Application of Graph Separators to the Effcient

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• Application of Graph Separators to the Effcient
• Division-Free Computation of Determinant
• Anna Urbanska
• Institute of Computer Science
• Warsaw University, Poland

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Determinant
Let A be the n x n integer matrix. The
determinant of A, det(A), is defined as
S
sgn(s) weight(s)

n
det(A) (-1)
s
where the sum ranges over all permutations s of
the permutation group on 1, 2, ..., n sgn(s) is
(-1) , where k is the number of cycles in cycle
decomposition of s and the weight of s is
weight(s) A1,s(1) A2,s(2) ... An,s(n)
k
Planar Graphs Planar graph is a graph
which can be embedded in the plane, i.e., it can
be drawn on the plane in such a way
that its edges intersect only at their endpoints.

• Each planar graph has only O(n) edges

• Each planar graph has a small separator

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• Gaussian elimination is the classical algorithm
  for computing the determinant
• It needs O(n )
• additions
• subtractions
• multiplications
• divisions
• Determinant is the sum of n! products - it can be
  computed without divisions
• Avoiding divisions seems attractive when working
  over a commutative ring which is not a field
• integers
• polynomials
• rational
• more complicated expressions
• M. Mahajan and V. Vinay, Determinant
  Combinatorics, Algorithms, and Complexity, 1997,
  time O(n )

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• In this paper we
• present a special version of Mahajan and Vinay's
  algorithm for the case of planar graphs
• our algorithm is based on a novel algebraic view
  of Mahajan and Vinay's algorithm introduced in
  our earlier paper a relation to a
  pseudo-polynomial dynamic-programming algorithm
  for the knapsack problem
• show how to implement Mahajan and Vinay's
  algorithm for matrices whose graphs are planar in
  time O(n ) without divisions
• present the analogous results for
• characteristic polynomial
• adjoint

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