J.R. Gilbert and R.E. Tarjan, The analysis of a nested dissection algorithm, Numer. Math. 50 (4), 377-404, (1987).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....positive diagonal entries. For genera] sparse linear systems, the bounds of Gaussian elimination are at least 12(n2) However, for sparse linear systems that have sparsity graphs with separators of size s(n) the work bound to construct a sparse LL factorization using nested dissection orderings [10,11] is O(n s(n) For example, for matrices with planar sparsity graphs, where s(n) O(v) the work bounds for sparse LL factorization are O(n 2) and for dimension grids (that is an n node grid graph in d dimensions) where s(n) n d x d, the work bounds for sparse LL factorization are ....

....In comparison, previously Vaidya [1] gave for this case a bound of O (n(r log n) for 0. Also note that these iterative methods are much faster than direct methods (for example nested dissection) that compute a sparse LLT factorization; in contrast the work bound of nested dissection [10,11] is lower bounded as f(n 2) for SPD matrices with planar sparsity graphs. 5. PROOF OF OUR RESULTS FOR SPARSE MATRICES WITH NO MAGNITUDE BOUNDS 5.1. Reduction from Sparse to Bounded Degree Matrices Let A be a symmetric nonsingular DD (and thus SPD) n x n matrix with m = O(n) nonzeros. The ....

J.R. Gilbert and R.E. Tarjan, The analysis of a nested dissection algorithm, Numer. Math. 50 (4), 377-404, (1987).

.... For general sparse linear systems, the bounds of Gaussian elimination are at least Omega Gamma n 2 ) However, for sparse linear systems that have sparsity graphs with separators of size s(n) the work bound to construct a sparse LL T factorization using nested dissection orderings [LRT 79, GT 87] is O(n s(n) For example, for matrices with planar sparsity graphs, where s(n) O( p n) the work bounds for sparse LL T factorization are O(n =2 ) and for d Gammadimension grids (that is an n node grid graph in d dimensions) where s(n) n d Gamma1 d ; the work bounds for ....

.... Vaidya [V 91] gave for this case a bound of O( n( log n) 1 fl ) for fl 0: Also note that these iterative methods are much faster than direct methods (for example nested dissection) that compute a sparse LL T factorization; in contrast the work bound of nested dissection [LRT 79, GT 87] is lower bounded as Omega Gamma n =2 ) for SPD matrices with planar sparsity graphs. 5 Proof of our Results For Sparse Matrices with No Magnitude Bounds 5.1 Reduction from Sparse to Bounded Degree Matrices Let A be a symmetric nonsingular DD (and thus SPD) n Theta n matrix with m = O(n) ....

J.R. Gilbert and R.E. Tarjan, The analysis of a nested dissection algorithm, Numer. Math. 50, 4 (1987), 377--404.

....in O(n) time. Other interesting results include separator theorems for the class of graphs of bounded genus [6, 8, 1] the class of graphs of excluded minor [2] and classes of geometric graphs [17] Separator theorems have applications in solving eciently large sparse systems of linear equations [14, 9], for developing algorithms for VLSI layout design [4, 12] for shortest path problems [7] in parallel computing [10] and in computational complexity [16] A class of problems for whose solutions separator theorems are especially well suited is data partitioning and load balancing for parallel ....

John R. Gilbert and Robert E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377-404, 1987.

....induced simple chordless cycle of length 4 or more. In the minimum chordal completion problem we are required to construct a chordal graph with the fewest edges. This bounds the space required to solve a sparse linear system. This problem has received considerable attention from researchers; see [9, 15, 17, 18, 19, 21, 22, 25, 29, 37, 36, 41, 42, 50, 54, 57], amongst many others. Most of these approaches use one of two methods, minimum degree heuristic (see [37, 42, 54] for example) or nested dissection (see [17, 19, 36] Hybrid algorithms (see [2, 27] are also used. The space requirement is not the only measure of interest in solving sparse ....

....inputs b. The minimum number of rounds required if rows can be eliminated in parallel, the the elimination height, is also important in this context. The problem of minimizing operation count has not been studied as extensively as the problem of minimizing ll in. Some results are shown in [26, 36, 25]. Minimizing the elimination height has been studied with regard to eliminating more than one row in parallel. The elimination tree was rst de ned in [51] although several algorithms prior to this paper used the idea. A considerable number of other heuristics have been proposed for this problem ....

