Liu, Joseph W.H. "Modification of the Minimum-degree Algorithm by Multiple Elimination," ACM Transaction on Mathematical Software, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....fill reduction strategies, these blocks are also relabeled using a different reordering technique, typically one that is based on a minimum degree type ordering. The tests shown in the experiments reorder these blocks with the well known Multiple Minimum Degree algorithm of George and Liu [17]. We refer to this combination as ND ARMS. ND ARMS also incorporates diagonal dominance selection. The weights (17) are computed for all the rows belonging to the leaves of the ND tree. Those rows among this set that do not satisfy the diagonal dominance criterion, are rejected to the complement ....

Joseph W-H Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11:141--153, 1985.

....algorithm. This is called mass elimination in MD implementations. At the beginning of the algorithm, all vertices are supernodes of size one. Then during the algorithm, indistinguishable supernodes are merged together as they are detected. It is common to use the external degrees of supernodes [7]: the external degree of a supernode is the number of vertices adjacent to it that belong to other supernodes. The weight of a supernode is the number of vertices that are absorbed in it. 2 2 1 2 1 2 4 2 4 4 2 5 6 5 6 3 4 4 5 3 3 3 5 6 4 5 6 3 6 6 5 3 3 1 2 reach (1) ....

....quotient graphs and use the tools described in Section 2.4. Since we use the external degree of a supernode, the computed ordering might not in some cases correspond to a strict minimum degree ordering. However, the use of external degree tends to give better results than exact degree in practice [7]. 3.1 Original Minimum Degree The original MD algorithm, enhanced by the techniques mentioned in Section 2.4, is presented in Figure 3. We only discuss the details of the most time consuming steps. Asymptotically, the costliest operation in MD is the degree update. After a vertex has been ....

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J. W. H. Liu, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. Math. Software, 11 (1985), pp. 141--153.

....3: Load balance results. For both algorithms, the preprocessing steps are the same. These include a step to permute large entries on the diagonal (using the routine MC64 [3] followed by a symetric permutation to preserve the sparsity (using multiple minimum degree algorithm applied on A A [13]) and the symbolic factorization to get the structures of L and U.Only the numerical factorization phase is di#erent in the two approaches. This includes the matrix distribution and the actual factorization. After the preprocessing steps, SLUD distributes the data among processors using a 2D ....

J. W. Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Trans. Math. Software, 11:141--153, 1985.

....the point of view of incomplete factorization preconditioning, some orderings are less sensitive to the departure from symmetry than others. There are many possible ways of implementing the (reverse) Cuthill McKee and minimum degree reorderings. We used Liu s multiple minimum degree algorithm [34], and for the Cuthill McKee reorderings, we used an implementation which chooses a pseudoperipheral node as starting node and sorts nodes in the same level set by increasing degree [29] Other strategies are possible as well, and di#erent choices may lead to somewhat di#erent results. We note ....

J. W. Liu, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. Math. Software, 11 (1985), pp. 141--153.

....4. For each matrix, the table shows the parallel run time and the number of nonzeros in the Cholesky factor L of the resulting matrix for 16, 32, and 64 processors. On p processors, the ordering is computed by using nested dissection for the first log p levels, and then multiple minimum degree [20] (MMD) is used to order the submatrices stored locally on each processor. Matrix T16 ILl T3: ILl T64 ILl BRACK2 2.9 7788096 2.5 7690143 1.8 7687988 CANT 4.4 29818759 2.8 28854330 2.2 28358362 COPTER2 2.6 12905725 2.1 12835682 1.6 12694031 CYLINDER93 3.5 15581849 2.2 15662010 1.7 15656651 ....

J.W.-H. Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11:141 153, 1985.

....is apparently O(log n) Thus the total running time is O(n log n) 5. 3 Numerical Experiments In order to measure the performance of the modified nested dissection algorithm, we compare its fills and LU factorization time with that of regular nested dissection and that of minimum degree algorithm [12, 7]. Fills 1 192 6348 3636 1836 2034 1980 2 375 15285 25722 9342 10269 8586 3 648 30060 89199 31041 28872 26145 4 882 42384 196641 47286 50481 42462 5 1029 52131 254718 74880 73845 57537 6 1176 60360 332766 96381 88785 71685 7 1344 69888 487701 117900 112365 91971 8 1536 82956 657612 149652 ....

