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Data Distribution

  The fundamental data object in the LU factorization algorithm presented in Section 4.1 is a block-partitioned matrix. In this section, we describe the block-cyclic method for distributing such a matrix over a two-dimensional mesh of processes, or template. In general, each process has an independent thread of control, and with each process is associated some local memory directly accessible only by that process. The assignment of these processes to physical processors is a machine-dependent optimization issue, and will be considered later in Section 7.

An important property of the class of data distribution we shall use is that independent decompositions are applied over rows and columns. We shall, therefore, begin by considering the distribution of a vector of M data objects over P processes. This can be described by a mapping of the global index, m, of a data object to an index pair (p,i), where p specifies the process to which the data object is assigned, and i specifies the location in the local memory of p at which it is stored. We shall assume tex2html_wrap_inline2067 and tex2html_wrap_inline2069.

Two common decompositions are the block and the cyclic decompositions [48, 30]. The block decomposition, which is often used when the computational load is distributed homogeneously over a regular data structure such as a Cartesian grid, assigns contiguous entries in the global vector to the processes in blocks.
where tex2html_wrap_inline2071, and tex2html_wrap_inline2073 denotes the greatest integer less than or equal to x. The cyclic decomposition (also known as the wrapped or scattered decomposition) is commonly used to improve load balance when the computational load is distributed inhomogeneously over a regular data structure. The cyclic decomposition assigns consecutive entries in the global vector to successive different processes,
Examples of the block and cyclic decompositions are shown in Figure 8.

Figure 8: Examples of block and cyclic decompositions of M=10 data objects over P=3 processes.The global index m is mapped to the process number, p, and local index, i.

The block cyclic decomposition is a generalization of the block and cyclic decompositions in which blocks of consecutive data objects are distributed cyclically over the processes. In the block cyclic decomposition the mapping of the global index, m, can be expressed as tex2html_wrap_inline2089, where p is the process number, b is the block number in process p, and i is the index within block b to which m is mapped. Thus, if the number of data objects in a block is r, the block cyclic decomposition may be written,
where T=rP. It should be noted that this reverts to the cyclic decomposition when r=1, with local index i=0 for all blocks. A block decomposition is recovered when r=L, in which case there is a single block in each process with block number b=0. The inverse mapping of the triplet (p,b,i) to a global index is given by,
where B=p+bP is the global block number. The block cyclic decomposition is one of the data distributions supported by High Performance Fortran (HPF) [38], and has been previously used, in one form or another, by several researchers (see [1, 4, 5, 9, 22, 26, 44, 46, 47] for examples of its use). The block cyclic decomposition is illustrated with an example in Figure 9.

Figure 9: An example of the block cyclic decomposition of M=23 data objects over P=3 processes for a block size of r=2. (a) shows the mapping from global index, m, to the triplet (p,b,i), and (b) shows the inverse mapping.

The form of the block cyclic decomposition given by Eq. 9 ensures that the block with global index 0 is placed in process 0, the next block is placed in process 1, and so on. However, it is sometimes necessary to offset the processes relative to the global block index so that, in general, the first block is placed in process tex2html_wrap_inline2133, the next in process tex2html_wrap_inline2135, and so on. We, therefore, generalize the block cyclic decomposition by replacing m on the righthand side of Eq. 9 by tex2html_wrap_inline2139 to give,
Equation 10 may also be generalized to,
where now the global block number is given by tex2html_wrap_inline2141. It should be noted that in processes with tex2html_wrap_inline2143, block 0 is not within the range of the block cyclic mapping and it is, therefore, an error to reference it in any way.

In decomposing an tex2html_wrap_inline1863 matrix we apply independent block cyclic decompositions in the row and column directions. Thus, suppose the matrix rows are distributed with block size r and offset tex2html_wrap_inline2133 over P processes by the block cyclic mapping tex2html_wrap_inline2153, and the matrix columns are distributed with block size s and offset tex2html_wrap_inline2157 over Q processes by the block cyclic mapping tex2html_wrap_inline2161. Then the matrix element indexed globally by (m,n) is mapped as follows,
The decomposition of the matrix can be regarded as the tensor product of the row and column decompositions, and we can write,
The block cyclic matrix decomposition given by Eqs. 13 and 14 distributes blocks of size tex2html_wrap_inline2165 to a mesh of tex2html_wrap_inline2167 processes. We shall refer to this mesh as the process template, and refer to processes by their position in the template. Equation 14 says that global index (m,n) is mapped to process (p,q), where it is stored in the block at location (b,d) in a two-dimensional array of blocks. Within this block it is stored at location (i,j). The decomposition is completely specified by the parameters r, s, tex2html_wrap_inline2133, tex2html_wrap_inline2157, P, and Q. In Figure 10 an example is given of the block cyclic decomposition of a tex2html_wrap_inline2189 matrix for block size tex2html_wrap_inline2191, a process template tex2html_wrap_inline2193, and a template offset tex2html_wrap_inline2195. Figure 11 shows the same example but for a template offset of (1,2).

The block cyclic decomposition can reproduce most of the data distributions commonly used in linear algebra computations on parallel computers. For example, if Q=1 and tex2html_wrap_inline2201 the block row decomposition is obtained. Similarly, P=1 and tex2html_wrap_inline2205 gives a block column decomposition. These decompositions, together with row and column cyclic decompositions, are shown in Figure 12. Other commonly used block cyclic matrix decompositions are shown in Figure 13.

next up previous
Next: Parallel Implementation Up: No Title Previous: Derivation of a Block

Jack Dongarra
Sun Feb 9 10:05:05 EST 1997