3,168 research outputs found
Observed strategies in the freehand drawing of complex hierarchical diagrams
Chunk decomposition and assembly strategies have been found in the drawing of complex hierarchical diagrams (spe- cifically AVOW diagrams). Analysis of 40 diagrams pro- duced by five participants provided evidence for the strategies based on the duration of pauses between drawn elements. The strategies were initially discovered using a new visualiza- tion technique developed to allow the detailed examination of the sequential order of diagram drawing in conjunction with information about the durations of pauses associated with drawn elements
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
The boundary value problem for discrete analytic functions
This paper is on further development of discrete complex analysis introduced
by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying
in the complex plane and having quadrilateral faces. A function on the vertices
is called discrete analytic, if for each face the difference quotients along
the two diagonals are equal.
We prove that the Dirichlet boundary value problem for the real part of a
discrete analytic function has a unique solution. In the case when each face
has orthogonal diagonals we prove that this solution uniformly converges to a
harmonic function in the scaling limit. This solves a problem of S. Smirnov
from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L.
Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G.
Ciarlet-P.-A. Raviart for rhombic lattices.
In particular, our result implies uniform convergence of the finite element
method on Delaunay triangulations. This solves a problem of A. Bobenko from
2011. The methodology is based on energy estimates inspired by
alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized,
several assertions added, minor correction in the proofs of Lemma 2.5, 3.3,
Example 3.6, Corollary 5.
- …