Fast query structures in anisotropic media

Transcription

1 Fast query structures in anisotropic media Radwa El Shawi 1,2 and Joachim Gudmundsson 1,2 1 School of Information Technology, University of Sydney 2 NICTA, Sydney, Australia. {radwa.elshawi,joachim.gudmundsson}@sydney.edu.au Abstract. We study the minimum cost path problem in an environment where the cost is direction-dependent (anisotropic). This problem arise in sailing, robotics, aircraft navigation, and routing of autonomous vehicles, where the cost is affected by the direction of waves, winds or slope of the terrain. We present an approximation algorithm to find a minimum cost path for a point robot moving in a planar subdivision, where each face is assigned a translational flow that reflects the cost of travelling within this face. In addition we consider the nearest neighbor problem in an anisotropic media that is given a planar subdivision together with a set Z of m sites, a query point q, find the nearest site in Z from q. Our main contribution is a data structure that given a subdivision with translational flows returns a (1 + )-approximate minimum cost path in the subdivision between any two query points in the plane. 1 Introduction The (geometric) shortest path problem is one of the fundamental problems studied in computer science. An important special case is to determine the minimum cost path between a source point x and a destination point t in a geometric environment. In many cases the environment is modelled as a triangular subdivision. Different metrics may be used in all faces of the subdivision to represent some additional mechanical constraints such as friction, flow or steepness. In this paper we assume that the problem is given as a planar triangular subdivision where each face r defined by the subdivision is assigned a translational flow defined by a vector f r. We will use the same notations and definitions as in [18]. That is, each face r is also assigned a non-negative real number b r giving the maximum Euclidean norm of the control velocity that the robot can apply within r. We define ρ r to be the ratio between b r and f r. Assume we are considering the movement of a robot. The robot is a point with a given initial position and also a given final position. Within each face r of the subdivision, the robot can apply, at each time τ and in any direction, a translational control velocity vector v(τ) of bounded Euclidean norm v(τ) = b r. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

2 However, the actual velocity of the robot at time τ is given by the sum of its control velocity vector v(τ) and the translational flow velocity f r of region r, see Fig. 1. In this paper we focus on the flow path optimization problem [18] which aims to find an optimal path of movement of the robot from the initial to the final position of minimum cost (time duration). In addition we will also consider the nearest neighbor queries and its variant include the k nearest neighbor query. The flow path optimization problem has a wide range of applications (see also [18]), for example: Navigating a vessel on the ocean through regions with different currents. In particular finding a path with minimum fuel consumption from a source point to a destination point. Finding a quickest path on a terrain where going up or down affects the maximum speed. Finding a cheapest path (in terms of fuel consumption) for an aircraft moving through regions with different wind conditions. The direction-dependent structure of the problem results in an asymmetric cost function, where the cost of traversing a straight line segment ab, is not necessarily equal to that of a reversed link ba. Thus, the cost function is not a metric, consequently restricting the set of mathematical tools available to us. The second problem we consider is the nearest neighbor query. This problem is well studied in the literature, and its variant include the k-nearest neighbor query [5]. The nearest neighbor query in a anisotropic media is formally defined as follows. Given a query point q R 2, a set Z of m sites or service stations, and a planar triangular subdivision T. The problem is to find a closest site z Z q with minimum cost from q. 1.1 Previous work Optimal path planning problems have been studied for a long time. In the following we discuss the most relevant work and refer the interested reader to the survey by Mitchell [14]. In the weighted subdivision problem [15], a point robot moves within a planar subdivision, each face f of the subdivision is assigned a weight w > 0. The cost of a path within a face f is the length of this path multiplied by w. Let n denote the number of vertices of the triangular subdivision. The first approximation algorithm for the weighted subdivision problem was presented by Mitchell and Papadimitriou [15]. Their algorithm used a continuous Dijkstra method to find an optimal path between any source point and destination point. The complexity of their algorithm is O(n 8 M), where M is a function in several parameters including a parameter specifying the degree of the precision. In 1997 Mata and Mitchell [13] presented a (1+) approximation algorithm. The algorithm is based on constructing a relatively sparse graph, a pathnet, that links selected pairs of subdivision vertices (and critical points of entry ) with locally optimal paths. 2

