Reconstructing Set Partitions

Colloque avec actes et comité de lecture.We study the following combinatorial search problem: Reconstruct an unknown partition of the set [n]={1,...,n} into at most K disjoint non-empty subsets (classes) by making queries about subsets $Q_i \subseteq [n]$ such that the query returns the number of classes represented in $Q_i$. The goal is to reconstruct the whole partition with as few queries as possible. We also consider a variant of the problem where a representative of each class should be found without necessarily reconstructing the whole partition. Besides its theoretical interest, the problem has practical applications. Paper \cite{SBAW-RECOMB97} considers a physical mapping method based on hybridizing probes to chromosomes using the FISH technology. Under some reasonable assumptions, the method amounts to finding a partition of probes determined by the chromosome they hybridize to, and actually corresponds to the above formalization

Efficient reconstruction of partitions

AbstractWe consider the problem of reconstructing a partition x of the integer n from the set of its t-subpartitions. These are the partitions of the integer n-t obtained by deleting a total of t from the parts of x in all possible ways. It was shown (in a forthcoming paper) that all partitions of n can be reconstructed from t-subpartitions if n is sufficiently large in relation to t. In this paper we deal with efficient reconstruction, in the following sense: if all partitions of n are t--reconstructible, what is the minimum number N=N-(n,t) such that every partition of n can be identified from any N+1 distinct subpartitions? We determine the function N-(n,t) and describe the corresponding algorithm for reconstruction. Superpartitions may be defined in a similar fashion and we determine also the maximum number N+(n,t) of t-superpartitions common to two distinct partitions of n

Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification minors: classes of linear or affine functions over nonassociative semirings are shown to be weakly reconstructible. Moreover, affine functions of sufficiently large arity over finite fields are reconstructible.Comment: 18 pages. Int. J. Algebra Comput. (2014

On the reconstruction of linear codes

For a linear code over GF (q) we consider two kinds of `subcodes' called residuals and punctures. When does the collection of residuals or punctures determine the isomorphism class of the code? We call such a code residually or puncture reconstructible. We investigate these notions of reconstruction and show that, for instance, selfdual binary codes are puncture and residually reconstructible. A result akin to the edge reconstruction of graphs with sufficiently many edges shows that a code whose dimension is small in relation to its length is puncture reconstructible

Methodology for automatic recovering of 3D partitions from unstitched faces of non-manifold CAD models

Data exchanges between different software are currently used in industry to speed up the preparation of digital prototypes for Finite Element Analysis (FEA). Unfortunately, due to data loss, the yield of the transfer of manifold models rarely reaches 1. In the case of non-manifold models, the transfer results are even less satisfactory. This is particularly true for partitioned 3D models: during the data transfer based on the well-known exchange formats, all 3D partitions are generally lost. Partitions are mainly used for preparing mesh models required for advanced FEA: mapped meshing, material separation, definition of specific boundary conditions, etc. This paper sets up a methodology to automatically recover 3D partitions from exported non-manifold CAD models in order to increase the yield of the data exchange. Our fully automatic approach is based on three steps. First, starting from a set of potentially disconnected faces, the CAD model is stitched. Then, the shells used to create the 3D partitions are recovered using an iterative propagation strategy which starts from the so-called manifold vertices. Finally, using the identified closed shells, the 3D partitions can be reconstructed. The proposed methodology has been validated on academic as well as industrial examples.This work has been carried out under a research contract between the Research and Development Direction of the EDF Group and the Arts et Métiers ParisTech Aix-en-Provence