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J. R. Gilbert and R. E. Tarjan. \The Analysis of a Nested Dissection Algorithm". In Numerische Mathematik vol 50, pages 377-404, (1987).

....elimination may introduce fill in solving sparse linear systems. The amount of fill depends on the ordering of the rows and columns of the sparse matrix. Such an ordering is called an elimination ordering. The problem of finding an optimal ordering is known to be NP hard. Nested dissection [1, 12, 13, 16, 26] is one of the provably good methods for finding an elimination ordering for sparse matrices. The nested dissection algorithm divides the graph of a linear system by first finding a small separator, that is, a set of vertices whose removal divides the rest of the graph into two disjoint subgraphs ....

....the inverse of the Ackerman function. They also showed that with the elimination tree one can count fill (and get al..l the vertex degrees in the filled graph) in another O(n#(n) time. For proofs of these statements and many other fascinating facts about elimination trees and filled graphs consult [14, 15, 16, 28, 29]. Therefore, we have the following corollary. Corollary 5.1. If the graph of a symmetric sparse matrix A is a well shaped mesh, then a provably good nested dissection ordering of A, its elimination tree, and its fill information can be found in O(n log log n) time. 5.2. 3D point location. One ....

J. R. Gilbert and R. E. Tarjan, The analysis of a nested dissection algorithm, Numer. Math., 50 (1987), pp. 377--404.

.... This problem can be solved in O(nm) time [12] Last problem could be solved for planar graphs in linear time [3] Another approach is the nested dissection [1] It is known that this problem can be solved more efficiently than for graphs in general for planar graphs and for bounded degree graphs [9]. In so far, it is reasonable to develop also a minimal elimination ordering algorithm that runs for bounded degree graphs more efficiently than for graphs in general. We show the following. Theorem 1 For bounded degree graphs, a minimal elimination ordering can be determined in O(n)ff(n) time. ....

J. Gilbert, R. Tarjan, The Analysis of a Nested Dissection Algorithm, Numerische mathematik 50 (1987), pp. 377-404.

.... i ) 2 In theory as well as in practice, the condition of a family of f bisectors can be relaxed to the condition of a family of f separators, that is, every subgraph of G of s vertices has a ffi bisection, for some constant ffi , of cost bounded by f(s) Lipton and Tarjan [21] and Gilbert [14] showed that if a graph has a family of f separators, then it has a family of O(f) bisectors. The following are some classes of graphs that have a family of small separators and hence have a family of small bisectors as well. 6 H. D. SIMON AND S. H. TENG ffl Planar graphs [21] have a family of ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50(4):377--404, 1987. 10 H. D. SIMON AND S. H. TENG

....and the communication cost is minimized, where the communication costed is modeled by the total number (or weights) of edges whose endpoints are in different components. Thus, the partitioning problem has applications in both direct and iterative methods for sparse linear and non linear systems [5, 6, 13, 14, 12, 15, 20, 21, 22, 24, 27]. It is also an important subproblem in run time and compiler time optimization for high performance parallel processing [4, 6] and for VLSI layout [18] The most commonly used approach for p way partitioning is to recursively bisect the graph 1 , i.e. it first optimally divides the graph ....

....i ) The condition of family of f(n) bisection property can be relaxed to the following condition of family of f(n) separator property: every subgraph of G of m vertices has an f(m) edge separator that ffi splitts, for some constant ffi. Using an argument due to Lipton and Tarjan [19] and Gilbert [12], we can show that if a graph has a family of f(m) separator, then it has a family of O(f(m) bisectors. Here is a list of graphs that have a family of small separators (and hence bisectors) 7 ffl Planar graph [19] has a family of O( p m) separators. ffl Bounded genus graph [10] has a family ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50(4):377--404, 1987.

....edge (j; k) and for each i 2 BCol(k) there is an oriented edge (k; i) This graph is the quotient graph of G by the partition P (see figure 2. 2) We can associate the above block Cholesky factorization algorithm with partitions induced by a nested dissection ordering of the vertices of G [18, 21]. By using the partition of separators, we can exploit a first level of parallelism which is induced by sparsity; the ordering straightforwardly leads to a large block elimination tree. Now, we can take 1 array = 2 graph commutative 3 function S[i:1 . dimension] x: node) node; 4 var id : ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377--404, 1987.

