Algorithms of informatics, vol.2.. applications (2007)(ISBN by Ivanyi A. (ed.)

By Ivanyi A. (ed.)

Ivanyi A. (ed.) Algorithms of informatics, vol.2.. purposes (2007)(ISBN 9638759623)

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A graph that matters, at a given point in an execution, is the one induced by the processors that have not crashed till this step of the execution. 27 Let f < n be a pair of positive integers. ) imposes monotonicity on the required subgraphs. Observe that graph P (R) is connected, even if R is not, since its diameter is nite. The following result shows that graphs satisfying property R(n, f ) can be constructed, and that their degree is not too large. 28 For each f < n, there exists a graph G(n, f ) satisfying property R(n, f ).

So entry ki + 1 is not majorised by K , and since all subsequent entries, including the one for instruction x, can have only larger coordinates, the entries are not majorised by K either. But, x happens before instruction number kj , so entry kj can only have lager coordinates than respective coordinates of the entry corresponding to x, and so V Tj [kj ] cannot be majorised by K either. This contradicts the assumption that V Tj [kj ] is majorised by K . Therefore, (k1 , . . , kn ) must be a consistent cut.

The proof, for any algorithm, is based on constructing certain executions of the algorithm on rings of size n/2. Then two rings of size n/2 are pasted together in such a way that the constructed executions on the smaller rings are combined, and Θ(n) additional messages are received. 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.

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