Isorropia: Partitioning, Coloring, and Ordering

Version 3.0

Introduction

Isorropia is a package for combinatorial scientific computing, with focus on partitioning (load balancing), but also supports coloring and ordering. Its main purpose is to assist with redistributing objects such as matrices and matrix-graphs in a parallel execution setting, to allow for more efficient computations. Isorropia produces good maps for Epetra matrices/graphs. Isorropia should be called after the matrix/graph is filled, so the sparsity pattern is known.

In addition to matrix/graph partitioning, Isorropia has the capability to partition 1, 2 or 3-dimensional real coordinates into balanced spatial regions. The coordinates are represented as an Epetra_MultiVector.

Isorropia's coloring capability can also be employed for the purpose of probing structurally symmetric matrices.

Isorropia is primarily an interface to the Zoltan toolkit, which is now required. Zoltan became a Trilinos package in 9.0 and is automatically enabled by Isorropia. To enable Isorropia in your Trilinos build, provide the argument '--enable-isorropia' to configure. If you wish to use third-party libraries (such as ParMetis or PT-Scotch) with Isorropia via Zoltan, see the configure options for Zoltan.

Overview of Isorropia.

Isorropia currently contains algorithms for three separate problems. In the case of sparse matrices, these problems can each be viewed as a combinatorial problem based on graphs:

Isorropia contains separate classes to solve each of these problems, all of which derive from Isorropia::Operator and Isorropia::Epetra::Operator. Note that an operator in Isorropia is different from an operator in Epetra!

Partitioning and Load Balancing

We focus on the partitioning (load balancing), which is the most heavily used feature. Isorropia's load balancing user interface consists primarily of the Isorropia::Partitioner, Isorropia::Redistributor and Isorropia::CostDescriber classes, as well as the Isorropia::Epetra::create_balanced_copy functions.

The Isorropia::Epetra::create_balanced_copy functions are free-standing functions (not class members) which simply take an input object (Epetra_CrsGraph, Epetra_CrsMatrix, etc.) and return a copy which is balanced across processes (parts). The reference-counted pointer class Teuchos::RCP is used.

In many use cases, such as performing one partitioning and then using it to redistribute multiple objects, it is more efficient to use the Isorropia::Partitioner and Isorropia::Redistributor classes. The general usage model is to:

  1. Create a Isorropia::Partitioner instance, providing the objects to be partitioned, and optionally weights for those objects. Weights can be defined with an Isorropia::CostDescriber for graphs and hypergraphs, or with an Epetra_MultiVector for geometric coordinates.
  2. Create a Isorropia::Redistributor object (providing as input a Isorropia::Partitioner instance)
  3. Use the Isorropia::Redistributor to redistribute one or more objects to the new partitioning.

Isorropia was designed such that these steps can usually be done in three lines of code. For example, the Partitioner will by default compute the partitioning at construction time.

Isorropia currently supports partitioning/redistributing of several Epetra objects, including Epetra_CrsGraph, Epetra_RowMatrix (which includes Epetra_CrsMatrix and Epetra_VbrMatrix) and Epetra_MultiVector. (The Epetra_MultiVector is interpreted as 1, 2 or 3 dimensional coordinates and is partitioned into balanced spatial regions using Zoltan's geometric partitioning methods.)

Several Isorropia classes and methods accept Teuchos::ParameterList objects, allowing the user to control certain behavior. Isorropia parameters can be either lower or upper case.

Example Programs

A number of example programs which demonstrate using Isorropia are provided in the 'packages/isorropia/example' subdirectory.

matrix_1.cpp demonstrates creating a balanced copy of Epetra_CrsGraph and Epetra_CrsMatrix objects using Isorropia::Epetra::create_balanced_copy functions.

part_redist.cpp demonstrates repartitioning and redistributing the contents of an Epetra_LinearProblem object, using the Isorropia::Partitioner and Isorropia::Redistributor classes. This program does not use user-specified weights/costs.

Other programs in the example subdirectory demonstrate the use of weights/costs data to influence the partitioner. See vert_weights.cpp, graphedge_weights.cpp and hgedge_weights.cpp.

example_rcb.cpp demonstrates the use of Isorropia to partition spatial coordinates using Zoltan's Recursive Coordinate Bisection algorithm.

Coloring

Isorropia supports vertex coloring of a graph via Zoltan. Both distance-1 and distance-2 coloring is supported. See the Zoltan User's guide (available at the Zoltan web site) for details. Currently, coloring is limited to structurally symmetric matrices but we expect to support nonsymmetric matrices (e.g. Jacobians) in the future.

Ordering

Isorropia currently supports a subset of the graph (sparse matrix) ordering features in Zoltan. The intended main use is to produce a permutation that can be used as fill-reducing ordering for direct solvers. Currently Zoltan relies on third-party libraries (ParMetis and PT-Scotch) for this.

Probing

Isorropia's distance-2 graph vertex capability enables matrix probing, the re-construction (or approximation) of a matrix from matrix-vector products with specially chosen vectors. These vectors are combinations of columns of the identity which are carefully chosen such that the columns probed by such a vector are structurally orthogonal. The most common use case of probing is when the matrix is not explicitly stored, but the matrix entries are necessary for some other procedure (e.g. preconditioning). In certain applications, an exact reconstruction of the matrix is needed. In others, an approximation based on a priori knowledge of the ``significant'' matrix entries suffices. Isorropia's probing functionality works in both cases. See Probing for an example of how probing is used.

Currently, Isorropia's probing functionality is limited to structurally symmetric matrices but we expect to support nonsymmetric matrices (e.g. Jacobians) in the future.