Construction of non-intersecting colored flows through a planar cellular figure

we give a linear time algorithm which, given a simply connected figure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed flow lines which match inputs and outputs according to the colors, in such a way that each edge of any cell is crossed by at most one line. The main tool is the notion of height function, previously introduced for the tilings. This notion appears as the extension of the notion of potential of a flow in a planar graph.