Abstract
An important problem in any structured P2P overlay network is what routes are available between nodes. Understanding the structure of routes helps to solve problems related to routing performance, security, and scalability. In this chapter, we propose a theoretical approach for describing routes and their structures. It is based on recent results in the linear Diophantine analysis. A route aggregates several P2P paths that packets follow. A commutative context-free grammar describes the forwarding behavior of P2P nodes. Derivations in the grammar correspond to P2P routes. Initial and final strings of a derivation define packet sources and destinations, respectively. Based on that we construct a linear Diophantine equation system, where any solution counts forwarding actions in a route representing certain integral properties. Therefore, P2P paths and their composition into routes are described by a linear Diophantine system—a Diophantine model of P2P routes. Its finite basis of the solution set defines the structure of available routes.
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- 1.
For brevity we ignore the case when \(\alpha {\Rightarrow }^{{_\ast}}\varkappa \beta \) is not simple. It also satisfies (9.4) but requires extracting all simple cycles and moving them to \({x}^{\epsilon }\), so leading to \({x}^{\epsilon }\neq 0\).
- 2.
The sign “ + ” in \({b}_{i}^{+}\) means that packets arrive to d i .
- 3.
The sign “ − ” in \({b}_{i}^{-}\) means that packets leave s i .
- 4.
- 5.
This problem is of deciding, given a CCF-grammar and strings α, β, if \(\alpha {\Rightarrow }^{{_\ast}}\beta \).
- 6.
Theory says that there is a polynomial algorithm for finding a non-zero solution of an arbitrary homNLDE system, not necessarily a homANLDE system. Unfortunately, to the best of our knowledge, we know no implementation appropriate for practical large-scale applications.
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Korzun, D., Gurtov, A. (2013). Diophantine Routing. In: Structured Peer-to-Peer Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5483-0_9
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