3 The running time of this algorithm is O( n3 N 2 w max w min ), where N is the maximum coordinate of the vertices and w max (w min ) is the maximum (minimum) weight of a triangular region. Other algorithms for the weighted subdivision problem discretize the polygonal subdivision by placing Steiner points along the edges of the subdivision and then finding a minimum cost path in a graph whose nodes are Steiner points or vertices of the subdivision and whose edges are line segments. Then an optimal path is computed on the resulting graph. In particular, Lanthier et al. [12] presented a (1 + )-approximation algorithm based on uniform discretization for the polygonal regions which adds m = O(n 2 ) points on each edge and then they constructed a spanner graph in which they computed the approximate path. The time complexity of this algorithm is O( n3 log n), as the graph has O(n3 ) vertices and O(n 3 /) edges. This was later improved by Aleksandrov et al. [2] who proposed a logarithmic discretization scheme. The running time of their algorithm L is O(kn log kn), where k = O(log t r ) and L denotes the length of the longest edge in the subdivision. Here r is used to represent the minimum distance from w any point to the boundary of the regions adjacent to it and t = 1+ max w min sin(θ, min) where θ min is the minimum angle in the subdivision. In the above weighted subdivision model metrics are isotropic, while we in many cases would like to model anisotropic costs for example the effect of current, wind or any other types of forces. Only a handful papers considered the problem of finding the minimum cost path in anisotropic media. Papadakis and Perakis [17] gave heuristics for related problems such as minimal time vessel routing among ocean currents. Rowe [19] discussed optimal path planning for a mobile robot with direction dependent forces (friction and gravity) using a state space representation. However, no theoretical bounds were given in [19]. Reif and Sun [18] gave an approximation algorithm for the motion planning in the presence of uniform flows. The anisotropy was introduced as a uniform flow assigned to each face in the subdivision. Then, the actual velocity of an object is defined to be the sum of a flow vector and a control velocity. The complexity of their algorithm is O( n c skew ( c skew + log n) ), where C skew is defined as follows. Let λ = max{c f /c f : adjacent faces f and f }, where c f and c f are the maximum control velocities applied in regions f and f respectively. Then C skew = Θ( λ(wmin+1) θ ). min(w min 1) In 2008 Cheng et al. [8] considered the case when distances in each face of the subdivision are measured using a asymmetric convex function that has the following property: its unit disk contains a unit Euclidean disk, and is contained in a Euclidean disk with radius ρ. The running time of their algorithm is O( ρ2 n 3 log ρ log ρn 2 ). Note that this bound does not depend on any other parameters; in particular it does not depend on the minimum angle in the subdivision. For the shortest path query version one usually considers two settings. In the fixed source query version, the source is fixed and the query specifies the destination or vice versa. The second version is what is sometimes called the two-point query problem, the query specifies both the source and the destination. 3

4 In the case of the weighted region problem on polyhedral surfaces, Lanthier et al. [12] presented a data structure that can answer a two-point query in O(log n) time. However the approximation error is additive O(W L), where W is the maximum weight of the faces and L is the length of the longest edge in the surface. Aleksandrov et al. [1] presented improved results for answering fixed-source and two-point queries on weighted polyhedral surfaces of arbitrary genus. The result was recently improved again by Djidjev and Sommer [9]. They presented a data structure that answers (1 + )approximate distance queries using O( n log 2 n 3/2 ) log2 1 )) space, O( n log 3 n 2 log2 1 ) preprocessing time, and O( 1 log 1 + log log n) query time. Cheng et al. [8] presented a data structure to answer approximate shortest path queries from a fixed source in a planar subdivision. Distances in a face are measured using a possibly asymmetric convex distance function, and the convex distance functions can be different for different faces. Given a real number (0, 1), a (1 + )-approximate minimum cost query can be answered in O(log ρn ) time. The data structure can be constructed in O( ρ2 n 3 (log ρn 2 )2 ) time using O( ρ2 n 3 log ρn 2 ) space. To the best of our knowledge this paper presents the first (1+)-approximate for the two-point query problem in anisotropic regions and presents the first (1 + )-approximate for the nearest neighbor query problem. 1.2 Problem formulation In the following we describe the problem setting and the algorithm by Reif and Sun [18]. For any two points p, q R 2, we denote by pq the closed, oriented straight line segment from p to q. We denote by pq the Euclidean distance between p and q. And let τ(p, q) denote the minimum cost path between two points p and q in the Euclidean plane. We next show how to compute an optimal path from a source point u to a destination point u both lying inside a face r of the subdivision. Reif and Sun [18] showed that an optimal path is simple, piecewise linear and it only changes direction on the boundary between two faces. Let r = ABC be a triangular face with flow f r and let β = BAC, as shown in Fig. 1. Let u be a point on AB with distance d to A and let u be a point on AC with distance d to A. Let α be the angle between BA and f r and let θ be the angle u ua. For a robot to travel along a minimum cost path from a source point u to a destination point u, it needs to apply a control velocity v r with magnitude b r. Referring to Fig. 1 we can draw a virtual triangle uu u such that u uu = α θ, uu = τ(u, u ) f r and u u = τ(u, u ) b r. In the triangle uu u, the vector uu represents the transactional flow velocity, u u represents the control velocity of a robot in region r and uu represents the composite vector of the robot movement. The following lemma (refer to Fig. 1) describes how an optimal path is computed. Lemma 1 (Adapted from Lemma 1 in [18]). The face-wise optimal path from u to u can be achieved by adopting a control velocity with maximum magni- 4