Joseph W. H. Liu. Modification of minimum-degree algorithm by multiple elemination. ACM Transaction on Mathematical Software, 11(2), June 1985.

....factorization is markedly less efficient if an input ordering is given since different logic 1 is used than in the case of the native ordering. The standard ordering used by MUMPS is the approximate minimum degree (AMD) ordering [1] while SuperLU uses the multiple minimum degree ordering (MMD) [24] on this symmetrized pattern. However, in the experiments using a minimum degree code in the following sections, we consider only the AMD ordering since both codes can generate this using the HSL routine MC47, it is usually far quicker than MMD, and it produces a symbolic factorization close to ....

J. W. H. Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11(2):141--153, 1985.

....algorithms instead choose the vertex that would create the minimum number of fill in elements. A nice comparison of many of these different approaches can be found in [11] 7 5 Implementation Our GGCL based implementation of MMD closely follows the algorithmic descriptions of MMD given, e.g. in [15, 13]. The implementation presently includes the enhancements for mass elimination, incomplete degree update, multiple elimination, and external degree. In addition, we use a quotient graph representation. Some particular details of our implementation are given below. Prototype The prototype for our ....

Joseph W. H. Liu. Modification of the minimum-degree algorithm by multiple elimination. ACM Transaction on Mathematical Software, 11(2):141--153, 1985.

....all vertices that belong to a supernode I V k can be replaced by a single logical node with weight jIj. This reduces the size of the elimination graphs and, therefore, the runtime of the minimum degree algorithm. Closely related to the concept of supernodes is the notion of external degrees [42]. With our notations, the external degree of a vertex v in supernode I is j adj G k (I)j. Instead of using true degrees, the vertex to be eliminated next is selected according to its external degree. The motivation is that the only edges added by the elimination of v 2 I are between vertices in ....

....in each step. Once the new elimination graph has been built, the degree of all vertices that were adjacent to the newly eliminated vertex have to be updated. The most time consuming part of the minimum degree algorithm is this degree update step. In Liu s multiple minimum degree (MMD) algorithm [42] an independent set of minimum degree vertices is eliminated in each step. This multiple elimination technique reduces the amount of degree updates and leads to a significant acceleration of the minimum degree algorithm. The efficiency of the minimum degree algorithm can be further improved when ....

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J.W.H. Liu, Modification of the minimum-degree algorithm by multiple elimination, ACM Trans. Math. Software, Vol. 11, No. 2, 141--153, 1985.

....the symbolic factorization is markedly less efficient if an input ordering is given since different logic is used in this case. The default ordering used by MUMPS is approximate minimum degree (AMD) Amestoy, Davis and Duff 1996a) while the default for SuperLU is multiple minimum degree (MMD) (Liu 1985). However, in our experiments using a minimum degree ordering, we considered only the AMD ordering since both codes can generate this using the subroutine MC47 from HSL (2000) It is usually far quicker than MMD and produces a symbolic factorization close to that produced by MMD. We also use ....

Liu, J. W. H. (1985), `Modification of the minimum degree algorithm by multiple elimination', ACM Trans. Math. Softw. 11(2), 141--153.

....and 4 irregular sparse matrices from the Harwell Boeing sparse matrix test set [6] The 2 D and 3 D grid matrices are pre ordered using nested dissection [11] which gives asymptotically optimal orderings for these problems. The Harwell Boeing matrices are pre ordered using multiple minimum degree [15], which is considered the best for most irregular sparse matrices with respect to sequential operation count and fill. Note that the floating point operation counts listed in the table are from the best known sequential sparse factorization algorithm. All Mflops numbers presented in this paper are ....

