D. Rose and R. Tarjan. Algorithmic Aspects of Vertex Elimination on Directed Graphs. SIAM J. Appl. Math., 34(1):176--197, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....before performing the factorization. Reducing fill reduces the amount of memory that the factorization uses and the number of floating point operations that it performs. There is no e#cient algorithm for finding an optimal ordering. This problem has been shown by Rose and Tarjan to be NP complete[22]. There are, however, several classes of algorithms that work well in practice, like minimum degree orderings and vertex separator based orderings. Minimum degree is a greedy heuristic which, at each elimination step, eliminates the vertex with the fewest uneliminated neighbors, that is the vertex ....

D. J. Rose and R. E. Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM Journal on Applied Mathematics, 34:176--197, 1978.

....elimination trees, by showing that the symmetric structure theory of elimination trees can be obtained as special cases of the elimination dag results. In Section 5, we apply elimination dags to sparse symbolic LU factorization. We briefly review two algorithms Fill1 and Fill2 by Rose and Tarjan [12] that compute fill for sparse unsymmetric matrices. We then formulate a new symbolic factorization scheme based on the pair of elimination dags and compare its performance with that of Fill1 and Fill2. Our experimental results show that the new algorithm performs much better than either on ....

....directed graph G(A) from vertex i to vertex j, then there is some 1 k n such that the directed graph G(U ) has a path from vertex i to vertex k, and the directed graph G(L) has a path from vertex k to vertex j. Proof. This can be proved by induction using the path lemma of Rose and Tarjan [12], which characterizes G(L U ) in terms of G(A) Another proof, however, is to interpret the equation A Gamma1 = U Gamma1 L Gamma1 in terms of paths and transitive closures. Theorem 3.3 implies that G (A) G(A Gamma1 ) G(U Gamma1 L Gamma1 ) G(U Gamma1 ) Delta G(L ....

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D. J. Rose and R. E. Tarjan. Algorithmic aspects of vertex elimination of directed graphs. SIAM Journal on Applied Mathematics, 34:176--197, 1978.

....nonzero diagonal, and the graph in question is the directed graph of A. The square matrix L U Gamma I represents the entire factorization. Not all nonsingular matrices have LU factorizations without pivoting. In a later subsection we consider factorization with partial pivoting. Theorem 4. 1 ([36]) Let a structure G(A) be given, with nonzero diagonal elements. i) If values are chosen for which A has an LU factorization as above, then G(L U Gamma I) G (A) ii) There exist values for the nonzeros of A such that G(L U Gamma I) G (A) Rose and Tarjan [36] gave an ....

....Theorem 4.1 ( 36] Let a structure G(A) be given, with nonzero diagonal elements. i) If values are chosen for which A has an LU factorization as above, then G(L U Gamma I) G (A) ii) There exist values for the nonzeros of A such that G(L U Gamma I) G (A) Rose and Tarjan [36] gave an algorithm for computing G (A) from A in O(nm) time, where A is n by n with m nonzeros. They also showed that G (A) can be computed in time asymptotically the same as that to compute G (A) so a faster algorithm to compute G (A) would give a faster algorithm to compute ....

Donald J. Rose and Robert Endre Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM Journal on Applied Mathematics, 34:176--197, 1978.

....Gaussian elimination without pivoting. We now briefly review a graphtheoretic model of LU factorization without row or column interchanges. The undirected version of this model is due to Parter [27] and was developed extensively by Rose [30] the directed version was developed by Rose and Tarjan [31]. George and Liu [11] is a good source for the undirected model. Gilbert [14] surveys these and related results. If G = G(A) is a directed or undirected graph, we define the deficiency of a vertex v of G as the set of edges fhr; ci : v 2 AdjG (r) c 2 AdjG (v) and c = 2 AdjG (r)g : The ....

....(A) in terms of paths in the graph of A, without actually computing all the elimination graphs. In the following theorem, the paths can be interpreted as directed paths for nonsymmetric matrices and either directed or undirected paths for symmetric matrices. Lemma 2. 11 (Rose, Tarjan, and Lueker [31, 32]) Let G be a directed or undirected graph whose vertices are the integers 1 through n, and let G be its filled graph. Then hx; yi is an edge of G if and only if there is a path in G from x to y whose intermediate vertices are all smaller than min(x; y) Paths from x to y whose ....

Donald J. Rose and Robert Endre Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM Journal on Applied Mathematics, 34:176--197, 1978.

.... we have described elsewhere [11] computes structures identical to those of the classic algorithm, but operates by conducting breadth first searches in the underlying graph of G(A T L ) S G(AU ) This algorithm arises from a theorem, cited below, which is an extension of the fill path theorem [12], originally developed in the context of complete factorizations. The fill path theorem states that fill edges can only exist between vertices joined by a fill path. Definition 1 A fill path is a path joining two vertices v and w, all of whose interior vertices are numbered lower than the minimum ....

....graph of the factor F and the matrix A be identical. This constraint permits the determination of all communication patterns prior to actual factorization phases, as discussed below. Within each subdomain we require that interior nodes be ordered before boundary nodes. By the fill path theorem [12], this ensures that, if an edge f i;j arises during factorization and crosses subdomain boundaries, then nodes i and j must have been boundary nodes in the graph of A. In other words, ordering interior nodes first in the graph of A ensures that interior nodes can not be converted to boundary nodes ....

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D. J. Rose and R. E. Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM J. Appl. Math., 23(1):176--197, 1978.

....the L(F 2 i ) rows are all zero and that U(F i ) Gamma U(F 1 i ) U(F 2 i ) The breaking of F i into two independent diagonal blocks implies that the matrix was not in block upper triangular form. 2 A more elegant proof of Theorem 7. 32 is possible using the results of Rose and Tarjan [128, 127] but a different graph formulation of the matrix would be required. The results of Theorem 7.32 are very powerful in that they provide for a number of simplifications for lost pivot recovery, which are defined in the following theorems. Theorem 7.33 (E T Sufficient For Lost Pivot Recovery) ....

D. J. Rose and R. E. Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM J. Appl. Math., 34(1):176--197, 1978.

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D. Rose and R. Tarjan. Algorithmic Aspects of Vertex Elimination on Directed Graphs. SIAM J. Appl. Math., 34(1):176--197, 1978.

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D. J. Rose and R. E. Tarjan, Algorithmic aspects of vertex elimination on directed graphs, SIAM J. Appl. Math., 23 (1978), pp. 176-197.

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