Do-all computing in distributed systems by C Georgiou; Alex Allister Shvartsman

By C Georgiou; Alex Allister Shvartsman

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There may be multiple coordinators in subsequent phases. The number of processors that assume the coordinator role is determined by the martingale principle: if none of the expected coordinators survive through the entire phase, then the number of coordinators for the next phase is doubled. Whenever at least one coordinator survives a given phase, the number of coordinators for the next phase is reduced to one. If at least one processor acts as a coordinator during a phase and it completes the phase without failing, we say that the phase is attended , the phase is unattended otherwise.

When failures of coordinators disrupt the progress of the computation, the number of coordinators is increased; when the failures subside, a single coordinator is appointed. The algorithm is tolerant of f crash-failures (f < p). It has total-work (available processor steps) complexity1 S = O((n + p log p/ log log p) log f ) and message complexity M = O(n + p log p/ log log p + f p). ) The algorithm assumes that the communication is reliable. If a processor sends a message to another operational processor and when the message arrives at the destination the processor is still operational, then the message is 1 The expression “log f ” stands for 1 when f < 2 and log2 f otherwise.

P ∂ = c/fi (log p − log fi )2 , c/log The first derivative over fi is ∂fi fi ∂2 p and its second derivative is c/log = 2c/fi2 (log p − log fi )3 − 2 ∂fi fi c/fi2 (log p − log fi )2 . Observe that the second derivative is negative in the domain considered (assuming p > 16). Hence the first derivative is decreasing (with fi ). In this case, given any two fi , fj where fi > fj , the adversarial pattern obtained by replacing fi with fi − ǫ and fj by fj + ǫ (where ǫ < (fi − fj )/2) results in increased work.

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