Liu, J., "Modification of the minimum degree algorithm by multiple elimination", ACM Transactions on Mathematical Software, 12(2): 127-148, 1986. 15

....intensive part of the RMCD strategy is the updating of the column degrees, just as in the minimum degree algorithms used in the context of fill reduction. In the fill reduction context, there has been considerable effort (e.g. Duff and Reid, 1983; Eisenstat et al. 1981; George and Liu, 1980a,b; Liu, 1985) spent on reducing the work required to keep track of the degrees, and recently work (e.g. Gilbert et al. 1992; Davis and Duff, 1997; Davis et al. 1996) has 17 concentrated on using approximations of (generally upper bounds on) the degrees, in order to further reduce computational ....

Liu, J. W. H., Modification of the minimum degree algorithm by multiple elimination. ACM Trans. Math. Softw., 11, 141--153 (1985).

....a Sun Microsystems Ultra 30 with the UltraSPARC II 296MHz microprocessor. For these experiments, GGCL is 5 to 7 times faster than LEDA. 5. 2 Comparison to Special Purpose Library In additionn, we demonstrate the performance of a GGCL based implementation of the multiple mininum degree algorithm [13] using selected matrices from the Harwell Boeing collection [9] and the University of Florida s sparse matrix collection [2] Our tests compare the execution time of our implementation against that of the equivalent SPARSPAK Fortran algorithm (GENMMD) 7] For each case, our implementation and ....

Liu, J. W. H. Modification of the minimum-degree algorithm by multiple elimination. ACM Transaction on Mathematical Software 11, 2 (1985), 141--153.

....jAdj Gk (x j )j) time take to compute the true degree. The true degree d c (j) jStruct(A 0 j )j = jAdj Gk (x j )j is the degree of node x j in the implicitly represented elimination graph, G k [19] If indistinguishable uneliminated nodes are present in the quotient graph (as used in [28], for example) both of these time complexity bounds are reduced, but computing the true degree still takes much more time than computing our approximate degree. We now describe how we compute our degree bound, d c (j) in an amortized time of O(jA k j j jC j j) We compute the external column ....

....algorithm [8] MUPS always attempts to preserve symmetry. It does not permute the matrix to block upper triangular form. SSGETRF is a classical multifrontal method in the Cray Research, Inc. library (version 1.1) installed on the CRAY YMP. It uses Liu s multiple minimum degree (MMD) algorithm [28] on the pattern of A A T . It includes a threshold partial pivoting test. It is not specified in the documentation, but from our results we conclude that SSGETRF always uses a maximum transversal algorithm. We base this conclusion on the observation that MUPS and SSGETRF obtain similar fill in ....

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J. W. H. Liu, Modification of the minimum-degree algorithm by multiple elimination, ACM UNSYMMETRIC-PATTERN MULTIFRONTAL METHOD 21 Trans. Math. Softw., 11 (1985), pp. 141--153.

....algorithms for semiconductor device simulations and expected main memory requirements for sparse direct methods in the 10 100 Gbyte range for a simulation with 100 000 vertices and 300 000 unknowns. However, they used only local fill in reduction methods like minimum degree based algorithms [4] and the sparse direct solver did not exploit the memory hierarchy of the computing architecture. The rapid and widespread acceptance of shared memory multiprocessors, from the desktop to the high end server, has now created a demand for parallel device and process simulation on such shared ....

J.W.H. Liu. Modification of the Minimum-Degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11(2):141--153, 1985.

....quotient graph representation, mass elimination, incomplete degree update, multiple elimination, and external degree. See [11] for a historical survey of the minimum degree algorithm. The GGCL implementation of the Minimum Degree algorithm closely follows the algorithmic descriptions of the one in [11, 16]. The implementation presently includes the enhancements for mass elimination, incomplete degree update, multiple elimination, and external degree. In particular, I create a graph representation to improve the performance of the algorithm. It is based on a templated vector of vectors. The vector ....

....on a Sun Microsystems Ultra 30 with the UltraSPARC II 296MHz microprocessor. For these experiments, GGCL is 5 to 7 times faster than LEDA. 6. 2 Comparison to Special Purpose Library In addition, we demonstrate the performance of a GGCL based implementation of the multiple minimum degree algorithm [16] using selected matrices from the Harwell Boeing collection [12] and the University of Florida s sparse matrix collection [2] Our tests compare the execution time of our implementation against that of the equivalent SPARSPAK Fortran algorithm (GENMMD) 9] For each case, our implementation and ....

