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**1 - 2**of**2**### Self-Stabilizing Repeated Balls-into-Bins

"... Abstract We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary way. Then, in every subsequent round, one ball is chosen according to some fixed strategy (random, FIFO, etc) from each non-empty bin, and ..."

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Abstract We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary way. Then, in every subsequent round, one ball is chosen according to some fixed strategy (random, FIFO, etc) from each non-empty bin, and re-assigned to one of the n bins uniformly at random. This process corresponds to a non-reversible Markov chain and our aim is to study its self-stabilization properties with respect to the maximum (bin) load and some related performance measures. We define a configuration (i.e., a state) legitimate if its maximum load is Oplog nq. We first prove that, starting from any legitimate configuration, the process will only take on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). Further we prove that, starting from any configuration, the process converges to a legitimate configuration in linear time, w.h.p. This implies that the process is self-stabilizing w.h.p. and, moreover, that every ball traverses all bins in Opn log 2 nq rounds, w.h.p. The latter result can also be interpreted as an almost tight bound on the cover time for the problem of parallel resource assignment in the complete graph.

### Stabilizing Consensus with Many Opinions

, 2015

"... We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must con-verge toward a consensus state in which all, or almost all nodes, hold the same opinion. M ..."

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We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must con-verge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k 6 nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to o( n) nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ|> 2.