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Register a new userA New Algorithm for Euclidean Shortest Paths in the Plane
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Added by
Haitao Wang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Haitao Wang
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Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri [SIAM J. Comput. 1999] gave an algorithm of $O(nlog n)$ time and $O(nlog n)$ space, where $n$ is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suris algorithm, Wang [SODA 2021] reduced the space to $O(n)$ while the runtime of the algorithm is still $O(nlog n)$. In this paper, we present a new algorithm of $O(n+hlog h)$ time and $O(n)$ space, provided that a triangulation of the free space is given, where $h$ is the number of obstacles. The algorithm, which improves the previous work when $h=o(n)$, is optimal in both time and space as $Omega(n+hlog h)$ is a lower bound on the runtime. Our algorithm builds a shortest path map for a source point $s$, so that given any query point $t$, the shortest path length from $s$ to $t$ can be computed in $O(log n)$ time and a shortest $s$-$t$ path can be produced in additional time linear in the number of edges of the path.
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Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. The previous best algorithm was given by Hershberger and Suri [FOCS 1993, SIAM J. Comput. 1999] and the algorithm runs in $O(nlog n)$ time and $O(nlog n)$ space, where $n$ is the total number of vertices of all obstacles. The algorithm is time-optimal because $Omega(nlog n)$ is a lower bound. It has been an open problem for over two decades whether the space can be reduced to $O(n)$. In this paper, we settle it by solving the problem in $O(nlog n)$ time and $O(n)$ space, which is optimal in both time and space; we achieve this by modifying the algorithm of Hershberger and Suri. Like their original algorithm, our new algorithm can build a shortest path map for a source point $s$ in $O(nlog n)$ time and $O(n)$ space, such that given any query point $t$, the length of a shortest path from $s$ to $t$ can be computed in $O(log n)$ time and a shortest path can be produced in additional time linear in the number of edges of the path.
Let $mathcal{P}$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $mathcal{P}$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ gateways for $s$ and a set of $n_t$ gateways for $t$ such that there exist a shortest $s$-$t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s$-$t$ path using these gateways. Previous algorithms all take quadratic $O(n_scdot n_t)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(n_s + n_t log n_s)$ time. As a consequence, we construct a data structure of $O(n+(h^2log^3 h/loglog h))$ size in $O(n+(h^2log^4 h/loglog h))$ time such that each query can be answered in $O(log n)$ time.
We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $mathcal{P}$ of $h$ pairwise-disjoint rectilinear polygonal obstacles with a total of $n$ vertices in the plane, a minimum-link rectilinear path between two points is a rectilinear path that avoids all obstacles with the minimum number of edges. In this paper, we present a new algorithm for finding minimum-link rectilinear paths among $mathcal{P}$. After the plane is triangulated, with respect to any source point $s$, our algorithm builds an $O(n)$-size data structure in $O(n+hlog h)$ time, such that given any query point $t$, the number of edges of a minimum-link rectilinear path from $s$ to $t$ can be computed in $O(log n)$ time and the actual path can be output in additional time linear in the number of the edges of the path. The previously best algorithm computes such a data structure in $O(nlog n)$ time.
We consider the online search problem in which a server starting at the origin of a $d$-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which each point on the $d$-dimensional unit sphere can be seen are within a constant factor of each other. We show that this length is in $Omega(d)cap O(d^{3/2})$.
This paper presents an optimal $Theta(n log n)$ algorithm for determining time-minimal rectilinear paths among $n$ transient rectilinear obstacles. An obstacle is transient if it exists in the scene only for a specific time interval, i.e., it appears and then disappears at specific times. Given a point robot moving with bounded speed among transient rectilinear obstacles and a pair of points $s$, $d$, we determine a time-minimal, obstacle-avoiding path from $s$ to $d$. The main challenge in solving this problem arises as the robot may be required to wait for an obstacle to disappear, before it can continue moving toward the destination. Our algorithm builds on the continuous Dijkstra paradigm, which simulates propagating a wavefront from the source point. We also solve a query version of this problem. For this, we build a planar subdivision with respect to a fixed source point, so that minimum arrival time to any query point can be reported in $O(log n)$ time, using point location for the query point in this subdivision.
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