Subobject

Abstract: An approach is described for modifying subobjects of geometry objects based on per-subobject objects. A per-subobject object is associated with a sequence of components, such as a modifier stack, that are used to modify a geometry object. The per-subobject object may take one or more actions with respect to subobjects of the mesh object, such as applying data to specified faces. After a subsequent modification of the geometry object, the per-subobject object reapplies data to the faces that result from the modification. For example, the per-subobject object may specify the data to apply to faces that result from splitting faces during the modification or to faces that result from merging faces during the modification. The data for the faces may be accessed using a data channel that is associated with the per-subobject object.

Abstract: We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object.

Abstract: Two natural kinds of problems about structured collections of symbols can be generally referred to as the LARGEST COMMON SUBOBJECT and the SMALLEST COMMON SUPEROBJECT problems. which we consider here as the dual problems of interest. For the case of rooted binary trees where the symbols occur as leaf-labels and a subobject is defined by label-respecting hereditary topological containment, both of these problems are NP-complete, as are the analogous problems for sequences (the well-known LONGEST COMMON SUBSEQUENCE and SHORTEST COMMON SUPERSEQUENCE problems). When the trees are restricted by allowing each symbol to occur as a leaf-label at most once (which we call a phylogenetic tree or p-tree), then the LARGEST COMMON SUBTREE problem, better known as the MAXIMUM AGREEMENT SUBTREE (MAST) problem, is solvable in polynomial time. We explore the complexity of the basic subobject and superobject problems for both sequences and binary trees when the inputs are restricted to p-trees and p-sequences (p-sequences are sequences where each symbol occurs at most once). We prove that the sequence analog of MAST can be solved in polynomial time. The SHORTEST COMMON SUPERSEQUENCE problem restricted to inputs consisting of a collection of p-sequences (pSCS) is NP-complete; we show NP-completeness of the analogous SMALLEST COMMON SUPERTREE problem restricted to p-trees (pSCT). We also show that both problems are hard for the parameterized complexity classes W[1] where the parameter is the number of input objects. We prove fixed-parameter tractability for pSCS and pSCT when the k input objects are restricted to be complete: every symbol of Σ occurs exactly once in each object and the question is whether there is a common superobject of size bounded by | Σ | + r and the parameter is the pair (k, r). We show that without this restriction, both problems are harder than DIRECTED FEEDBACK VERTEX SET, for which parameterized complexity is famously unresolved.