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We present the first polynomial time approximation algorithm for computing
shortest paths in weighted three-dimensional domains. Given a polyhedral domain
$\D$, consisting of $n$ tetrahedra with positive weights, and a real number
$\eps\in(0,1)$, our algorithm constructs paths in $\D$ from a fixed source
vertex to all vertices of $\D$, whose costs...
Contexts in source publication
Context 1
... the out-angle ϕ + is the acute angle between the normal and the segment s + . The angles θ − and θ + are the acute angles between the orthogonal projections of s − and s + with a reference direction in the plane containing the face f , respectively (see Figure 1). ...
Context 2
... define the weight ˙ w ′ j+1 , and the bending points Y , and y, analogously with respect to b i,j+1 and the edge vicinity D ε (e j+1 ). It follows that the point x precedes y along˜πalong˜ along˜π 3 (a i,j , b i,j+1 ). We define the portion of the path˜πpath˜ path˜π 4 joining p i,j and q i,j+1 by˜πby˜ by˜π 4 (p i,j , q i,j+1 ) = {(p i,j , x), ˜ π 3 (x, y), (y, q i,j+1 )} and estimate its cost. ...
Context 3
... standard greedy approach for solving the SSSP problem works as follows: a subset, S, of nodes to which the shortest path has already been found and a set, E(S), of edges connecting S with S a ⊂ V \ S are maintained. The set S a consists of nodes not in S but adjacent to S. In each iteration, an optimal edge e(S) = (u, v) in E(S) is selected, with source u in S and target v in S a (see Figure 10). The target vertex v is added to S and E(S) is updated correspondingly. ...
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... our discussion below, we refer to these segments, including the edge of D, as Steiner segments. We further partition the group of edges associated with the triple (b, b 1 , f) into subgroups corresponding to pairs of Steiner segments (ℓ, ℓ 1 ) on b and b 1 , respectively, see Figure 11 (a). In this way, the edges of G ε are partitioned into groups corresponding to ordered triples (ℓ, ℓ 1 , f), where ℓ and ℓ 1 are Steiner segments parallel to f on two neighboring bisectors sharing f. ...
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... µ (Figure 13 (b)), the curve a(y) is a half-circle centered at the point (0, ν 0 ) with radius ν − . The curve a 1 (y) is symmetric with respect to the ν-axis. ...
Context 6
... is convex and approaches the µ-axis at infinity. In the case where ν − + ν − ≤ ν 0 ( Figure 13 (a)), the curves a(y) and a 1 (y) have a common part -a horizontal segment that projects at (−µ 0 , µ 0 ) on the µ-axis. For |µ| > µ 0 , the curves are the same as in the case ν − + ν − > ν 0 . ...
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Citations
... The proposed method exploits Snell's law of physical refraction and is able to return a path in a reasonable time that is very close to the optimum weighted shortest path. Furthermore, Aleksandrov et al. [21] have presented a (1 + )approximation algorithm ( ∈ (0, 1)) for computing shortest paths in a weighted threedimensional environment. Given n tetrahedra with positive weights in a polyhedral domain D, their algorithm constructs (1 + )-approximation paths in D from a fixed source vertex to all vertices of D in O(C(D) n 2.5 log n log 3 1 ) time, where C(D) is a geometric parameter related to the aspect ratios of tetrahedra. ...
In this paper, we consider the problem of path planning in a weighted polygonal planar subdivision. Each polygon has an associated positive weight which shows the cost of path per unit distance of movement in that polygon. The goal is to find a minimum cost path under the Manhattan metric for two given start and destination points. First, we propose an $$O(n^2)$$ O ( n 2 ) time and space algorithm to solve this problem, where n is the total number of vertices in the subdivision. Then, we improve the time and space complexity of the algorithm to $$O(n \log ^2 n)$$ O ( n log 2 n ) and $$O(n \log n)$$ O ( n log n ) , respectively, by applying a divide and conquer approach. We also study the case of rectilinear regions in three dimensions and show that the minimum cost path under the Manhattan metric is obtained in $$ O(n^2 \log ^3 n) $$ O ( n 2 log 3 n ) time and $$ O(n^2 \log ^2 n) $$ O ( n 2 log 2 n ) space.
