Minimum confluent cut of a directed graph and its application to the block replacement problem in VLSI design

Assume a situation in which a block is required to change its relative position at the step where placement and routing in the VLSI layout design has been completed. We formulate a problem of minimizing the number of steps for modification, assuming that we need a step of one unit for rerouting one channel. Then we obtain a solution. For routing, we assume a channel routing style performed channel by channel on the grid determined by a design rule, and that a block movement is required by such slight increase of the width that a new wiring requirement is generated between blocks. The problem is reduced to that of finding an A‐confluent cut with total minimum weights for an arbitrarily specified edge set A in a directed graph with weighted edges. By an A‐confluent cut we mean a cut such that the orientation of each constituent edge is the same with respect to edges in A. Although it is a fundamental notion in graph theory, it appears to have been given little consideration.