254,482 research outputs found
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
Any-Angle Pathfinding for Multiple Agents Based on SIPP Algorithm
The problem of finding conflict-free trajectories for multiple agents of
identical circular shape, operating in shared 2D workspace, is addressed in the
paper and decoupled, e.g., prioritized, approach is used to solve this problem.
Agents' workspace is tessellated into the square grid on which any-angle moves
are allowed, e.g. each agent can move into an arbitrary direction as long as
this move follows the straight line segment whose endpoints are tied to the
distinct grid elements. A novel any-angle planner based on Safe Interval Path
Planning (SIPP) algorithm is proposed to find trajectories for an agent moving
amidst dynamic obstacles (other agents) on a grid. This algorithm is then used
as part of a prioritized multi-agent planner AA-SIPP(m). On the theoretical,
side we show that AA-SIPP(m) is complete under well-defined conditions. On the
experimental side, in simulation tests with up to 200 agents involved, we show
that our planner finds much better solutions in terms of cost (up to 20%)
compared to the planners relying on cardinal moves only.Comment: Final version as submitted to ICAPS-2017 (main track); 8 pages; 4
figures; 1 algorithm; 2 table
Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs (With erratum)
For given a pair of nodes in a graph, the minimum non-separating path problem
looks for a minimum weight path between the two nodes such that the remaining
graph after removing the path is still connected. The balanced connected
bipartition (BCP) problem looks for a way to bipartition a graph into two
connected subgraphs with their weights as equal as possible. In this paper we
present an algorithm in time for finding a minimum weight
non-separating path between two given nodes in a grid graph of nodes with
positive weight. This result leads to a 5/4-approximation algorithm for the
BCP problem on grid graphs, which is the currently best ratio achieved in
polynomial time. We also developed an exact algorithm for the BCP problem
on grid graphs. Based on the exact algorithm and a rounding technique, we show
an approximation scheme, which is a fully polynomial time approximation scheme
for fixed number of rows.Comment: With erratu
Path design and optimization with obstacle avoidance via reinforcement learning
For the last couple of decades, finding an optimized drilling path has been one of the key concerns for drilling engineers. It takes a couple of months to plan a well for a large number of people. The motive of this thesis is to find the optimal drilling path based on coordinates. To trace the optimal path, this thesis will apply the reinforcement learning algorithm in Matlab.
Another approach for this thesis is to find the shortest path by avoiding collision in a threedimensional
grid view
Hamiltonian orthogeodesic alternating paths
AbstractLet R be a set of red points and let B be a set of blue points. The point set P=R∪B is called equitable if ||B|−|R||⩽1 and it is called general if no two points are vertically or horizontally aligned. An orthogeodesic alternating path on P is a path such that each edge is an orthogeodesic chain connecting points of different color and such that no two edges cross. We consider the problem of deciding whether a set of red and blue points admits a Hamiltonian orthogeodesic alternating path, that is, an orthogeodesic alternating path visiting all points. We prove that every general equitable point set admits a Hamiltonian orthogeodesic alternating path and we present an O(nlog2n)-time algorithm for finding such a path, where n is the number of points. On the other hand, we show that the problem is NP-complete if the path must be on the grid (i.e., vertices and bends have integer coordinates). Further, we show that we can approximate the maximum length of an orthogeodesic alternating path on the grid by a factor of 3, whereas we present a family of point sets with n points that do not have a Hamiltonian orthogeodesic alternating path with more than n/2+2 points. Additionally, we show that it is NP-complete to decide whether a given set of red and blue points on the grid admits an orthogeodesic perfect matching if horizontally aligned points are allowed. This contrasts a recent result by Kano (2009) [9] who showed that this is possible on every general point set
Towards use of Dijkstra Algorithm for Optimal Navigation of an Unmanned Surface Vehicle in a Real-Time Marine Environment with Results from Artificial Potential Field
The growing need of ocean surveying and exploration for scientific and industrial application has led to the requirement of routing strategies for ocean vehicles which are optimal in nature. Most of the optimal path planning for marine vehicles had been conducted offline in a self‐made environment. This paper takes into account a practical marine environment, i.e. Portsmouth Harbour, for finding an optimal path in terms of computational time between source and end points on a real time map for an USV. The current study makes use of a grid map generated from original and uses a Dijkstra algorithm to find the shortest path for a single USV. In order to benchmark the study, a path planning study using a well‐known local path planning method artificial path planning (APF) has been conducted in a real time marine environment and effectiveness is measured in terms of path length and computational time
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