Self-Stabilizing and Self-Organizing Virtual Infrastructures for Mobile Networks

Abstract

Self-stabilizing algorithms can be started in any arbitrary state to exhibit a desired behavior following a convergence period. The class of self-organizing distributed algorithms is regarded here as a subclass of the self-stabilizing class of algorithms, where convergence is sub-linear in the size of the system and local perturbation of state is handled locally converging faster than the convergence from an arbitrary state. The chapter starts with a short overview of several virtual infrastructures and fitting self-stabilizing and self-organizing techniques:

• Group communication by random walks [23]

• Polygon-based stateless infrastructure [19]

• Geographic quorum systems [1, 16]

• Autonomous virtual node [18]

• Secret swarm units [20]

• Spanners, spanning expanders

The last design, which is based on expanders and short random walks, is described in detail.

Partially supported by EU ICT-2008-215270 FRONTS, Rita Altura Trust Chair in Computer Sciences, and the Lynne and William Frankel Center for Computer Sciences.

Definitions Summary

Definition

For a graph $G=(V,E)$, given two sets of nodes, $V_1, V_2$, define $E(V_1, V_2) = \{ e = (v_1, v_2) \in E | v_1 \in V_1 \land v_2 \in V_2\}$ (e.g., the set of edges between V ₁ and V ₂). We also define $\overline{V_1} = V \setminus V_1$, the set of nodes not in V ₁.

Definition

For a graph $G=(V,E)$ and a given set of nodes, S, define $\varGamma(S) = \{u\in V | \exists s\in \land (u,s) \in E\}$.

Definition

A graph $G=(V,E)$ is an edge expander if there exists a constant c, such that for each set S of vertices (where $|S|<|V|/2$) it follows that $|E(S,\overline{S})|/|S| > c$.

Definition

An expander is considered “good” if it has a constant expansion parameter.

Definition

A graph $G=(V,E)$ is a vertex expander if there exists a constant c, such that for each set S of vertices (where $|S|<|V|/2$) it follows that

$$P\left[\min_{S\subset V, |S|\le\dfrac{n}{2}}{ \dfrac{|\varGamma(S)\setminus S|}{|S|} < c }\right] < o(1)$$

Definition

Given a graph $G=(V,E)$, a spander, $\mathcal{S} = (V,E')$, is a spanning subgraph of G if there exists a constant $p>0$, such that $|E'| \leq p|E|$ and the edge expansion of $\mathcal{S}$ is at worst p times the edge expansion of G.

Definition

The mixing rate of a graph is the measure of how fast a random walk on the graph converges to its stationary distribution.

Definition

The mixing time of a graph gives the time scale (in steps) for a random walk to reach the stationary distribution.

Definition

A task is defined by a set of legal executions.

Definition

A fair execution is an execution of the system in which every node makes steps infinitely often.

Definition

A configuration c is a safe configuration for a system and a set of legal executions LE if every fair execution that starts in c is in LE.

Definition

A system is self-stabilizing for a task and a set of legal executions LE if every infinite execution reaches a safe configuration in relation to LE.

Definition

A communication round (or just a round) is a sequence of atomic steps such that each node has taken at least one atomic step during this sequence. If this atomic step involves a send operation of a message m over link l, then we require that the atomic step which corresponds to receiving a message from l, which was sent during this sequence of atomic steps, will also appear in the sequence.

Definition

A distributed algorithm is termed self-organizing ([24]) if it satisfies the following properties: (1) the algorithm is self-stabilizing, (2) convergence time to a safe configuration, $s(n)$, is in $o(n)$, and (3) after reaching a safe configuration, convergence time following a dynamic change, $d(n)$, is in $o(s(n))$.

Definition

A distributed algorithm is termed snap-stabilizing if the algorithm stabilizes following the first request by any node and before, or simultaneously with, a notification arriving to the requesting node at the completion of the request (for more information, see [7]).