OPTIMIZATION SEMINAR



The Expected Time for a Widely Used Min-Plus Algebra Algorithm is Infinite




Dave Fisher
University of Colorado at Denver
Department of Mathematics

Tuesday, Dec. 2, 1997, 12:00 noon
CU-Denver Bldg., Room 626


ABSTRACT:

In min-plus algebra, minimization replaces addition, and addition replaces multiplication. For example, let A be the matrix 
                                    2 0 2
                                A = 4 2 3
                                    3 0 3
By iterating A^n = A A^{n-1} where matrix multiplication is done in min-plus algebra, we get 
                    4 2 3            6 3 5           7 5 6
              A^2 = 6 3 5      A^3 = 7 5 6     A^4 = 9 6 8
                    4 2 3            6 3 5           7 5 6
Note A^4 = 3 A^2 (using "scalar addition") allowing us to conclude that A^n= 3 A^(n-2) for all n >= 4. It can be shown that we can always find a recursion with this procedure.

How long does it take to find a recursion? In many applications, the recursion is found with only a few matrix multiplications (sublinear in the matrix size). However when coefficients of a 2x2 matrix are chosen from a uniform distribution, the expected time to find a recursion is infinite.

This talk will start with an introduction to Min-Plus Algebra (with several applications) and conclude with calculations verifying the title result.