5 C fr b r τ(u, u ) u φ u d B τ f r u α θ d β A Fig. 1. Illustrating the notations used in Lemma 1. tude b r and an angle of φ = arcsin( sin(α θ) ρ r ) from uu. Further, the cost τ(u, u ) l of this path is b r(cos φ+cos(α θ)/ρ, where l = r) uu = d 2 + d 2 2dd cos β. In this paper we make three contributions. First we show that a small modification to the algorithm by Reif and Sun [18] improves the running time of their algorithm by roughly a factor of c skew (Section 2). However, the main contribution of the paper is a data structure for the query version of the problem (Section 3). That is, preprocess the input such that an approximate minimum cost path between two query points x and t can be answered efficiently. The third contribution is an approximation algorithm to answer the nearest neighbor query (Section 4), that is given a subdivision with translational flows returns a (1 + ɛ)-approximate nearest site with minimum cost path. We conclude with some remarks and open problems in Section 5. 2 An efficient (1 + ) approximation algorithm We consider a simple improvement of Reif and Sun s [18] construction. In their paper they transform the geometric continuous problem into a graph problem on a directed graph G(V, E) as follows. Place Steiner points on the boundary of the triangular regions (this step will be described in more details below). The vertex set V of the graph G corresponds to the set of Steiner points plus the given start and endpoints, that is, for every Steiner point (or start and endpoint) p there is a vertex v p V. Sometimes we will abuse the notation and refer to v p as both a vertex in V and a Steiner point in the plane, or p as a Steiner point in the plane and a vertex in V. Two vertices v p and v q in V are connected by a directed edge of weight τ(p, q) if and only if p and q belong to the same face of the subdivision. Hence G has a quadratic number of edges. On the resulting graph, a minimum cost path is computed using Dijkstra s algorithm. The path was proven, in [18], to be an approximate shortest path. In order to reduce the complexity of the graph, we show how one can use the well-separated pair decomposition (WSPD) [2] to determine which pairs should 5

6 be connected. Thus instead of a quadratic number of edges only a linear number is needed, which results in an improvement of the running time. We show that there exists a path between v x and v t in G with cost (1 + γ) times the cost of an optimal path in G between v s and v t, where γ is a positive constant given as part of the input. u 1,2 u 1,j e v 1 v 2 u 1,1 u 1,j 1 u e Fig. 2. Adding Steiner points on a boundary edge. 2.1 Placing Steiner points To compute a γ-good path i.e., a path whose cost is at most (1 + γ) times the cost of an optimal path, we use the same logarithmic discretization schema used by Reif and Sun [18] to place Steiner point along the boundary segments of the subdivision. Note that if the robot is traveling on the shared boundary between two regions, it can be considered as traveling inside either region, whichever is more favorable; the robot can always move by an infinitesimal distance into that region, and then travel inside that region along the boundary before eventually moving back onto the boundary. Let τ min (u) denote the minimum cost of travelling along a straight line path between u on a boundary edge and any point on a boundary edge not incident to u. For any boundary edge e, let u e denote the point on the boundary edge e with maximum τ min (u). This discretization scheme places a higher number of Steiner points in the portions of e closer to the endpoints, and a smaller number of Steiner points in portions closer to u e. The intuition is that if a segment along an optimal path intersects an edge e at a point close to u e, the segment will be relatively long. Therefore, it is always possible to find an approximate segment (a segment connecting two Steiner points) that neighbours the optimal segment. Further, 6

7 the cost of this approximate segment is not more than (1 + γ) times the cost of an optimal segment. Next we describe in more detail how the Steiner points are placed on the boundary edges. For a more thorough description see [18]. Let b e be the lesser of the maximum composite velocities of a robot travelling in either direction on e. For any vertex v, we use R v to give a lower bound on the cost of travelling (on any possible path) between v and any point on a boundary edge not incident to v. For any boundary edge e = v 1 v 2, u e divides edge e into two segments v 1 u e and u e v 2, as shown in Fig. 2. The first Steiner point u i,1 is placed on segment v i u e with distance b e R vi γ to vertex v i. The subsequent Steiner point u i,j is placed between u i,j 1 and u e with a distance γb e τ min (u i,j 1 ) to u i,j 1. We continue adding Steiner points until no more Steiner point can be added on v i u e. That is, if u i,j is the last Steiner point added, no more Steiner point is inserted on segment v i u e if v i u i,j + γb e τ min (u i,j ) v i u e. Finally, we add u e as a Steiner point on e. Throughout this paper we use σ = c skew γ. Theorem 1 (Adapted from Theorem 5 in [18]). For the discretization scheme in [18], the total number of Steiner points added to the triangular subdivision is O(n σ log σ). 2.2 Constructing the graph Given a source point s, a destination point t, a positive real value, and a triangular subdivision T, we will show how to build a graph G = (V, E ). Consider the face r T containing s. Triangulate r into three (possibly degenerate) triangles with apex at s. Then place Steiner points in geometric progression along the boundary of the subdivision (as described in Section 2.1), including the three new edges. Once the Steiner points are placed, we can construct a weighted directed graph G = (V, E ). The vertex set V of the graph G corresponds to the set of Steiner points, that is, for every Steiner point (or start and endpoint) p there is a vertex v p V. Let G = (V, E) denote the graph constructed by Reif and Sun [18]. In this graph two vertices v p and v q in V are connected by a directed edge of weight τ(p, q) if and only if p and q belong to the same face of the subdivision. Hence G has a quadratic number of edges. Our aim is to construct a graph G that closely approximates the distance in G while only having a linear number of edges in each region. We will show how the WSPD [7] can help us to achieve the goal. Definition 1 ([7]). Let s > 0 be a real number, and let A and B be two finite sets of points in R d. We say that A and B are well-separated with respect to s, if there are two disjoint d-dimensional balls C A and C B, having the same radius, such that (i) C A contains the bounding box R(A) of A, (i) C B contains the bounding box R(B) of B, and (ii) the minimum distance between C A and C B is at least s times the radius of C A. The parameter s will be referred to as the separation constant. The next lemma follows easily from Definition 1. 7

