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

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 ).

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

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

  2. Anna Urbańska, Warsaw University Determinant Let A be the n x n integer matrix. The determinant of A, det(A), is defined as Σ sgn(σ) weight(σ) n det(A) = (-1) σ where the sumranges over all permutations σof the permutation group on {1, 2, ..., n} sgn(σ) is (-1) , where k is the number of cycles in cycle decomposition of σand the weight of σ is weight(σ) = A[1,σ(1)] A[2,σ(2)] ... A[n,σ(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

  3. Anna Urbańska, Warsaw University • 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 ) 3 4

  4. Anna Urbańska, Warsaw University In this paper we: • present a special version of Mahajanand Vinay's algorithm for the case of planar graphs • our algorithm is based on a novel algebraic view of Mahajanand Vinay's algorithmintroducedin our earlier paper:a relation to a pseudo-polynomial dynamic-programming algorithm for the knapsack problem • show how to implement Mahajanand Vinay's algorithm for matrices whose graphs are planar in time O(n )withoutdivisions • present the analogous results for: • characteristic polynomial • adjoint 2.5

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