Graph Theory

4-Quasiperiodic Functions on Graphs and Hypergraphs by Rudenskaya O.G.

By Rudenskaya O.G.

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Next, in Chapter 4, we give a brief survey of basic algorithms used in more general networks. Specifically, we describe some algorithms used to solve such fundamental problems as electing a leader, conducting a breadth-first search, finding shortest paths, finding a minimum spanning tree, and finding a maximal independent set of nodes. Typical forms of uncertainty here are unknown UIDs and an unknown network graph. Then, in Chapters 5 and 6, we consider problems of reaching consensus in a distributed network.

In such a network, a single "token" circulates around the network, giving its current owner the sole right to initiate communication. ) Sometimes, however, the token may be lost, and it becomes necessary for the processes to execute an algorithm to regenerate the lost token. This regeneration procedure amounts to electing a leader. 1). We often count mod n, allowing 0 to be another name for process n, n + 1 another name for process 1, and so on. The processes associated with the nodes of G do not know their indices, nor those of their neighbors; we assume that the message-generation and transition functions are defined in terms of local, relative names for the neighbors.

A synchronizer is a system component that enables asynchronous networks (without failures) to simulate the synchronous networks of Chapters 24 (those without failures). We give efficient implementations and contrast these implementations with a lower bound result that seems to say that any such simulation must be inefficient. The apparent contradiction turns out to depend on the type of problem being solved. The second technique, described in Chapter 17, is the simulation of the asynchronous shared memory model by the asynchronous network model.

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