8 Lemma 2 ([7]). Let A and B be two finite sets of points that are well-separated w.r.t. s, let x and p be points of A, and let y and q be points of B. Then (i) xy (1 + 2/s) xq, (i) xy (1 + 4/s) pq, and (ii) px (2/s) pq. Definition 2 ([7]). Let S be a set of n points in R d, and let s > 0 be a real number. A well-separated pair decomposition (WSPD) for S with respect to s is a sequence of pairs of non-empty subsets of S, (A 1, B 1 ),..., (A m, B m ), such that 1. A i B i =, for all i = 1,..., m, 2. for any two distinct points p and q of S, there is exactly one pair (A i, B i ) in the sequence, such that (i) p A i and q B i, or (ii) q A i and p B i, 3. A i and B i are well-separated w.r.t. s, for 1 i m. The integer m is called the size of the WSPD. Callahan and Kosaraju showed that a WSPD of size m = O(s d n) can be computed in O(s d n + n log n) time. u q u φ q x α φ p f r f r p α θ Fig. 3. Illustrating the proof of Lemma 3. The following lemma is straight-forward but included for completeness. Lemma 3. Let A and B be two finite sets of Steiner points that are wellseparated w.r.t. s and let x and p be points of A, and let q be a point of B (see figure 3). Then τ(x, p) 4 s τ(p, q) and τ(p, x) 4 s τ(p, q). Proof. Let u p be the control velocity applied to travel from x to p within the face r and let u q be the control velocity applied to travel from p to q within r. Let b r be the maximum Euclidean norm of the control velocity that the robot can apply within r. In the triangle xu p we have τ(x, p) = u p b r and τ(p, q) = u q b r. (1) From (1) we have τ(x, p) τ(p, q) = u p u q (2) 8

9 In the triangle pu q, u q is greater than u p which follows from the definition of ρ r (see section 1.2). From (1) and (2) we get τ(x, p) τ(p, q) 2 xp pq 4 s pq = 4 pq s thus τ(x, p) 4 τ(p, q). s This concludes the proof of the first part of the lemma. The second part can be shown using similar arguments. Let V j be the set of Steiner points along the boundary of face r j, 1 j m. For each face r j, compute a WSPD {(A i, B i )} k i=1 of V j with respect to a separation constant s = 32 γ. Next, construct the graph G = (V, E ), where E is constructed as follows. For each well-separated pair {(A i, B i )} in r j, pick two arbitrary points a A i and b B i as representative points. Add the directed edges (a j, b j ) and (b j, a j ) to E with weights τ(a j, b j ) and τ(b j, a j ) respectively. Lemma 4. The graph G has O(σn log σ) vertices and O(s d σn log σ) edges and can be built in O((s d σn log σ) log(σn log σ)) time. Proof. The graph G has O(σn log σ) vertices according to the discretization scheme used in [18]. The number of edges is linear with respect to the number of vertices. The WSPD can be computed in time O((s d σn log σ) log(σn log σ)). By simply running Dijkstra s algorithm [10], implemented using Fibonacci heaps, on G gives the following theorem. Theorem 2. A minimum cost path between s and t in G can be computed in O(s d (σn log σ) log(σn log σ)) time using O(σn log σ) space. Our algorithm improves the time complexity of the algorithm by Reif and Sun algorithm [18] by roughly a factor of c skew γ. The reason for the improvement is the use of the WSPD as it reduces the complexity of the number of edges from quadratic to linear within a region. 2.3 Bounding the approximation error Let G and G be the two graphs as described above. Recall that V is the set of Steiner points on the boundary of the subdivision and E is constructed by connecting every pair of Steiner points on the same face of the subdivision (including s and t). Also recall that E is constructed by connecting every pair of representative points in the WSPD in the same region. Here we analyze how well a shortest path in G approximates a shortest path in G from a given source point s to a given destination point t. Let P be an optimal path between s and t in the continuous space with cost of D(P) and let P 1 be a minimal cost path in G between v s and v t in V. Denote the cost of P 1 by D(P 1 ) and let S = {b 1, b 2,..., b m 1, b m } be the vertices visited by P 1 in order of occurrence. Let δ G (u, v) and δ G (u, v) denote be the cost of an optimal path in G and G, respectively, between two vertices u and v in V. The following theorem is adapted from Theorem 6 in [18]. 9

