Self-Stabilizing Algorithms for graph parameters Phd student : - - PowerPoint PPT Presentation

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RESCOM 2014 UMR 5205 Self-Stabilizing Algorithms for graph parameters Phd student : Brahim NEGGAZI 1 Laboratoire d'InfoRmatique en Image et Systmes d'information LIRIS UMR 5205 CNRS/INSA de Lyon/Universit Claude Bernard Lyon 43, boulevard

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UMR 5205

RESCOM’14 15/05/2014

Self-Stabilizing Algorithms for graph parameters

Phd student : Brahim NEGGAZI

Team: GrAMA

1 Laboratoire d'InfoRmatique en Image et Systèmes d'information

LIRIS UMR 5205 CNRS/INSA de Lyon/Université Claude Bernard Lyon 43, boulevard du 11 novembre 1918 — F-69622 Villeurbanne Cedex

RESCOM 2014

Associate professor: Mohammed Haddad Professor : Hamamache Kheddouci

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Self-stabilization was introduced by E. Dijkstra en 1974. A system is “self-stabilizing” if it can start from any possible configuration and converges to a desired configuration in finite time by itself without any external intervention.

Self-stabilization

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Self-stabilization presents many advantages: § Self-recovering. § No initialization. § Dynamic topology adaptation. However, there are of course some disadvantages of self- stabilization which cannot be ignored: § High complexity. § No termination detection.

Self-stabilization: advantages & inconvenients

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Self-Stabilization properties

Illegitimate configurations Desired configurations

Convergence Closure

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Goal of my thesis

Proposing distributed and self-stabilizing algorithms for graph decompositions. These algorithms are very useful for

rganization and optimization protocols in large scale

systems/networks. Challenges and originality of the research work

Focus on the problems of decomposition of graphs

subgraphs (triangles, stars, chains, ...)

Proving convergence of self-stabilizing algorithms,
Providing distributed algorithms with low complexities.
Using One-hop knowledge (i.e. each node can read only

states of its neighbors.

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First contribution Triangle decomposition problem for arbitrary graphs

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Decomposition into triangles

Instance graph G = (V,E) |V| = 3n Question Can the vertices of G be partitioned into n disjoint Sets V1, V2, …, Vn such that each Vi contains exactly 3 vertices forming a triangle in G? Finding the maximum number

f node disjoint triangles (k)

in graph is NP-Hard. Problem called Node Disjoint Triangle Packing [Albertto & Rizzi 2002]

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Since perfect partitioning does not always exist for an arbitrary graph, and finding the maximum number of disjoint triangles is hard, we consider the local maximization of this decomposition.

Maximal decomposition into triangles

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9 (a) (b) (c) Maximal

Maximal Maximal

Maximal decomposition into triangles

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Results

First step: We propose a first distributed and self-stabilizing algorithm for maximal graph decomposition into disjoint triangles . The complexity of the first algorithm is O(n4) where n is the number of nodes in the graph (More details can be found in the published

paper: Self-stabilizing Algorithm for Maximal Graph Partitioning into Triangles. 14th International Symposium on Stabilization, Safety, and Security of Distributed Systems, 2012, Toronto, Canada).

Second step: A second algorithm is proposed that stabilizes within O(m) where m is the number of edges in graph (Submited).

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Second contribution P-Star decomposition problem

f arbitrary graphs

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p-Star decomposition problem

A p-star has one center node and p leaves where p ≥ 1. A p-star decomposition subdivides a graph into p-stars Variant of generalized matchings and general graph factor problems that were proved to be NP-Complete [D. Kirkpatrick et al. in

ST0C 78] , Journ. Comp. 83] Center node Leaf node

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p-Star Decomposition

f General Graphs

Graph G = (V,E) p=3

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p-Star Decomposition

f General Graphs

Graph G = (V,E) p=3

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p-Star Decomposition

f General Graphs

Graph G = (V,E) p=3

Star 1 Star 2 Star 3 Star 4 Maximal p-star Decomposition

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Results

We propose the first distributed and self-stabilizing algorithm for decomposing a graph into p-stars. The algorithm operates under a Distributed Scheduler and stabilizes within O(n) rounds (More details can be found in the

published paper: Self-stabilizing Algorithm for Maximal p-star decomposition of arbitrary graph. 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems 2013, Osaka, Japan)

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Self-Stabilizing Algorithms for graph parameters

RESCOM 2014

Self-stabilization was introduced by E. Dijkstra en 1974. A system is “self-stabilizing” if it can start from any possible configuration and converges to a desired configuration in finite time by itself without any external intervention.

Self-stabilization

Self-stabilization presents many advantages: § Self-recovering. § No initialization. § Dynamic topology adaptation. However, there are of course some disadvantages of self- stabilization which cannot be ignored: § High complexity. § No termination detection.

Self-stabilization: advantages & inconvenients

Self-Stabilization properties

Convergence Closure

Goal of my thesis

Proposing distributed and self-stabilizing algorithms for graph decompositions. These algorithms are very useful for

systems/networks. Challenges and originality of the research work

subgraphs (triangles, stars, chains, ...)

states of its neighbors.

First contribution Triangle decomposition problem for arbitrary graphs

Decomposition into triangles

Instance graph G = (V,E) |V| = 3n Question Can the vertices of G be partitioned into n disjoint Sets V1, V2, …, Vn such that each Vi contains exactly 3 vertices forming a triangle in G? Finding the maximum number

in graph is NP-Hard. Problem called Node Disjoint Triangle Packing [Albertto & Rizzi 2002]

Since perfect partitioning does not always exist for an arbitrary graph, and finding the maximum number of disjoint triangles is hard, we consider the local maximization of this decomposition.

Maximal decomposition into triangles

Maximal decomposition into triangles

Results

First step: We propose a first distributed and self-stabilizing algorithm for maximal graph decomposition into disjoint triangles . The complexity of the first algorithm is O(n4) where n is the number of nodes in the graph (More details can be found in the published

Second step: A second algorithm is proposed that stabilizes within O(m) where m is the number of edges in graph (Submited).

Second contribution P-Star decomposition problem

p-Star decomposition problem

A p-star has one center node and p leaves where p ≥ 1. A p-star decomposition subdivides a graph into p-stars Variant of generalized matchings and general graph factor problems that were proved to be NP-Complete [D. Kirkpatrick et al. in

p-Star Decomposition

p-Star Decomposition

p-Star Decomposition

Results

We propose the first distributed and self-stabilizing algorithm for decomposing a graph into p-stars. The algorithm operates under a Distributed Scheduler and stabilizes within O(n) rounds (More details can be found in the

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