... 11,27 There are also successful discretization schemes whose running time is linear in the input size and dependent on some geometric parameter of the polygonal domain. 5,33 In contrast, only one algorithm for the weighted region problem in 3D has been proposed (Aleksandrov et al. 6 ). The authors present a (1 + ε)-approximation algorithm whose running time is O(Knε −2.5 log n ε log 3 1 ε ), where K is asymptotically at least the cubic power of the maximum aspect ratio of the tetrahedra in the worst case. ...
... Thus, there exists a constant C dependent on ρ but independent of T such that if κ ≤ 1 C log log n + O(1), the running time is polynomial in n, 1 Navigating Weighted Regions with Scattered Skinny Tetrahedra 15 time is linear in n. In comparison, the running time of the algorithm by Aleksandrov et al. 6 has the advantage of being independent from N and W , but K can be arbitrarily large even if there are only O(1) skinny tetrahedra. Putting their result in our model, K is a function of N and n in the worst case, and K can be Ω( 1 n N 3 + 1). ...
... Our algorithm discretizes T and builds an edge-weighted graph G so that the shortest path in G is a 1+O(ε) approximation. This approach is also taken by Aleksandrov et al. 6 in 3D. Unlike their approach, in order to allow for skinny tetrahedra, we discretize the fat tetrahedra only, and the edges in G represent approximate shortest paths that may not lie within a single tetrahedron. ...
We propose an algorithm for finding a (1 + ϵ)-approximate shortest path through a weighted 3D simplicial complex τ. The weights are integers from the range [1,W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in τ. Let ρ be some arbitrary constant. Let Κ be the size of the largest connected component of tetrahedra whose aspect ratios exceed Κ. There exists a constant C dependent on Κ but independent of such that if Κ ≤ 1 Cloglog n + O(1), the running time of our algorithm is polynomial in n, 1/ and log(NW). If Κ = O(1), the running time reduces to O(nϵ-O(1)(log(NW))O(1)).
Automatic path planning is a classical research field, which has been studied for decades with a wide range of applications in multiple research communities. Given a graph, finding a shortest path between two vertices of that graph, with the popular algorithms such as Dijkstra, Bellman-Ford and Floyd-Warshall can be considered as the most basic instance of path planning. In practice, path planning is applied in different environments, which can be more complex than only on graphs. In this thesis, we consider two real world problems of path planning. They are (1) given a set of obstacles in an environment, we are asked to find a Euclidean shortest path between two points avoiding the obstacles, and (2) given a set of regions where each region is assigned a unit weight or cost, we are asked to find a weighted shortest path between two points.
For (1), if the obstacles are represented as polygons in a two-dimensional (2D) environment, the problem is well-studied, and there exist polynomial time algorithms that can find an exact optimal path. However, when the environment is expanded to a three-dimensional (3D) environment, where the obstacles are represented as polyhedrons, the problem is known to be NP-hard. For (2), the problem is believed to be much more difficult. That is, even in a 2D environment, there is no known polynomial or exponential time algorithm for finding an exact optimal path for (2). So far, all practical solutions for (1) in 3D and (2) both in 2D and 3D are approximations. Currently, the main approaches of these approximation methods focus on discretizing the continuous space of the problem into a graph or grid before using a traditional shortest path graph algorithm to find the final result, or using heuristics. Only a few algorithms attempt to find a shortest path directly in the original continuous environment of the problem.
In this thesis, we propose new algorithms for four specific instances related to (1) and (2).
