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163 lines
8.8 KiB
<!-------- @HEADER
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! Zoltan Toolkit for Load-balancing, Partitioning, Ordering and Coloring
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------->
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<!doctype html public "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; SunOS 5.7 sun4u) [Netscape]">
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<meta name="sandia.approved" content="SAND99-1376">
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<meta name="author" content="william mitchell, william.mitchell@nist.gov">
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<title> Zoltan Developer's Guide: Refinement Tree</title>
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</head>
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<body bgcolor="#FFFFFF">
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<div ALIGN=right><b><i><a href="dev.html">Zoltan Developer's Guide</a>
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| <a href="dev_hsfc.html">Next</a>
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| <a href="dev_phg.html">Previous</a></i></b></div>
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<h2>
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Appendix: Refinement Tree</h2>
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<h3>
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Overview of structure (algorithm)</h3>
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The refinement tree based partitioning algorithm was developed and implemented
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by <a href="https://math.nist.gov/~mitchell">William Mitchell</a> of the National Institute of Standards and Technology.
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It is similar to the Octree method except that it uses a tree representation
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of the refinement history instead of a geometry based octree. The method
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generates a space filling curve which is cut into <i>K</i> appropriately-sized pieces
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to define contiguous parts, where the size of a piece is the sum of the
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weights of the elements in that piece. <i>K</i>, the number of parts, is not
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necessarily equal to <i>P</i>, the number of processors. It is an appropriate load balancing
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method for grids that are generated by adaptive refinement when the refinement
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history is available. This implementation consists of two phases: the
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construction of the refinement tree, and the definition of the parts.
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<h4>
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Refinement Tree Construction</h4>
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The refinement tree consists of a root node and one node for each element
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in the refinement history. The children of the root node are the elements
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of the initial coarse grid. The children of all other nodes are the elements
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that were formed when the parent element was refined. Upon first invocation,
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the refinement tree is initialized. This creates the root node and initializes
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a hash table that maps global IDs into nodes of the refinement tree.
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It also queries the user for the elements of the initial grid and creates
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the children of the root node. Unless the user provides the order
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through which to traverse the elements of the initial grid, a path is
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determined through the initial elements along with the "in" vertex and
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"out" vertex of each element, i.e., the vertices through which the path
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passes to move from one element to the next.
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This path can be determined by a Hilbert space filling curve, Sierpinski
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space filling curve (triangles only), or an algorithm that attempts to make
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connected parts (connectivity is guaranteed for triangles and
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tetrahedra).
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The refinement tree is required to have all initial coarse grid elements,
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not just those that reside on the processor. However, this requirement is not
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imposed on the user; a communication step fills in the elements from other
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processors. This much of the tree persists throughout execution of the
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program. The remainder of the tree is reconstructed on each invocation of
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the refinement tree partitioner. The remainder of the tree is built through
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a tree traversal. At each node, the user is queried for the children of the
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corresponding element. If there are no children, the user is queried for
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the weight of the element. If there are children, the order of the children
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is determined such that a tree traversal produces a space filling curve.
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The user indicates what type of refinement was used to produce the children
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(bisection of triangles, quadrasection of quadrilaterals, etc.). For each
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supported type of refinement, a template based ordering is imposed. The
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template also maintains an "in" and "out" vertex for each element
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which are used by the template to determine the beginning and end of the space
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filling curve through the children. If the
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refinement is not among the types supported by templates, an exhaustive
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search is performed to find an appropriate order, unless the user provides
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the order.
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<h4>
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Partition algorithm</h4>
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The algorithm that determines the parts uses four traversals of the
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refinement tree. The first two traversals sum the weights in the tree.
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In the first traversal, each node gets the sum of the weights of all the
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descendant nodes that are assigned to this processor. The processors then
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exchange information to fill in the partial sums for the leaf elements
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that are not owned by this processor. (Note that an unowned leaf on one
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processor may be the root of a large subtree on another processor.)
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The second traversal completes the summation of the weights. The root
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now has the sum of all the weights, which, in conjunction with an array
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of relative part sizes, determines the desired weight of each part.
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Currently the array of part sizes are all equal, but in the future
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the array will be input to reflect heterogeneity in the system. The third
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traversal computes the partition by adding subtrees to a part
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until the size of the part meets the desired weight, and counts
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the number of elements to be exported. Finally, the fourth traversal
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constructs the export list.
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<h3>
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Data structures</h3>
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The implementation of the refinement tree algorithm uses three data
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structures which are contained in <i>reftree/reftree.h</i>. <i>Zoltan_Reftree_data_struct</i>
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is the structure pointed to by <i>zz->LB.Data_Structure</i>. It contains a pointer
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to the refinement tree root and a pointer to the hash table.
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<i>Zoltan_Reftree_hash_node</i> is an entry in the hash table. It consists of a global ID,
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a pointer to a refinement tree node, and a "next" pointer from which
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linked lists at each table entry are constructed to handle collisions.
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<i>Zoltan_Reftree_Struct</i> is
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a node of the refinement tree. It contains the global ID, local ID,
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pointers to the children, weight and summed weights, vertices of the
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element, "in" and "out" vertex, a flag to indicate if this element is
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assigned to this processor, and the new part number.
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<h3>
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Parameters</h3>
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There are two parameters. <a href="../ug_html/ug_alg_reftree.html">REFTREE_HASH_SIZE</a> determines the size of
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the hash table.
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<a href="../ug_html/ug_alg_reftree.html">REFTREE_INITPATH</a> determines which
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algorithm to use to find a path through the initial elements.
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Both are set by <b>Zoltan_Reftree_Set_Param</b> in the file <i>reftree/reftree_build.c</i>.
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<h3>
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Main routine</h3>
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The main routine is <b>Zoltan_Reftree_Part</b> in file <i>reftree/reftree_part.c</i>.
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<p>
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<hr WIDTH="100%">
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<br>[<a href="dev.html">Table of Contents</a>
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| <a href="dev_hsfc.html">Next: Hilbert Space-Filling Curve (HSFC)</a>
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| <a href="dev_phg.html">Previous: Hypergraph Partitioning</a> | <a href="https://www.sandia.gov/general/privacy-security/index.html">Privacy and Security</a>]
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</body>
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</html>
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