10 Theorem 3 (Adapted from Theorem 6 in [18]). For any piecewise linear path P from a source point s V to a destination point t V, there exist a discrete path P 1 from s to t in G such that D(P 1 ) (1 + γ) D(P). Consider each segment b i b i+1, 1 i < m, along P 1. Recall that b i, b i+1 must be vertices in V but they may not be connected by an edge in G. However, for every pair b i, b i+1 there exists a well-separated pair (A, B) such that a i, b i A, a i+1, b i+1 B and, a i and a i+1 are connected by a directed edge in G. In the next lemma we will show that for every segment along P 1 there exists a path in G between b i and b i+1 that is almost as good as the direct path between b i and b i+1. Lemma 5. Let v p and v q be any pair of vertices in V whose corresponding Steiner points p and q lie in the same face r, it holds that δ G (v p, v q ) ( s ) δ G (v p, v q ), where s > 16 is the separation constant of the WSPD. Proof. The proof is done by induction on the Euclidean distance between p and q. We know that there exists a well-separated pair (A, B) such that p A and q B, and that there exists an edge from v p to v q in G with p A and q B. If the separation constant for the WSPD is s then we have that pp 2 s p q and q q 2 s p q, which follows from Lemma 2. Base case: Assume that (p, q) is the closest pair of Steiner points. In this case there exists a well-separated pair {(A, B)} such that A = {p} and B = {q}, otherwise (p, q) could not be the closest pair. Hence the claim holds since p = p and q = q, thus, there must be an edge in G from v p to v q. Induction hypothesis: Assume that the lemma holds for all Steiner point pairs closer than pq to each other. Induction step: According to Lemma 2, pp < p q and q q < p q. According to the induction hypothesis there is a path δ G (v p, v p ) with cost at most (1 + 32/s) δ G (v p, v p ) and a path δ G (v q, v q ) with cost at most (1 + 32/s) δ G (v q, v q ). Also, recall that the cost of the edge (v p, v q) in G is δ G (v p, v q ) = τ(p, q ). We get: δ G (v p, v q ) δ G (v p, v p ) + δ G (v p, v q ) + δ G (v q, v q ) < (1 + 32/s) δ G (v p, v p ) + (δ G (v p, v p ) + δ G (v p, v q ) + δ G (v q, v q )) +(1 + 32/s) δ G (v q, v q ) (2 + 32/s) τ(p, p ) + τ(p, q) + (2 + 32/s) τ(q, q) 4/s (2 + 32/s) τ(p, q) + τ(p, q) + 4/s (2 + 32/s) τ(p, q) (1 + 16/s + 256/s 2 ) δ G (v p, v q ) < (1 + 32/s) δ G (v p, v q ) From Lemma 5, we can now establish the following theorem: Theorem 4. For any piecewise linear path P from a source point p V to a destination point q V, we have δ G (p, q) (1 + ) D(P). 10

11 Proof. m δ G (p, q) (1 + 32/s) δ G (b j, b j+1 ) j=1 = (1 + 32/s) δ G (p, q) (1 + 32/s)(1 + γ)d(p) By setting s = /32 and γ = /4 the theorem follows since < 1. 3 Shortest Path Queries In this section we turn our attention to the query version. We present a data structure that, given two query points x and t, and a positive real value, returns a path in T between x and t of cost at most (1 + ) times the cost of an optimal path between x and t. We will start with the simpler case, when t is already known in advance and we are only given the source point x and > 0 as part of the query. Then, in Section 3.3 we show how to generalize it to the general case. 3.1 The preprocessing and the query In this subsection we will present the data structure, describe the preprocessing and show how a query is answered. Preprocessing In the preprocessing step we need to build three data structures, denoted M, N and P. M[ ] : In Section 2.2 we showed how to build a graph G given a triangular subdivision and two points x and t. Build the same graph, again denoted G (V, E ), but without including the source point x. Then compute the minimum cost path in G, using Dijkstra s shortest path algorithm [10], from every vertex in V to t. Note that this can be done by a single call to Dijkstra s algorithm from t to all other vertices in V provided that the directions of all edges have been reversed. The costs are stored in a vector M, such that for a vertex v V the entry M[v] stores the cost of the minimum cost path in G from v to t. According to Lemma 4 the complexity of G is linear with respect to the number of vertices, thus it takes O(s d (σn log σ) log(σn log σ)) time and requires O(σn log σ) space to build M[ ] [10]. For reasons that will become clear below we set s = max{16, /64} and γ = /4. N r [, ] : The second structure is an angle restricted nearest neighbor querying structure for each face r in the subdivision. Let κ 9 be a constant (to be defined below) and let θ = 2π/κ. If we rotate the positive x-axis by iθ, 0 i < κ, then we get κ rays, denoted r 1,..., r κ. Each pair of successive rays r i and r i+1, 11