They are (i) a weighted region problem in a 2D environment, (ii) a weighted region problem on the surface of a polyhedron in a 3D environment, (iii) an additional constraint Euclidean shortest path avoiding vertical obstacles in a 3D environment, and (iv) a Euclidean and weighted shortest path crossing a given sequence of skew segments in a 3D environment. For (i), (ii) and (iv), we present a very-close optimal solution rather than a fast approximation solution. For (iii), we develop a flexible solution that can be adapted to multiple different optimization criteria. Without discretizing the search space or using heuristics, we solve the four instances in the original continuous environment. Moreover, our algorithms do not require any complex data structures or expensive geometrical computations like unfolding faces on the surface of a polyhedron in 3D. Along with theoretical analyses, the efficiency and practicality of our algorithms are also verified by extensive experiments, both in artificial and real environments.
The d-dimensional weighted region shortest path problem asks for finding a shortest path between two given points s and t in a d-dimensional polyhedral structure consisting of polyhedral cells having individual positive weights. It is a generalization of the d-dimensional unweighted (Euclidean) shortest path problem for polyhedral structures.
In the unweighted (Euclidean) setting, a shortest path visits, due to cell convexity, each polyhedral cell at most once. Surprisingly, this is no longer true for the weighted setting, which is a strong motivation for studying the complexity of weighted shortest paths in polyhedral structures.
Previously, Ω(n2), respectively Ω(n3) lower bounds on the maximum number of cell crossings for weighted shortest paths in 2-dimensional, respectively 3-dimensional polyhedral structures have been obtained.
In this paper, a new lower bound of Ω(nd) is derived on the maximum number of cell crossings a weighted shortest path could take in d-dimensional polyhedral structures consisting of a linear number of O(n) polyhedral cells and cell faces. This new result is a generalization and sharpening of the formerly known lower bounds and has been a long-standing open problem for the general d-dimensional case.
A new lower bound of \(\varOmega (n^3)\) on the maximum number of cell crossings for weighted shortest paths in 3-dimensional polyhedral structures consisting of a linear number of \(\mathcal {O}(n)\) polyhedral cells and cell faces is derived. This is a generalization and sharpening of the formerly known \(\varOmega (n^2)\) lower bound on the maximum number of cell crossings for weighted shortest path in 2-dimensional polyhedral structures and has been a long-standing open problem for the 3-dimensional case.
Computing paths over a terrain that are highly occluded with respect to observers is an important problem in GIS. Given a fast algorithm for computing the visibility map, the path-planning step becomes the bottleneck. In this paper, we present an approach for quickly computing occluded paths over a terrain using a sparse network, a sparse 1-dimensional network over the terrain. We present different strategies for constructing the sparse network. Experimental results show that our approach results in significantly improved time for computing highly occluded paths between two query points, and that the different strategies offer a tradeoff between higher-quality paths and lower preprocessing times. Furthermore, there are strategies that achieve near-optimal paths with small preprocessing cost.
We propose an algorithm for finding a \((1+\varepsilon )\)-approximate shortest path through a weighted 3D simplicial complex \(\mathcal T\). The weights are integers from the range [1, W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in \(\mathcal T\). Let \(\rho \) be some arbitrary constant. Let \(\kappa \) be the size of the largest connected component of tetrahedra whose aspect ratios exceed \(\rho \). There exists a constant C dependent on \(\rho \) but independent of \(\mathcal T\) such that if \(\kappa \le \frac{1}{C}\log \log n + O(1)\), the running time of our algorithm is polynomial in n, \(1/\varepsilon \) and \(\log (NW)\). If \(\kappa = O(1)\), the running time reduces to \(O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})\).
Understanding the locations of highly occluded paths on a terrain is an important GIS problem. In this paper we present a model and a fast algorithm for computing highly occluded paths on a terrain. It does not assume the observer locations to be known and yields a path likely to be occluded under a rational observer strategy. We present experimental results that examine several different observer strategies. The repeated visibility map computations necessary for our model is expedited using a fast algorithm for calculating approximate visibility maps that models the decrease in observational fidelity as distance increases. The algorithm computes a multiresolution approximate visibility map and makes use of a graphics processing unit (GPU) to speed up computation. We present experimental results on terrrain data sets with up to 144 million points.
