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By Ivanyi A. (ed.)

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Extra resources for Algorithms of informatics. Applications

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This construction strategy yields the desired logarithmic multiplicative overhead. 3-1 Show that the simplied algorithm has Ω(n2 ) message complexity, by appropriately assigning identiers to processors on a ring of size n, and by determining how to delay processors and messages. 3-2 Show that the algorithm Bully has Ω(n lg n) message complexity. 4. Fault-tolerant consensus The algorithms presented so far are based on the assumption that the system on which they run is reliable. Here we present selected algorithms for unreliable distributed systems, where the active (or correct) processors need to coordinate their activities based on common decisions.

That is, it possible to postpone or advance the instants when instructions are executed in such a way, that each instruction x that has been assigned a value tx of the logical clock, is executed exactly at the instant tx of the global clock, and that the resulting execution is a valid one, in the sense that it can actually occur when the algorithm is run with the modied delays. The Logical-Clock algorithm assigns logical time to each instruction. Each processor has a local variable called counter.

The values of logical time form a total order, where pairs are compared lexicographically. This logical time is also called Lamport time. We dene tx to be a quotient counter + 1/(id + 1), which is an equivalent way to represent the pair. 21 For any execution, logical time satises three conditions: (i) if an instruction x is performed by a processor before an instruction y is performed by the same processor, then the logical time of x is strictly smaller than that of y , (ii) any two distinct instructions of any two processors get assigned dierent logical times, (iii) if instruction x sends a message and instruction y receives this message, then the logical time of x is strictly smaller than that of y .

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