Shortest paths in simple polygons with polygon-meet constraints

International Journal of Computational Geometry & Applications, 1999

This paper presents a simple O(n+k) time algorithm to compute the set of knon-crossing shortest paths between k source-destination pairs of points on the boundary of a simple polygon of n vertices. Paths are allowed to overlap but are not allowed to cross in the plane. A byproduct of this result is an O(n) time algorithm to compute a balanced geodesic triangulation which is easy to implement. The algorithm extends to a simple polygon with one hole where source-destination pairs may appear on both the inner and outer boundary of the polygon. In the latter case, the goal is to compute a collection of non-crossing paths of minimum total cost. The case of a rectangular polygonal domain where source-destination pairs appear on the outer and one inner boundary12 is briefly discussed.

Computational Geometry, 2013

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We assume that a simple n-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in O(n 2 ) time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow s words of working space. After a preprocessing of O(n 2 ) time, we are able to solve shortest path queries between any two points inside the polygon in O(n 2 /s) time.

2008

We study a constrained version of the shortest path problem in polygonal domains, in which the path must visit a given target polygon. We provide an efficient algorithm for this problem based on the wavefront propagation method and also present a method to construct a subdivision of the domain to efficiently answer queries to retrieve the constrained shortest paths from a single-source to the query point.

Computational Geometry

Consider two axis-aligned rectilinear simple polygons in the domain consisting of axisaligned rectilinear obstacles in the plane such that the bounding boxes, one for each obstacle and one for each polygon, are disjoint. We present an algorithm that computes a minimumlink rectilinear shortest path connecting the two polygons in O((N + n) log(N + n)) time using O(N + n) space, where n is the number of vertices in the domain and N is the total number of vertices of the two polygons.

International Journal of Computational Geometry & Applications, 1996

We present a data structure that allows to preprocess a rectilinear polygon with n vertices such that, for any two query points, the shortest path in the rectilinear link or L 1 -metric can be reported in time O(log n + k) where k is the link length of the shortest path. If only the distance is of interest, the query time reduces to O(log n). Furthermore, if the query points are two vertices, the distance can be reported in time O(1) and a shortest path can be constructed in time O(1 + k). The data structure can be computed in time O(n) and needs O(n) storage. As an application we present a linear time algorithm to compute the diameter of a simple rectilinear polygon w.r.t. the L 1 -metric.

2013

In our quest for understanding the geometric structure of the minimum diffuse reflection paths vis-a-vis shortest paths and minimum link paths, we define a new kind of diffuse reflection path called a constrained diffuse reflection path where (i) the path is simple, (ii) it intersects only the eaves of the Euclidean shortest path between $s$ and $t$, and (iii) it intersects each eave exactly once. For computing a minimum constrained diffuse reflection path from $s$ to $t$, we present an $O(n(n+\beta))$ time algorithm, where $\beta =\Theta (n^2)$ in the worst case. Here, $\beta$ depends on the shape of the polygon. We also establish some properties relating minimum constrained diffuse reflection paths and minimum diffuse reflection paths. Constrained diffuse reflection paths introduced in this paper provide new geometric insights into the hitherto unknown structures and shapes of optimal reflection paths.

1998

Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let P be a convex polygon with n vertices. We study the collision-free, optimal path-planning problem for B moving between two configurations inside P (a configuration specifies both a location and a direction of travel). We present an O(n 2 log n) time algorithm for determining whether a collision-free path exists for B between two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles.

2003

Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk 2 log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NP-hard.

2017

We can use a Dijkstra algorithm to calculate shortest paths on a polygon mesh. However, if the number of vertices of a polygon mesh is large, the efficiency of conventional path searching method that using Dijkstra algorithm is low. In order to solve this problem, this paper proposes a method to search shortest distance and path by using the A* algorithm. According to an experiment with the bunny model with 2,503 vertices, the efficiency of searching the shortest path is improved approximately 80% than conventional methods.

Siam Journal on Computing, 2002

Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let P be a convex polygon with n vertices. We study the collision-free, optimal path-planning problem for B moving between two configurations inside P (a configuration specifies both a location and a direction of travel). We present an O(n 2 log n) time algorithm for determining whether a collision-free path exists for B between two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles.

1993

We present the first polynomial-time algorithm that finds the shortest route in a simple polygon such that all points of the polygon is visible from some point on the route. This route is sometimes called the shortest watchman route, and it does not allow any restrictions on the route or on the simple polygon. Our algorithm runs in O(n 3) time.

2005

Abstract In this paper, we study the problem of finding the shortest path from a given source point in a simple polygon to some point visible from a given query point. We will present an algorithm based on the notion of funnels in simple polygons. The algorithm preprocesses the input containing a simple polygon and a source point to produce a data structure to answer the queries in logarithmic time. The time and space required for preprocessing is quadratic in size of the simple polygon.

Computational Geometry, 1997

A diagonal of a planar, simple polygon P is an open line segment that connects two nonadjacent vertices and lies in the relative interior of P. We present a linear time algorithm for finding a shortest diagonal (in the L2 norm) of a simple polygon, improving the previous best result by a factor of log n. Our result provides an interesting contrast to a known Y2(n log r 0 lower bound for finding a closest pair of vertices in a simple polygon-observe that a shortest diagonal is defined by a closest pair of vertices satisfying an additional visibility constraint.

Proceedings of the third annual symposium on Computational geometry - SCG '87, 1987

The problem of finding a rectilinear shortest path amongst obstacles may be stated as follows: Given a set of obstacles in the plane find a shortest rectilinear (L 1) path from a point s to a point t which avoids all obstacles. The path may touch an obstacle but may not cross an obstacle. We study the rectilinear shortest path problem for the case where the obstacles are non-intersecting simple polygons, and present an O(n (logn) 2) algorithm for finding such a path, where n is the number of vertices of the obstacles. This algorithm requires O(nlogn) space. Another algorithm is given that requires O(n(logn) 3/2) time and space. We also study the case of rectilinear obstacles in three dimensions, and show that L 1 shortest paths can be found in O(n 2 (log n) 3) time.

SIAM Journal on Computing, 1999

We propose an optimal-time algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worst-case time O(n logn) and requires O(n logn) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an e cient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a xed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a signi cant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving non-point and multiple sources.