12 1 i < κ, defines a cone X i whose apex is at the origin. The cone obtained by translating X i such that its apex is at a point q is denoted X i (q). Given a set S of n points in the plane we build a data structure N[q, i] that given a query point q R 2 and an integer i, 0 i < κ, returns a point x of S within the cone X i (q) whose orthogonal projection onto the bisector of X i (q) is the smallest. It has been shown (see for example Section in [16] or Lemma 2 in [4]) that such a structure can be preprocessed in O(κn log n) time into a data structure of size O(κn) such that queries can be answered in O(log n) time. For our purposes we will set κ = 50/. P[ ] : Finally, the triangular subdivision is preprocessed for efficient point location queries as described in Chapter 6.1 in [3], which can be performed in time O(σn log σ) log(σn log σ)). That is, given a query point q the data structure P returns the triangle in the subdivision that contains q. Theorem 5. Given a positive constant, the preprocessing requires O(s d ( c skew n ) log( c skew n )) time and O( c skew n ) space. x s x θ s X i (x) Fig. 4. (a) Partitioning the plane into κ cones with apex at x. (b) Selecting the point within X i(x) whose orthogonal projection onto the bisector of X i(x) is the smallest. Query As a query we are given a point x in the plane. Perform a point location query P[x] to determine which face r contains x. If x and t lie in the same region or in adjacent regions, then the approximate path from x to t is the direct path between them and can be computed as in Section 1.2. Note that if x and t are neither in the same region nor in adjacent regions, then according to Lemma 14 in [18] the approximate path from x to t can not be the direct path (it must go through Steiner points) and in this case we do the following. For each i, 1 i < κ perform an angle constrained nearest neighbor query N r [x, i], as shown in Fig. 4. Consider a cone X i (x). We have two cases: (1) X i (x) contains a point of V, or (2) X i (x) is empty of Steiner points. 12

13 Recall that the data structure M stores the minimum cost path to t from every vertex in V. Thus, our task is to find a good vertex v p in V such that τ(x, p) + M[v p, t] approximates D(P). Case 1: For each non empty cone X i (x), 1 i < κ, let v(i) := N r [x, i], see Fig. 4. Next, compute the cost of the minimum cost path P i in G between v x and v t via v(i), that is, τ(x, v(i)) + M[v(i)]. Case 2: For each empty cone X i (x), 1 i < κ, let CB be a boundary edge of face r that intersects with X i (x), as shown in Fig. 5. Let r 1 be the region that shares a boundary edge CB with face r. Let c 1 and c 2 be the points of intersection between the two boundaries of X i (x) and CB, such that c 1 is closer, or as close, to x as c 2, as shown in Fig. 5. Note that c 1 and c 2 may or may not correspond to vertices in V. For each cone X i (c 1 ), 1 j < κ, let v(j) := N r [x, j]. Compute the cost of the minimum cost path in G between v x and v t via v(j), that is, τ(x, c 1 ) + τ(c 1, v(j))+m[v(j)]. The path of smallest cost among these paths is denoted P i,j, that is P i is the path that minimizes τ(x, c 1) + τ(c 1, v(j)) + M[v(j)] among all v(j). To conclude the query structure reports the path, denoted P, between x and t that has the minimum cost among all computed paths, that is, P is the path with the smallest cost among all paths P i, 1 i κ. The approximation bound and the query time will be proven in the next section. C D A b 1 c 1 x r X i r 1 b 4 a 2 a 1 c 2 b 2 b 3 B Fig. 5. Handling the case of the empty cone X i. 13

14 3.2 Approximation bound We establish the following lemma that provides a bound on the error of using a discrete path to approximate an optimal path. As mentioned earlier, we have two cases for the query either (1) the cone where the first edge along P lies in contains a Steiner point, or (2) the cone where the first edge along P lies in is empty. In the following we show that D(P ) (1 + ) D(P) for the two cases. C u u fr x 2 γ 2 x 1 x 3 B θ1 θ 2 γ 1 α x α θ 2 θ1 φ 2 φ 1 β A Fig. 6. Illustrating the notations in the proof of Lemma 6. Lemma 6. Let X i (x) be the cone where the first segment along P is contained. If X i (x) contains a Steiner point then the data structure returns a path P such that D(P ) (1 + ) D(P). Proof. We will prove the lemma by comparing the cost of P to the cost of a minimum cost path, denoted P 1, in G(V, E) similar to the proof of Lemma 5. We will need some notations to prove the lemma, see also Fig. 6 for an illustration. Let r = ABC be the triangular face containing the source point x. The flow in r is f r and it has a fixed maximum Euclidean norm b r. Let β = BAC and let xu be the first segment along P 1 and let xu be the first segment along P such that u and u lie in the cone X i (x). Let α be the angle between BA and fr, let θ 2 be the angle between BA and the ray containing xu and let θ 1 be the angle between BA and the ray containing xu. Let x 3 be the perpendicular projection of point u on xu if applicable. In the triangle xx 1 x 3 the vector x 1 x 3 represents the control velocity of a robot travelling from x to x 3 in region r. In the triangle xx 2 u the vector x 2 u represents the control velocity of a robot travelling from x to u within r. Let φ 1 = x 1 x 3 x and let φ 2 = x 2 u x. ). Let γ 1 = xx 1 u and γ 2 = xx 2 u. We have D(P 1 ) = τ(x, u )+δ G (u, t) and D(P ) = τ(x, u )+δ G (u, t). Since P is the path from u to t with the smallest cost we have (from Lemma 5 and According to Lemma 1, φ 1 = arcsin( sin(α θ1) ρ r ) and φ 2 = arcsin( sin(α θ2) ρ r 14

