Further, some of the arguments used in the proof may be of independent interest, as they give new insights, for example, on how to choose a small set of nodes in which to plant the rumor initially, to guarantee fast rumor spreading
To cite this version:George Giakkoupis, Philipp Woelfel. On the time and space complexity of randomized test-and-set. PODC -31st Annual ACM SIGACT-SIGOPS Symposium ABSTRACTWe study the time and space complexity of randomized TestAnd-Set (TAS) implementations from atomic read/write registers in asynchronous shared memory models with n processes. We present an adaptive TAS algorithm with an expected (individual) step complexity of O(log * k), for contention k, against the oblivious adversary, improving a previous (non-adaptive) upper bound of O(log log n) (Alistarh and Aspnes, 2011). We also present a modified version of the adaptive RatRace TAS algorithm (Alistarh et al., 2010), which improves the space complexity from O(n 3 ) to O(n), while maintaining logarithmic expected step complexity against the adaptive adversary. We complement this upper bound with an Ω(log n) lower bound on the space complexity of any TAS algorithm that has the nondeterministic solo-termination property (which is a weaker progress condition than wait-freedom). No non-trivial lower bounds on the space requirements of TAS were known prior to this work.
We study the relation between the rate at which rumors spread throughout a graph and the vertex expansion of the graph. We consider the standard rumor spreading protocol where every node chooses a random neighbor in each round and the two nodes exchange the rumors they know. For any n-node graph with vertex expansion α, we show that this protocol spreads a rumor from a single node to all other nodes in O(α −1 log 2 n √ log n) rounds with high probability. Further, we construct graphs for which Ω(α −1 log 2 n) rounds are needed. Our results complement a long series of works that relate rumor spreading to edgebased notions of expansion, resolving one of the most natural questions on the connection between rumor spreading and expansion. 2 An event E(n) holds with high probability (w.h.p.), if Pr E(n) = 1 − O(n −c) for an arbitrary constant c > 0.
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