.... i ) In theory as well as in practice, the condition of a family of f bisectors can be relaxed to the condition of a family of f separators; that is, every subgraph of G of s vertices has a # bisection, for some constant #, of cost bounded by f(s) Lipton and Tarjan [21] and Gilbert and Tarjan [14] showed that if a graph has a family of f separators, then it has a family of O(f) bisectors. The following are some classes of graphs that have a family of small separators and hence have a family of small bisectors as well. Planar graphs [21] have a family of O( # n) separators. Bounded ....

J. R. GILBERT AND R. E. TARJAN, The analysis of a nested dissection algorithm, Numer. Math., 50 (1987), pp. 377--404.

....linear time algorithm for finding an O Gamma k 1=d n 1 Gamma1=d Delta separator for a k nearest neighborhood graph in d dimensions. 1 Introduction Many problems in scientific computing and computational geometry require efficiently computing a small separator of the underlying graph [6, 7, 9, 10, 11, 18, 20, 28, 29, 30]. By small separator we mean a relatively small subset of vertices whose removal divides the rest of the graph into two disconnected pieces of approximately equal size [19] In a series of papers [25, 27, 22, 26, 23, 30, 34] Miller, Teng, Thurston, and Vavasis have developed a geometric ....

....is dual in our original disk system to a great sphere which cuts O(k 1=d n 1 Gamma1=d ) disks and ffi splits the disks, so it is the separator we require. 2 5 Applications The sphere separator theorem has many applications [30] especially in scientific computing and computational geometry [9, 10, 11, 13, 18, 28]. In this section, we focus on the application of the deterministic linear time sphere separator algorithm in computational geometry. 5.1 Separators for geometric graphs As shown in [26, 30] the sphere separator results lead to vertex separators for various geometric graphs including the ....

Gilbert, J. R. and R. E. Tarjan. "The analysis of a nested dissection algorithm". Numerische Mathematik, 50(4):377--404, 1987.

....data structures and special solution methods. The methods are either direct, that is, are modifications of Gaussian elimination with some special policies of elimination ordering that preserve sparsity during the computation (notably, Markowitz rule and nested dissection, George and Liu 1981, Gilbert and Tarjan 1987, Lipton et al. 1979, Pan 1993] or various iterative algorithms. The latter ones usually rely either on computing Krylov sequences [Greenbaum 1997] or on multilevel or multigrid techniques [McCormick 1987, Pan and Reif 1992, Fiorentino and Serra 1996] specialized for solving linear systems that ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

....to solve an n Theta n linear system defined by planar constraints. Note that this area bound contrasts sharply with the exponential lower bound of Eades and Garvan [17] for the area of Tutte drawings under the vertex resolution rule. Incidentally, there are fairly simple separator based methods [24, 30, 31] for achieving an O(n 1:5 ) bound for P (n) while much more sophisticated methods allow one to achieve an O(M(n 1=2 ) bound, where M(n) is the time needed to multiply two n Theta n matrices (the current best bound for M(n) is O(n 2:375 ) 10] Thus, by our template, we have the ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

....to solve an n Theta n linear system defined by planar constraints. Note that this area bound contrasts sharply with the exponential lower bound of Eades and Garvan [20] for the area of Tutte drawings under the vertex resolution rule. Incidentally, there are fairly simple separatorbased methods [25, 37, 38] for achieving an O(n 1:5 ) bound for P (n) while much more sophisticated methods allow one to achieve an O(M (n 1=2 ) bound, where M (n) is the time needed to multiply two n Theta n matrices (the current best bound for M (n) is O(n 2:375 ) 11] Thus, by our template, we have the ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

....the benefits of ideal efficiency and no precedence edges. Thus the above bound is overly optimistic. Even so, it is useful for deriving heuristic bounds to performance in some irregular graphs. e.g. Gilbert and Tarjan study nested dissection algorithms to solve sparse systems on planar graphs [14], where a problem of size N is divided into d = 2 subproblems, where each part is no bigger than 2N=3. No matter what strategy we use in the upper levels, we only need to go down to roughly (ffl) 1 lg(3=2) 1 Gamma log ffl log P ) 1:71(1 Gamma log ffl log P ) levels before the ....

