184,425 research outputs found
Path Puzzles: Discrete Tomography with a Path Constraint is Hard
We prove that path puzzles with complete row and column information--or
equivalently, 2D orthogonal discrete tomography with Hamiltonicity
constraint--are strongly NP-complete, ASP-complete, and #P-complete. Along the
way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional
Matching and Numerical 3-Dimensional Matching.Comment: 16 pages, 8 figures. Revised proof of Theorem 2.4. 2-page abstract
appeared in Abstracts from the 20th Japan Conference on Discrete and
Computational Geometry, Graphs, and Games (JCDCGGG 2017
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Distributed computation of persistent homology
Persistent homology is a popular and powerful tool for capturing topological
features of data. Advances in algorithms for computing persistent homology have
reduced the computation time drastically -- as long as the algorithm does not
exhaust the available memory. Following up on a recently presented parallel
method for persistence computation on shared memory systems, we demonstrate
that a simple adaption of the standard reduction algorithm leads to a variant
for distributed systems. Our algorithmic design ensures that the data is
distributed over the nodes without redundancy; this permits the computation of
much larger instances than on a single machine. Moreover, we observe that the
parallelism at least compensates for the overhead caused by communication
between nodes, and often even speeds up the computation compared to sequential
and even parallel shared memory algorithms. In our experiments, we were able to
compute the persistent homology of filtrations with more than a billion (10^9)
elements within seconds on a cluster with 32 nodes using less than 10GB of
memory per node
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