## Metis.jl

Julia interface to Metis graph partitioning
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41 Stars
Updated Last
1 Year Ago
Started In
April 2013

# Metis

Build Status

Metis.jl is a Julia wrapper to the Metis library which is a library for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices.

## Graph partitioning

`Metis.partition` calculates graph partitions. As an example, here we partition a small graph into two, three and four parts, and visualize the result:

`Metis.partition(g, 2)` `Metis.partition(g, 3)` `Metis.partition(g, 4)`

`Metis.partition` calls `METIS_PartGraphKway` or `METIS_PartGraphRecursive` from the Metis C API, depending on the optional keyword argument `alg`:

• `alg = :KWAY`: multilevel k-way partitioning (`METIS_PartGraphKway`).
• `alg = :RECURSIVE`: multilevel recursive bisection (`METIS_PartGraphRecursive`).

## Vertex separator

`Metis.separator` calculates a vertex separator of a graph. `Metis.separator` calls `METIS_ComputeVertexSeparator` from the Metis C API. As an example, here we calculate a vertex separator (green) of a small graph:

`Metis.separator(g)`

## Fill reducing permutation

`Metis.permutation` calculates the fill reducing permutation for a sparse matrices. `Metis.permutation` calls `METIS_NodeND` from the Metis C API. As an example, we calculate the fill reducing permutation for a sparse matrix `S` originating from a typical (small) FEM problem, and visualize the sparsity pattern for the original matrix and the permuted matrix:

`perm, iperm = Metis.permutation(S)`
`⠛⣤⢠⠄⠀⣌⠃⢠⠀⠐⠈⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠔⠀⠀⠖⠻⣦⡅⠘⡁⠀⠀⠀⠀⠐⠀⠁⠀⢂⠀⠀⠠⠀⠀⠀⠁⢀⠀⢀⠀⠀⠄⢣⡀⢤⣁⠉⠛⣤⡡⢀⠀⠂⠂⠀⠂⠃⢰⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠉⣀⠁⠈⠁⢊⠱⢆⡰⠀⠈⠀⠀⠀⠀⢈⠉⡂⠀⠐⢀⡞⠐⠂⠀⠄⡀⠠⠂⠀⢀⠀⠀⠀⠠⠀⠐⠊⠛⣤⡔⠘⠰⠒⠠⠀⡈⠀⠀⠀⠉⠉⠘⠂⠀⠀⠀⡐⢈⠀⠂⠀⢀⠀⠈⠀⠂⠀⣐⠉⢑⣴⡉⡈⠁⡂⠒⠀⠁⢠⡄⠀⠐⠀⠠⠄⠀⠁⢀⡀⠀⠀⠄⠀⠬⠀⠀⠀⢰⠂⡃⠨⣿⣿⡕⠂⠀⠨⠌⠈⠆⠀⠄⡀⠑⠀⠀⠘⠀⠀⡄⠀⠠⢀⠐⢲⡀⢀⠀⠂⠡⠠⠱⠉⢱⢖⡀⠀⡈⠃⠀⠀⠀⢁⠄⢀⣐⠢⠀⠀⠉⠀⠀⠀⢀⠄⠣⠠⠂⠈⠘⠀⡀⡀⠀⠈⠱⢆⣰⠠⠰⠐⠐⢀⠀⢀⢀⠀⠌⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠁⣀⡂⠁⠦⠈⠐⡚⠱⢆⢀⢀⠡⠌⡀⡈⠸⠁⠂⠀⠀⠀⠀⠀⠀⠀⣠⠴⡇⠀⠀⠉⠈⠁⠀⠀⢐⠂⠀⢐⣻⣾⠡⠀⠈⠀⠄⠀⡉⠄⠀⠀⠁⢀⠀⠀⠰⠀⠲⠀⠐⠀⠀⠡⠄⢀⠐⢀⡁⠆⠁⠂⠱⢆⡀⣀⠠⠁⠉⠇⣀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠆⠑⠀⠀⢁⠀⢀⡀⠨⠂⠀⠀⢨⠿⢇⠀⡸⠀⢀⠠⠀⠀⠀⠀⠄⠀⡈⢀⠠⠄⠀⣀⠀⠰⡘⠀⠐⠖⠂⠀⠁⠄⠂⣀⡠⠻⢆⠄⠃⠐⠁⠤⣁⠀⠁⠈⠀⠂⠐⠀⠰⠀⠀⠀⠀⠂⠁⠈⠀⠃⠌⠧⠄⠀⢀⠤⠁⠱⢆`
`⣕⢝⠀⠀⢸⠔⡵⢊⡀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠑⠀⠀⠑⢄⠀⠳⠡⢡⣒⣃⢣⠯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢒⠖⢤⡀⠑⢄⢶⡈⣂⠎⢎⠉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢋⠅⣂⡘⠳⠻⢆⡥⣈⠆⡨⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠈⠼⢸⡨⠜⡁⢫⣻⢞⢔⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⡭⡖⡎⠑⡈⡡⠐⠑⠵⣧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠈⠁⠃⠂⠃⠊⠀⠘⠒⠙⠛⢄⠀⠀⢄⠀⠤⢠⠀⢄⢀⢀⠀⡀⠀⠀⢄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠊⠀⣂⠅⢓⣤⡄⠢⠠⠀⠌⠉⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠑⢄⠁⣋⠀⢀⢰⢄⢔⢠⡖⢥⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠌⠜⡥⢠⠛⣤⠐⣂⡀⠀⡀⡁⠍⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣴⠀⢀⠰⢠⠿⣧⡅⠁⠂⢂⠂⠋⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⡉⠐⢖⠀⠈⠅⠉⢕⢕⠝⠘⡒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠐⣑⠄⠨⠨⢀⣓⠁⣕⢝⡥⢉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠜⣍⠃⡅⡬⠀⠘⡈⡅⢋⠛⣤⡅⠒⢕⠘⡂⠄⠀⠀⠁⠀⠀⡂⠀⢠⠀⢕⠄⢐⠄⠀⠘⠀⠉⢐⠀⠀⠁⡀⢡⠉⢟⣵`
`S` (5% stored values) `S[perm,perm]` (5% stored values)

