About randomised distributed graph colouring and graph partition algorithms

AbstractWe present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O(logn) with probability 1-o(n-1). Each message containing 1 bit, its bit complexity per channel is O(logn).From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree Δ and size n that uses at most Δ+1 colours and halts in time O(logn) with probability 1-o(n-1).We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O(logn) with probability 1-o(n-1). We study some parameters such as the number, the size and the radius of trees of the spanning forest

The Universe of Symmetry Breaking Tasks

Processes in a concurrent system need to coordinate using a shared memory or a message-passing subsystem in order to solve agreement tasks such as, for example, consensus or set agreement. However, often coordination is needed to “break the symmetry” of processes that are initially in the same state, for example, to get exclusive access to a shared resource, to get distinct names or to elect a leader. This paper introduces and studies the family of generalized symmetry breaking (GSB) tasks, that includes election, renaming and many other symmetry breaking tasks. Differently from agreement tasks, a GSB task is “inputless”, in the sense that processes do not propose values; the task specifies only the symmetry breaking requirement, independently of the system's initial state (where processes differ only on their identifiers). Among many various characterizing the family of GSB tasks, it is shown that (non adaptive) perfect renaming is universal for all GSB tasks

Bit Complexity of Breaking and Achieving Symmetry in Chains and Rings (Extended Abstract)

Ye m Dinitz Shlomo Moran Sergio Rajsbaum Abstract We consider a failure-free, asynchronous message passing network, with n processors arranged on a ring or a chain. The processes are identically programmed but have distinct identities, taken from f1; : : : ; Mg. We investigate the communication costs of three well studied tasks: Consensus, Leader, and MaxF ( nding the maximum identity, a restricted version of Leader). We show that in both chain and ring topologies, somewhat surprisingly, the message complexities of all three tasks are the same. Hence, we suggest as a ner measure of complexity the number of bits transmitted, BitC(). We show that in chains, w.r.t. this measure, Consensus is easier than Leader, which is easier than MaxF. More speci cally, we prove several new lower bounds (and some simple upper bounds) that imply the following results: For the two processors case, BitC(Consensus) = 2 and BitC(Leader) = BitC(MaxF) = 2 log 2 M O(1). For a chain, BitC(Consensus) = (n), and BitC(MaxF) = (n log M ). When the length is even BitC(Leader) = (n), while if the length is odd BitC(Leader) = (n + log M )