A constrained minimum spanning tree problem

In the classical general framework of the minimum spanning tree problem for a weighted graph we consider the case in which a predetermined vertex has a certain fixed degree. In other words, given a weighted graph G, one of its vertices v0 and a positive integer k, we consider the problem of finding the minimum spanning tree of G in which the vertex v0 has degree k, that is the number of edges coming out of v0. We recall that among the various methods for the solution of the unconstrained problem an efficient way to find the minimum spanning tree is based on the simple procedure of choosing one after the other an edge of minimum weight that has not be chosen yet and does not create cycles if added to the previously chosen edges. This technique is known as the \u201cgreedy algorithm\u201d. There are problems for which the greedy algorithm works and problems for which it does not. We prove that for the solution of the one degree constrained minimum spanning tree problem the classical greedy algorithm finds a right solution

Minimax passband group delay nonlinear FIR filter design without imposing desired phase response

In this paper, a nonlinear phase finite impulse response (FIR) filter is designed without imposing a desired phase response. The maximum passband group delay of the filter is minimized subject to a positivity constraint on the passband group delay response of the filter as well as a specification on the maximum absolute difference between the desired magnitude square response and the designed magnitude square response over both the passband and the stopband. This filter design problem is a nonsmooth functional ine-quality constrained optimization problem. To tackle this problem, first, the one norm functional inequality constraint of the optimization problem is approximated by a smooth function so that the nonsmooth functional inequality con-strained optimization problem is approximated as a noncon-vex functional inequality constrained optimization problem. Then, a modified filled function method is applied for find-ing the global minimum of the nonconvex optimization prob-lem. Computer numerical simulation results show that our designed nonlinear phase peak constrained FIR filter could achieve lower minimum passband group delay than those of existing designs