We can also visualize the sparsity pattern of the Cholesky factorization of the same matrix. It is here clear that using the fill reducing permutation results in a sparser factorization:

`⠙⢤⢠⡄⠀⣜⠃⢠⠀⠐⠘⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⡔⠀⠀⠀⠙⢦⡇⠾⡃⠰⠀⠀⠀⠐⠀⠃⠀⢂⠀⠀⠠⠀⠀⠀⠃⢀⠀⢀⠀⠀⠆⢣⠀⠀⠀⠀⠙⢼⣣⢠⠀⣂⣂⢘⡂⡃⢰⣋⡀⣔⢠⠀⠀⠀⡃⠈⠀⢈⠀⡄⣄⡋⠀⠀⠀⠀⠀⠀⠑⢖⡰⠉⠉⠈⠁⠁⢘⢙⠉⡊⢐⢐⢀⣞⠱⠎⠀⠌⡀⡣⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣴⢸⣴⡖⢠⣤⡜⢣⠀⠀⠛⠛⡜⠂⠀⢢⠀⡔⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣛⣛⣛⣛⣓⣚⡃⢠⣖⣒⣓⢐⢠⣜⠀⡃⢘⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣾⣿⣿⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿`
`⠑⢝⠀⠀⢸⠔⡵⢊⡀⡂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⢕⠀⠀⠑⢄⠀⠳⠡⢡⣒⣃⢣⠯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠀⠀⠀⠀⠑⢄⢶⡘⣂⡎⢎⡭⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠴⠀⠀⠀⠀⠀⠀⠙⢎⣷⣏⢷⣯⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢼⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢄⠀⠤⢠⠀⢄⢀⢀⠀⡀⠀⠀⢟⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠊⠀⣂⠅⢓⣤⡄⠢⠠⠀⠌⠉⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠉⣋⠀⢁⢰⢔⢔⢠⡖⢥⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠘⣶⡂⠠⡀⣡⠭⣤⢓⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡇⡇⣢⣢⠂⣯⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⢟⢝⣒⠭⠭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣿⣿⡭⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿`
`chol(S)` (16% stored values) `chol(S[perm,perm])` (6% stored values)

• `METIS_PartGraphRecursive`
• `METIS_PartGraphKway`
• `METIS_ComputeVertexSeparator`
• `METIS_NodeND`