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

By Ivanyi A. (ed.)

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The algorithm sends O(e) messages, where e is the number of edges in the graph. Proof The fact that each variable ki is eventually different from ⊥ follows from our model, because we assumed that instructions are eventually executed and messages are eventually received, so the messages will eventually reach all nodes. 6. Communication services 619 Suppose that (k1 , . . , kn ) is not a consistent cut. Then there is a processor pj such that instruction number kj + 1 or later sends a message other than , and the message is received on or before a processor pi executes instruction number ki .

Our goal now is to argue that logical clock provides to processors the illusion of global clock. Intuitively, the reason why such an illusion can be created is that we can take any execution of a deterministic algorithm, compute the logical time tx of each instruction x, and run the execution again delaying or speeding up processors and messages in such a way that each instruction x is executed at the instant tx of the global clock. Thus, without access to a hardware clock or other external measurements not captured in our model, the processors cannot distinguish the reading of logical clock from the reading of a real global clock.

This may be too costly. The algorithm called DistributedSnapshot avoids this cost. In the algorithm, a processor initiates the calculation of consistent cut by flooding the network with a special message that acts like a sword that cuts the execution of algorithm A consistently. In order to prove that the cut is indeed consistent, we require that messages are received by the recipient in the order they were sent by the sender. Such ordering can be implemented using sequence number. In the Distributed-Snapshot algorithm, each processor pi has a variable called counter that counts the number of instructions of algorithm A executed by the processor so far.

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