Application of Graph Separators to the Effcient Division-Free Computation of Determinant

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




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 )

33

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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

Application of Graph Separators to the Effcient Division-Free Computation of Determinant

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