3,230 research outputs found
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
Throughput-driven floorplanning with wire pipelining
The size of future high-performance SoC is such that the time-of-flight of wires connecting distant pins in the layout can be much higher than the clock period. In order to keep the frequency as high as possible, the wires may be pipelined. However, the insertion of flip-flops may alter the throughput of the system due to the presence of loops in the logic netlist. In this paper, we address the problem of floorplanning a large design where long interconnects are pipelined by inserting the throughput in the cost function of a tool based on simulated annealing. The results obtained on a series of benchmarks are then validated using a simple router that breaks long interconnects by suitably placing flip-flops along the wires
Map Matching with Simplicity Constraints
We study a map matching problem, the task of finding in an embedded graph a
path that has low distance to a given curve in R^2. The Fr\'echet distance is a
common measure for this problem. Efficient methods exist to compute the best
path according to this measure. However, these methods cannot guarantee that
the result is simple (i.e. it does not intersect itself) even if the given
curve is simple. In this paper, we prove that it is in fact NP-complete to
determine the existence a simple cycle in a planar straight-line embedding of a
graph that has at most a given Fr\'echet distance to a given simple closed
curve. We also consider the implications of our proof on some variants of the
problem
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
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