libzonotope README

This is a collection of scripts and algorithms for zonotope manipulations. The most interesting functions are the generic C++ algorithms implemented in the header files in the include/ subdirectory of this project.

A zonotope is a convex polytope that is the Minkowski sum of line segments. We write

Z(V) := [0,v_0] + ... + [0,v_{n-1}]

for the Minkowski sum of the line segments [0,v_i], i=0..n-1, where V=[v_0 .. v_{n-1}] is the d×n matrix with columns v_i, and 0 is the origin in d dimensions.

Features

In its current state, the project should be considered no more than a proof of concept. For more sophisticated vertex enumeration, we recommend MinkSum. The main reason someone might use the library right now is its small size, and the control it gives over the components used in its algorithms. We currently provide implementations that depend on the linear programming solver from CGAL, but it's fairly easy to adapt the code to a different solver.

The main files are:

• include/combination_traversal.cpp: A generic combination traversal algorithm for lexicographical traversal of combinations. Currently lacking is a proper definition of the concept Combination_container, though it's already implemented in Combination_kernel_container<NT>.

• include/combination_kernel_container.hpp: A combination container that implements incremental kernel updates.

• include/zonotope_output_functor.hpp: An example output functor for the combination traversal that constructs the H-representation of the zontope. The functor accepts all combinations, but forwards only the (d-2)-combinations it accepts to handle_event_points which generates a batch of halfspaces.

• include/event_point_2.hpp: Implements the function handle_event_points, which generates a batch of halfspaces in time O(n*log(n)).

• zonotope_halfspaces.hpp: Outputs the H-represenation by combining the appropriate output functor and combination container with the combination traversal algorithm.

• include/linalg.hpp: Implements in particular the function update_kernel for efficient kernel updates (used in Combination_kernel_container).

• include/reverse_search.hpp: Implements a completely generic reverse search algorithm by Avis and Fukuda, using functors for adjacency checks and local search.

• include/cell_enum.hpp: Implements the functors for reverse search to perform cell enumeration in arrangements, and a wrapper function that finds a root cell. By providing an appropriate output functor, this can be used for vertex enumeration in zonotopes. This algorithm is also from the original reverse search paper by Avis and Fukuda.

• include/vertex_enum.hpp: Implements vertex enumeration in zonotopes using depth-first-search in the dual arrangement with manual stack management, for comparison with reverse search (it can be concluded that we need to improve our implementation of reverse search).

TODO

• Special functions for two dimensions (i.e. zonogons).

• Implement O(n^{d-1}\log{n}) general position vertex enumeration.