15 Theorem 4): D(P ) τ(x, u ) + τ(u, x 3 ) + τ(x 3, u ) + (1 + 32/s)... δ G (u, t) (3) Now we need to compare τ(x, u ) + τ(u, x 3 ) against τ(x, x 3 ). In the triangle xx 1 x 3 we have x 1 x 3 = τ(x, x 3 ) b r, hence and in xx 2 u we have From (4) and (5) we have τ(x, x 3 ) = x 1x 3 b r (4) τ(x, u ) = x 2u b r and τ(u, x 3 ) < u x 3 b r. (5) τ(x, u ) + τ(u, x 3 ) τ(x, x 3 ) < x 2u x 1 x 3 + u x 3 x 1 x 3 (6) In the following we will evaluate x2u x 1x 3 and u x 3 x 1x 3. First we bound u x 3 x 1x 3. Since ρ r > 1, as mentioned in Section 1.2, and ρ r = br f r, it follows that in xx 1x 3 we have x 1 x 3 > xx 1. Again we have x 1 x 3 > 1 2 xx 3 and hence: u x 3 x 1 x 3 < 2 u x 3 xx 3 (7) By applying Law of Sine on xu x 3, we get the following: u x 3 xx 3 = sin(θ 2 θ 1 ) cos(θ 2 θ 1 ). (8) From (7), (8) we have u x 3 x 1 x 3 < 2 sin(θ 2 θ 1 ) cos(θ 2 θ 1 ). (9) Now we will evaluate x2u x 1x 3. By applying Law of Sine on xx 1x 3, we get: x 1 x 3 = xx 3 sin(α θ 1). (10) sin(γ 1 ) Similarly, for xx 2 u, we can use Law of Sine to get the following equation: x 2 u = xu sin(α θ 2). (11) sin(γ 2 ) 15

16 From (10) and (11) we have x 2 u x 1 x 3 = xu xx 3 sin(α θ 2) sin(γ 1 ) sin(α θ 1 ) sin(γ 2 ). (12) In the right angle triangle xx 3 u, we have which together with (12) gives: From (6), (9) and (14) we get τ(x, u ) + τ(u, x 3 ) τ(x, x 3 ) xx 3 = xu cos(θ 2 θ 1 ) (13) x 2 u x 1 x 3 = sin(α θ 2 ) sin(γ 1 ) sin(α θ 1 ) cos(θ 2 θ 1 ) sin(γ 2 ). (14) < sin(α θ 2 ) sin(γ 1 ) sin(α θ 1 ) cos(θ 2 θ 1 ) sin(γ 2 ) +2 sin(θ 2 θ 1 ) cos(θ 2 θ 1 ). (15) The term 2 sin(θ2 θ1) cos(θ 2 θ 1) is less than which follows from the fact that the spanning angle θ sin(α θ = θ 2 θ 1 < 2π/κ and κ = 50/. In addition the term ( 2) sin(γ 1) sin(α θ ) 1) cos(θ 2 θ 1) sin(γ 2) 1, hence τ(x, u ) + τ(u, x 3 ) < (1 + /2) τ(x, x 3 ). (16) C x 4 φ 3 u A fr x 5 x x 3 1 α φ 1 x θ α 1 Fig. 7. Comparing τ(x, u ) versus τ(x, x 3) + τ(x 3, u ) B It remains to proof that τ(x, u ) = τ(x, x 3 ) + τ(x 3, u ). In xu x 5, the vector x 5 u represents the control velocity of a robot travelling from x to u (see Fig. 7). In triangle x 3 x 4 u, x 4 u represents the control velocity of a robot travelling from x 3 to u. Let φ 3 = x 5 u x. According to Lemma 1 φ 3 = arcsin( sin(α θ1) ρ r ). As shown in Fig. 7, τ(x, x 3 ) and τ(x 3, u ) lie on the same straight line, that 16

