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

2. Application of Graph Separators to the Effcient Division-Free Computation of Determinant Anna Urbańska, Warsaw University where the sumranges over all permutations σ of the permutation group on {1, 2,..., n} where the sum ranges over all permutations σ of the permutation group on {1, 2,..., n} sgn( σ ) is (-1), where k is the number of cycles in cycle decomposition of σ and the weight of σ is weight( σ ) = A[1, σ (1)] A[2, σ (2)]... A[n, σ (n)] Σ σ sgn( σ ) weight( σ ) n det(A) = (-1) Let A be the n x n integer matrix. The determinant of A, det(A), is defined as k Determinant Planar Graphs Planar graph is a graph which can be embedded in the plane, i.e., it can be Planar graph is a graph which can be embedded in the plane, i.e., it can begraphembeddedgraphembedded drawn on the plane in such a way that its edges intersect only at their endpoints. drawn on the plane in such a way that its edges intersect only at their endpoints.  Each planar graph has a small separator  Each planar graph has only O(n) edges

3. Application of Graph Separators to the Effcient Division-Free Computation of Determinant 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. Application of Graph Separators to the Effcient Division-Free Computation of Determinant Anna Urbańska, Warsaw University 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 2.5