Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings

Agrawal, Ajit; Klein, Philip N.; Ravi, R.

doi:10.1007/978-1-4613-8369-7_2

Cited by 40 publications

(35 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Chung and Mumford [8] proved that every planar, and more generally, H-minor-free, n-vertex graph has a fill-in with O(n log n) edges, thus yielding an O(n log n)-approximation on these classes of graphs. Agrawal et al [1] gave an algorithm with the approximation ratio O(m 1.25 log 3.5 n/k + √ m log 3.5 n/k 0.25 ), where m is the number of edges and n the number of vertices in the input graph. For graphs of degree at most d, they obtained a better approximation factor O((nd + k) √ d log 4 n)/k).…”

Section: That Ismentioning

confidence: 99%

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

2013

View full text Add to dashboard Cite

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k 3/2 ) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log k) on planar graphs and O( √ k log k) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs. ACM Subject Classification G.2.2 Graph Theory -Graph Algorithms Keywords and phrases IntroductionA graph is chordal (or triangulated) if every cycle of length at least four has a chord, i.e. an edge between nonadjacent vertices of the cycle. In the Minimum Fill-in problem (also known as Minimum Triangulation and Chordal Graph Completion) the task is to check if at most k edges can be added to a graph such that the resulting graph is chordal. That isMinimum Fill-in Input: A graph G = (V, E) and a non-negative integer k.This is a classical computational problem motivated by, and named after, a fundamental issue arising in sparse matrix computations. During Gaussian eliminations of large sparse matrices, new non-zero elements -called fill -can replace original zeros, thus increasing storage requirements, the time needed for the elimination, and the time needed to solve the system after the elimination. The problem of finding the right elimination ordering * The research of Fedor V.

show abstract

Section: That Ismentioning

confidence: 99%

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

2013

View full text Add to dashboard Cite

show abstract

“…Approximation algorithms that incur fill within a polylog factor of the optimum fill have been designed by Agrawal, Klein and Ravi [1]; but since it involves finding approximate concurrent flows with uniform capacities, it is an impractical approach for large problems. A more recent approximation algorithm, due to Natanzon, Shamir and Sharan [57], limits fill to within the square of the optimal value; this approximation ratio is better than that of the former algorithm only for dense graphs.…”

Section: The Elimination Gamementioning

confidence: 99%

“…Once the elimination tree is computed, this algorithm can be [1][2][3][4][5][6][7][8] implemented in O(n + e) time, where e ≡ |E| is the number of edges in the original graph G(A). THEOREM 1.6 [56] A vertex i is the first node of a fundamental supernode if and only if i has two or more children in the elimination tree T , or i is a leaf of some row subtree of T .…”

Section: Supernodesmentioning

confidence: 99%

“…Robust algorithms adapt the factorization to complete [1][2][3][4][5][6][7][8][9][10][11][12] with the core memory available by performing the data movement and computations at a smaller granularity when necessary. They partition the submatrices corresponding to the supernodes and stacks used in the factorization into smaller units called panels to ensure that the factorization completes with the available memory.…”

Section: Scheduling Out-of-core Factorizationsmentioning

confidence: 99%

“…Given an edge (i, j) directed from i to j, we will call i the predecessor of j, and j the successor of i. The elimination ordering must eliminate vertices in a topological [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] ordering of the DAG such that all predecessors of a vertex must be eliminated before it can be eliminated. The requirement that each matrix in the product form of the inverse must have the same nonzero structure as the corresponding columns in the factor is expressed by the fact that the subgraph corresponding to the matrix should be transitively closed.…”

Section: Design Of Efficient Algorithms With Clique Treesmentioning

confidence: 99%

See 2 more Smart Citations

Elimination Structures in Scientific Computing

Pothen¹,

Toledo²

2004

Handbook of Data Structures and Applications

View full text Add to dashboard Cite

Wilkinson's inertia‐revealing factorization and its application to sparse matrices

Druinsky

Carlebach

Toledo

2017

Numerical Linear Algebra App

View full text Add to dashboard Cite

SummaryWe propose a new inertia-revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof-of-concept implementation, and present experimental results, studying the method's numerical stability and performance.

show abstract

Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings

Cited by 40 publications

References 32 publications

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Elimination Structures in Scientific Computing

Wilkinson's inertia‐revealing factorization and its application to sparse matrices

Contact Info

Product

Resources

About