application of graph separators to the effcient division-free computation of determinant
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Application of Graph Separators to the Effcient Division-Free Computation of Determinant Anna Urbańska Institute of Computer Science Warsaw University, Poland. Determinant. Let A be the n x n integer matrix. The determinant of A , det(A) , is defined as. Σ. sgn( σ ) weight( σ ). - PowerPoint PPT PresentationTRANSCRIPT
Application of Graph Application of Graph Separators to the Separators to the
EffcientEffcientDivision-Free Division-Free
Computation of Computation of DeterminantDeterminant
Anna UrbańskaAnna Urbańska
Institute of Computer ScienceInstitute of Computer Science
Warsaw University, PolandWarsaw University, Poland
Application of Graph Separators to the Effcient Division-Free Application of Graph Separators to the Effcient Division-Free
Computation of DeterminantComputation of Determinant
Anna Urbańska, Warsaw UniversityAnna Urbańska, Warsaw University
where the sumwhere the sum ranges over all permutations ranges over all permutations σσ of the permutation group on of the permutation group on {1, {1, 2, ..., n}2, ..., n}
sgn(sgn(σσ)) is is ((--1)1) , where , where k k is the number of cycles in cycle decomposition of is the number of cycles in cycle decomposition of σσ and theand the
weight weight of of σσ is is weight(weight(σσ) = A[1,) = A[1,σσ(1)] A[2,(1)] A[2,σσ(2)] ... A[n,(2)] ... A[n,σσ(n)](n)]
Σσ
sgn(sgn(σσ) weight() weight(σσ) ) n n
det(A) = (-1) det(A) = (-1)
Let Let AA be the be the n x nn x n integer matrix. The integer matrix. The determinantdeterminant of of A A, , det(A)det(A), is , is defined asdefined as
k
DeterminantDeterminant
Planar GraphsPlanar Graphs
Planar graphPlanar graph is a is a graph which can be which can be embedded in the plane, i.e., it can in the plane, i.e., it can be be drawn on the plane in such a way that its edges intersect only at their drawn on the plane in such a way that its edges intersect only at their endpoints. endpoints.
Each planar graph has a small Each planar graph has a small separatorseparator
V 1 V 2S
Each planar graph has only Each planar graph has only O(n)O(n) edgesedges
Application of Graph Separators to the Effcient Division-Free Application of Graph Separators to the Effcient Division-Free
Computation of DeterminantComputation of Determinant
Anna Urbańska, Warsaw UniversityAnna Urbańska, Warsaw University
Gaussian eliminationGaussian elimination is the classical algorithm for computing the is the classical algorithm for computing the determinant determinant
It needs It needs O(n )O(n ) additionsadditions subtractionssubtractions multiplicationsmultiplications divisionsdivisions
Determinant is the sum of Determinant is the sum of n!n! products - it can be computed products - it can be computed without divisionswithout divisions
Avoiding divisions seems attractive when working over a commutative ring Avoiding divisions seems attractive when working over a commutative ring which is not a fieldwhich is not a field
integersintegers polynomialspolynomials rational rational more complicated expressionsmore complicated expressions
M. Mahajan and V. Vinay, M. Mahajan and V. Vinay, Determinant: Combinatorics, Algorithms, and Determinant: Combinatorics, Algorithms, and ComplexityComplexity, 1997, time , 1997, time O(n )O(n )
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Application of Graph Separators to the Effcient Division-Free Application of Graph Separators to the Effcient Division-Free
Computation of DeterminantComputation of Determinant
Anna Urbańska, Warsaw UniversityAnna Urbańska, Warsaw University
In this paper we:In this paper we:
present a special version of Mahajanpresent a special version of Mahajan and Vinay's algorithm for the case of and Vinay's algorithm for the case of planar graphs planar graphs
our algorithm is our algorithm is based on based on a a novel algebraic view of Mahajannovel algebraic view of Mahajan and Vinay's and Vinay's algorithmalgorithm introducedintroduced in our earlier paper:in our earlier paper: a relation to a pseudo-a relation to a pseudo-polynomial dynamic-programming algorithm for the knapsack problempolynomial dynamic-programming algorithm for the knapsack problem
show how to implement Mahajanshow how to implement Mahajan and Vinay's algorithm for matrices and Vinay's algorithm for matrices
whose graphs are planar in time whose graphs are planar in time O(n O(n )) withoutwithout divisionsdivisions
present the analogous results for: present the analogous results for: characteristic polynomialcharacteristic polynomial adjointadjoint
22.5.5