Lower bound on the channel routing problems

A detailed analysis of how grid points are used in routing nets in the Manhattan model is presented. A new lower bound on the channel width that asymptotically matches the known upper bound d_m+O(f) and, therefore, is existentially tight is established. The bound is established by introducing the notion of `holes´ and developing arguments on the supply and the demand of such a commodity inside a chosen region. Realizing that the hole is the grid point where only one nontrivial net is allowed to jog, it is seen that the more nets use grid points separately (without sharing) the stronger the lower bound is