Algebraic Tools for the Construction of Colored Ows with Constraints Algebraic Tools for the Construction of Colored Ows with Constraints Algebraic Tools for the Construction of Colored Ows with Constraints

We give a linear time algorithm which, given a simply connected gure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed ow 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 ow in a planar graph. R esum e Nous donnons un algorithme en temps lineaire qui, etant donnee une gure sim-plement connexe du plan divisee en cellules, dont la frontiere est traversee par des arcs colores entrant ou sortant, exhibe des lignes de ots non intersectantes re-liant les entrees aux sorties avec respect des couleurs, de telle facon que n'importe quelle arete de n'importe quelle cellule soit coupee par au plus une ligne. L'outil principal est la notion de fonction de hauteur, introduite precedemment pour les pavages. Cette notion apparait comme une generalisation de la notion de potentiel d'unn ot dans un graphe planaire. Abstract We give a linear time algorithm which, given a simply connected gure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed ow 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 ow in a planar graph.