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**METIS Three Phases Coarsening Partitioning Uncoarsening**

G. Karypis, V. Kumar, “A fast and high quality multilevel scheme for partitioning irregular graphs,” International Conference on Parallel Processing, 1995.

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**METIS - Coarsening Maximal Matching**

A set of edges without common vertices An NP-Complete problem

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**METIS - Partitioning Two Steps Randomly Choose a root**

BFS to include the vertex leading less edge-cuts Root

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**METIS - Uncoarsening Key Idea Each super-node comprises a set of nodes**

Decrease the edge-cuts by moving a vertex to one partition to another

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**Parallel METIS Five Phases Initial Partition Coloring Coarsening**

Partitioning Uncoarsening Each processor keeps two pieces of Information: 1. Sub-graph 2. Adjacency List G. Karypis, V. Kumar, “Parallel Multilevel k-way Partitioning Scheme for Irregular Graphs,” IEEE/ACM Conference on Supercomputing, 1996.

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**Parallel METIS Coloring**

Adjacent vertices have different colors [Luby’s Algorithm] The number of distinct colors used is to be minimized

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**Parallel METIS Coarsening Phase Unilateral Matching**

Matching Conflicts? Why do we need coloring? Node.Match Remote Edge Iterative Fashion

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**Parallel METIS Partitioning Phase**

Since the coarsened graph has been relatively small, partition can be done Further parallelization is also possible Iterative Fashion G. Karypis, V. Kumar, “Parallel Multilevel k-way Partitioning Scheme for Irregular Graphs,” IEEE/ACM Conference on Supercomputing, 1996.

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**Parallel METIS Uncoarsening Phase**

This phase is broken up into c sub-phases, where c is the number of colors During the cth phase, all the vertices of color c are considered for movement Iterative Fashion G. Karypis, V. Kumar, “Parallel Multilevel k-way Partitioning Scheme for Irregular Graphs,” IEEE/ACM Conference on Supercomputing, 1996.

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