Equivalence of the left-edge and column-by-column channel-routing algorithms

In the absence of vertical constraints, the left-edge algorithm is known to give an optimal solution to the channel-routing problem. Recently, new wiring algorithms using a column-by-column approach have been proposed, which are capable of generating all the optimal solutions. In the letter the proof is given for the identity of the result of the left-edge algorithm and the column-by-column algorithm (implementing a specific criterion). Some ways of exploiting the flexibility of the column-by-column algorithms are also discussed.