from application polytope
Polytope propagation means to define a polytope inductively by assigning vectors to arcs of a directed graph. At each node of such a graph a polytope arises as the joint convex hull of the polytopes at the translated sources of the inward pointing arcs. For details see
Joswig: Polytope Propagation on Graphs.
Chapter 6 in Pachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.
Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.
We define an acyclic digraph on 8 nodes with a unique source and a unique sink. Each arc is equipped with a vector in Q^2.
> $Gamma = new GraphAdjacency<Directed>(8); > $Vectors = new EdgeMap<Directed, Vector>($Gamma); > $Vectors->edge(0,1) = [1,0]; $Vectors->edge(0,2) = [0,0]; > $Vectors->edge(1,3) = [1,0]; $Vectors->edge(1,4) = [0,1]; > $Vectors->edge(2,3) = [0,0]; $Vectors->edge(2,4) = [0,2]; > $Vectors->edge(3,5) = [1,0]; $Vectors->edge(3,6) = [0,1]; > $Vectors->edge(4,5) = [0,0]; $Vectors->edge(4,6) = [0,2]; > $Vectors->edge(5,7) = [0,0]; > $Vectors->edge(6,7) = [0,0];
Then we can define the propagated polytope implicitly:
> $P = new PropagatedPolytope(CONE_AMBIENT_DIM=>3, "SUM_PRODUCT_GRAPH.ADJACENCY"=>$Gamma, "SUM_PRODUCT_GRAPH.TRANSLATIONS"=>$Vectors); > print $P->VERTICES; 1 3 0 1 1 0 1 0 1 1 1 3 1 0 4
Note that it is necessary to specify the dimension of the (homogeneous) ambient space by setting CONE_AMBIENT_DIM.