J. W. H. Liu. Modification of the minimum-degree algorithm by multiple elimination. ACM Transaction on Mathematical Software, 11(2):141--153, 1985.

....columns of A.Anedge(i, j) # No accidental cancellations will occur during factorization if the numerical values in A are algebraic indeterminates. 2 Table 2 1 Algorithms that fit into the Minimum Priority family. Abbreviation Algorithm Name Primary Reference MMD Multiple Minimum Degree Liu [5] AMD Approximate Minimum Degree Amestoy, Davis and Du# [1] AMF Approximate Minimum Fill Rothberg [8] AMMF Approximate Minimum Mean Local Fill Rothberg and Eisenstat [9] AMIND Approximate Minimum Increase in Rothberg and Eisenstat [9] Neighbor Degree MMDF Modified Minimum Deficiency Ng and ....

....accounting of quotient graph features and optimizations. Most of the time is spent in the last three lines Fig. 2.2, and often they are tightly intertwined in implementations. 3. Design. To provide a basis for comparison, we briefly discuss the design and implementation characteristics of MMD [5] and AMD [1] Both implementations were written in Fortran 77 using a procedural decomposition. They have no dynamic memory allocation and implement no abstract data types in the code besides arrays. GENMMD is implemented in roughly 500 lines of executable source code with about 100 lines of com5 ....

[Article contains additional citation context not shown here]

Joseph W. H. Liu, Modification of the minimum-degree algorithm by multiple elimination, ACM Trans. Math. Software, 11 (1985), pp. 141--153.

....graph partitioning algorithm. Results from multilevel nested dissection algorithms have been obtained by Hendrickson and Rothberg (personal communication, 1994) Karypis and Kumar [64] and by Kumfert and Pothen [69] We compare three algorithms: the Multiple Minimum Degree ordering of Liu [76], an MLND algorithm due to Kumfert and Pothen [69] and MLNDW, a weighted variant of the MLND algorithm, due to Karypis and Kumar, included in MeTiS version 2.0. The multilevel nested dissection algorithms differ in the way the coarse graphs are constructed. Both variants coarsen the graph by ....

J. W. H. Liu, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. on Math. Software, 11 (1985), pp. 141-- 153.

....4. For each matrix, the table shows the parallel run time and the number of nonzeros in the Cholesky factor L of the resulting matrix for 16, 32, and 64 processors. On p processors, the ordering is computed by using nested dissection for the first log p levels, and then multiple minimum degree [19] (MMD) is used to order the submatrices stored locally on each processor. Matrix T 16 L T 32 L T 64 L BCSSTK31 1.7 5588914 1.3 5788587 1.0 6229749 BCSSTK32 2.2 7007711 1.7 7269703 1.3 7430756 BRACK2 2.9 7788096 2.5 7690143 1.8 7687988 CANT 4.4 29818759 2.8 28854330 2.2 28358362 ....

J. W.-H. Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11:141--153, 1985.

No context found.

Liu, Joseph W.H. "Modification of the Minimum-degree Algorithm by Multiple Elimination," ACM Transaction on Mathematical Software, 1985.

No context found.

J. W. H. Liu, Modification of the minimum degree algorithm by multiple elimination, ACM Trans. Math. Software, 11 (1985), pp. 141--153.

No context found.

J. W. H. Liu. Modification of the minimum-degree algorithm by multiple elimination. ACM Trans. Math. Software, 11(2):141--153, 1985.

No context found.

J. W. H. Liu, Modification of the minimum-degree algorithm by multiple elimination,ACM Trans. Math. Softw., 11(2) (Jun. 1985), pp. 141-153.

No context found.

J. W.-H. Liu. Modification of the minimum degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11:141--153, 1985.

No context found.

J. Liu. Modification of the minimum-degree algorithm by multiple elimination. ACM Trans. Math. Software, 11(2):141--153, June 1985.

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