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## Refinement Tree Partitioning (REFTREE)

The refinement tree based partitioning method is due to William Mitchell of the National Institute of Standards and Technology [Mitchell]. It is closely related to the Octree and Space-Filling Curve methods, except it uses the tree that represents the adaptive refinement process that created the grid. This tree is constructed through the tree-based query functions.

Each node of the refinement tree corresponds to an element that occurred during the grid refinement process. The first level of the tree (the children of the root of the tree) corresponds to the initial coarse grid, one tree node per initial element. It is assumed that the initial coarse grid does not change through the execution of the program, except that the local IDs, assignment of elements to processors, and weights can change. If any other aspect of the coarse grid changes, then the Zoltan structure should be destroyed and recreated. The children of a node in the tree correspond to the elements that were created when the corresponding element was refined. The children are ordered such that a traversal of the tree creates a space-filling curve within each initial element. If the initial elements can be ordered with a contiguous path through them, then the traversal creates a space-filling curve through all the elements. Each element has a designated "in" vertex and "out" vertex, with the out vertex of one element being the same as the in vertex of the next element in the path, in other words the path goes through a vertex to move from one element to the next (and does not go out the same vertex it came in).

The user may allow Zoltan to determine the order of the coarse grid elements, or may specify the order, which might be faster or produce a better path. If Zoltan determines the order, the user can select between an order that will produce connected parts, an order based on a Hilbert Space Filling Curve, or an order based on a Sierpinski Space Filling Curve. See the parameter REFTREE_INITPATH below. If the user provides the order, then the in/out vertices must also be supplied. Similarly, the user may specify the order and in/out vertices of the child elements, or allow Zoltan to determine them. If the user knows how to provide a good ordering for the children, this may be significantly faster than the default general algorithm. However, accelerated forms of the ordering algorithm are provided for certain types of refinement schemes and should be used in those cases. See ZOLTAN_CHILD_LIST_FN. If the user always specifies the order, then the vertices and in/out vertices are not used and do not have to be provided.

Weights are assigned to the nodes of the tree. These weights need not be only on the leaves (the elements of the final grid), but can also be on interior nodes (for example, to represent work on coarse grids of a multigrid algorithm). The default weights are 1.0 at the leaves and 0.0 at the interior nodes, which produces a partition based on the number of elements in each part. An initial tree traversal is used to sum the weights, and a second traversal to cut the space-filling curve into appropriately-sized pieces and assign elements to parts. The number of parts is not necessarily equal to the number of processors.

The following limitations should be removed in the future.

• For multicomponent weights, only the first component is used.
• Heterogeneous architectures are not supported, in the sense that the computational load is equally divided over the processors. A vector of relative part sizes is used to determine the weight assigned to each part, but they are currently all equal. In the future they should be input to reflect heterogeneity.

Another limitation is that refinement tree partitioning has not been modified to work with 64-bit global IDs. If 64-bit IDs are selected at configure time with either the autotools build or the CMake build, the method will fail.

 Method String: REFTREE Parameters: REFTREE_HASH_SIZE The size of the hash table to map from global IDs to refinement tree nodes. Larger values require more memory but may reduce search time. Default: REFTREE_HASH_SIZE = 16384 REFTREE_INITPATH Determines the method for finding an order of the elements in the initial grid. "SIERPINSKI" uses a Sierpinski Space Filling Curve and is most appropriate for grids consisting of triangles. It is currently limited to 2D. "HILBERT" uses a Hilbert Space Filling Curve and is most appropriate for grids consisting of quadralaterals or hexahedra. "CONNECTED" attempts to produce connected parts (guaranteed for triangles and tetrahedra), however they tend to be stringy, i.e., less compact than the SFC methods. It is most appropriate when connected parts are required. An invalid character string will invoke the default method. Default: REFTREE_INITPATH = "SIERPINSKI" if the grid contains only triangles REFTREE_INITPATH = "HILBERT" otherwise NOTE: In Zoltan versions 1.53 and earlier the default was "CONNECTED". To reproduce old results, use REFTREE_INITPATH = "CONNECTED". Required Query Functions: ZOLTAN_NUM_COARSE_OBJ_FN ZOLTAN_COARSE_OBJ_LIST_FN or ZOLTAN_FIRST_COARSE_OBJ_FN/ZOLTAN_NEXT_COARSE_OBJ_FN pair ZOLTAN_NUM_CHILD_FN ZOLTAN_CHILD_LIST_FN ZOLTAN_CHILD_WEIGHT_FN The following functions are needed only if the order of the initial elements will be determined by a space filling curve method: ZOLTAN_NUM_GEOM_FN ZOLTAN_GEOM_MULTI_FN or ZOLTAN_GEOM_FN

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