J. Gilbert and R. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377--404, 1987.

....theorem 1, G w =Vw1 is a chordal extension of Gw1 , and G w =Vw2 is a chordal extension of Gw2 . Therefore, we have mw1 mw2 mw (1) Now, following [11] we note K the chordal graph obtained from the elimination of the nodes of G in the computed nested dissection order. It is easy to show [9] that if fa,bg is an edge of K, and ff Gamma1 (a) ff Gamma1 (b) then there exists two words u and v such that b 2 S u and a 2 S uv . Now we count the edges whose higher numbered node is in Sw . Each such edge fx; yg where x 2 Sw and y 2 Swt (t 2 A ) comes from an edge fx; zg of G, ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

....some special data structures and special solution methods. The methods are either direct, that is, are modifications of Gaussian elimination with some special policies of elimination ordering that preserve sparsity during the computation (notably, Markowitz rule and nested dissection, GL81] GT87] LRT79] Pan93] or various iterative algorithms. The latter algorithms rely either on computing Krylov sequences [Saa95] or on multilevel or multigrid techniques [McC87] PR92] specialized for solving linear systems that arise from discretization of PDEs. An important particular class of ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

....as e M e S e M e D 1 e 1 1 oeP L N ; in the asymptotic model using P processors. We can also derive some heuristic bounds to the performance gap in some irregular graphs. e.g. Gilbert and Tarjan study nested dissection algorithms to solve sparse systems on planar graphs [62], where a problem of size N is divided into d = 2 subproblems, where each part is no bigger than 2N=3. No matter what strategy we use in the upper levels, we only need to go down roughly (ffl) lg(ffl=P ) lg(2=3) 1:71(1 Gamma log ffl log P ) lg P levels before the largest leaf is of size ....

J. Gilbert and R. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377--404, 1987.

.... of nested dissection [10] which does not include the separators in the recursive call, is actually sufficient for some special classes of the p n separable graphs: planar graphs, graphs of bounded genus or bounded excluded minor, and two dimensional finite element meshes of bounded aspect ratio [16]. As a result, our analysis in section 4.1 for the model problem uses George s nested dissection ordering. 4.1. The Model Problem. If George s nested dissection ordering [10] is applied to the model problem, then each internal node has at most two children. To count the number of nonzeros in Y i ....

J. R. Gilbert and R. E. Tarjan, The analysis of a nested dissection algorithm, Numerische Mathematik. 50(1987), pp. 377--404.

....to solve an n Theta n linear system defined by planar constraints. Note that this area bound contrasts sharply with the exponential lower bound of Eades and Garvan [20] for the area of Tutte drawings under the vertex resolution rule. Incidentally, there are fairly simple separator based methods [25, 37, 38] for achieving an O(n 1:5 ) bound for P (n) while much more sophisticated methods allow one to achieve an O(M (n 1=2 ) bound, where M (n) is the time needed to multiply two n Theta n matrices (the current best bound for M (n) is O(n 2:375 ) 11] Thus, by our template, we have the ....

J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numer. Math., 50:377--404, 1987.

.... [7] which does not include the separators in the recursive call, is actually sufficient for all of the specific classes of p n separable graphs named above: planar graphs, graphs of bounded genus or bounded excluded minor, and two dimensional finite element meshes of bounded aspect ratio [17]. The labeling of the vertices of G (A) yields a column permutation P for A. Now we count the nonzeros in the factors R, Q, and H of AP . We shall assume that the rows of AP have been permuted to make the diagonal nonzero; but to simplify notation we won t mention this row permutation ....

John R. Gilbert and Robert Endre Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377--404, 1987.

....94304. 1 Introduction A separator in a graph is a small set of vertices or edges whose removal divides the graph into pieces of approximately equal size. Graph separators have applications in partitioning for parallel computation [8] divide andconquer algorithms for sparse Gaussian elimination [7, 15], VLSI layout [13] geometric algorithms [17, 22] and computational complexity [4, 14] A class of graphs is said to satisfy an f(n) separator theorem (for vertices) if there are constants ff 1 and fi 0 such that every n vertex graph in the class has a set, called a separator, of at most ....

John R. Gilbert and Robert Endre Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377--404, 1987.

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J. Gilbert, R. Tarjan, The Analysis of a Nested Dissection Algorithm, Numerische Mathematik 50 (1987), pp. 427-449.

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J. Gilbert, R. Tarjan, The Analysis of a Nested Dissection Algorithm, Numerische Mathematik 50 (1987), pp. 427-449.

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J. Gilbert, R. Tarjan, The Analysis of a Nested Dissection Algorithm, Numerische Mathematik 50 (1987), pp. 427-449.

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