17 means that they share θ 1. Since φ is function in θ, then φ 1 = φ 3 as well, that means that x 1 x 3 and x 5 u are parallel and hence we have In the triangle x 3 x 4 u we have: From (17) and (18) we get: From (16) and (19) we get; From (3) and (20) we get: τ(x, x 3 ) = x 1x 3 b r = x 5x 4 b r. (17) τ(x 3, u ) = x 4u b r. (18) τ(x, x 3 ) + τ(x 3, u ) = τ(x, u ). (19) τ(x, u ) + τ(u, x 3 ) + τ(x 3, u ) (1 + /2) τ(x, u ). (20) D(P ) (1 + /2) τ(x, u ) + (1 + 16/s) δ G (u, t) < (1 + /2) D(P 1 ). (21) Putting together Theorem 3 and equation (21) we get D(P ) (1 + γ) (1 + /2) D(P ). And by putting γ = /4 we get D(P ) < (1 + ) D(P ). This finishes the proof. It remains to consider Case 2, if the cone where the first segment along P lies in is empty. Assume without loss of generality that b 1 and b 2 are the Steiner points closest to c 1 and c 2 respectively on the boundary edge CB as shown in Fig. 5. Let a 1,..., a m be the points along P where P bends. The following lemma is adapted from Lemma 14 in [18]. Recall that the construction of the set of Steiner points is described in Section 2.1. Lemma 7 (Adapted from Lemma 14 in [18]). Let r = BCD be a face containing a path segment a i a i+1 of an optimal path P. Assume that a i lies between two Steiner points b 1 and b 2 on the boundary edge CB and a i+1 lies between two Steiner points b 3 and b 4 on the boundary edge DB, as shown in Fig. 5. Recall that u e is the point on the boundary edge e with maximum τ min (u). Given a positive constant φ and a destination point t in V then, if b 1 lies between C and u e then τ(b 1, b 3 ) < (1 + φ) τ(a 1, a 2 ) and δ G (b 3, t) < (1 + φ) τ(a 2, t) otherwise τ(b 2, b 4 ) < (1 + φ) τ(a 1, a 2 ) and δ G (b 4, t) < (1 + φ) τ(a 2, t). Lemma 8. Let X i (x) be a cone that contains the first segment along P. If X i (x) is empty then there exists a path P in G such that D(P ) (1 + ) D(P). 17

18 Proof. According to the construction of P we have D(P ) = τ(x, c 1 )+τ(c 1, v(j))+ δ G (v(j), v t ) while the optimal path can be described as D(P) = τ(x, a 1 ) + τ(a 1, a 2 ) + τ(a 2, t). The proof is divided into parts. In the first part we prove that τ(x, c 1 ) (1+ ) τ(x, a 1 ) and in the second part we prove that τ(c 1, v(j))+δ G (v(j), v t ) (1+ ) (τ(a 1, a 2 ) + τ(a 2, t)). The first part can be proved using the same arguments used in Lemma 6 so we will omit it here. The second part can be proved as follows. Without loss of generality we assume that b 1 lies between C and u e, and using Lemma 7 we have: τ(b 1, b 3 ) < (1 + φ) τ(a 1, a 2 ) and δ G (v b3, v t ) < (1 + φ) τ(a 2, t). (22) Recall that to compute the second segment, denoted (c 1, v(j)), along P we partition the face r 1 into κ cones with apex at c 1. Then using the same arguments as in Lemma 6 together with (22), we can prove the following: τ(c 1, v(j)) < (1 + φ) τ(a 1, a 2 ). (23) From Lemma 7 and Theorem 4 we get: δ G (v b3, v t ) < (1 + ) τ(a 2, t). Using exactly the same argument we can prove that δ G (v b4, v t ) < (1 + ) τ(a 2, t), which also implies that δ G (v(j), v t ) < (1 + ) τ(a 2, t). (24) By setting φ = and then using (23) and (24) we get: τ(c 1, v(j)) + δ G (v(j), v t ) < (1 + ) (τ(a 1, a 2 ) + τ(a 2, t)) (25) Finally putting together the first and second part of the proof we obtain the desired result D(P ) (1 + ) D(P). Theorem 6. Given a planar triangular subdivision T of complexity n with a translational flow, a point t R 2 and a positive constant, one can preprocess T in O(( c skew n ) ) log( c skew n )) time using O(( c skew n ) ) space such that given a query point x R 2 a (1 + )-approximate minimum cost path between s and t can be calculated in O(1/ 2 log( c skew n )) time. 3.3 General case In this section we turn our attention to the general query version when we are given two query points s and t in R 2 and our goal is to find a minimum cost path between x and t. The idea is the same as in the previous section. That is we perform the exact same preprocessing steps as in the previous section (omitting the source point x and the destination point t), but with the exception that M contains all-pair shortest costs. Using Johnson s algorithm [11] the all-pairs shortest paths can be computed in O(s d ( c skew n O(( c skew n ) 2 ) space. 18 ) 2 ) log( c skew n )) using

19 The query is performed as in the previous section, however, for both x and t one needs to find a good next point along the path. That is, instead of only searching for a point in each cone X i (x), we also need to search for a good point in each X i (t) and then try all combinations of the two. This will add a factor of 1/ 2 to the bounds in Theorem 6. By putting together the results, we obtain the following theorem: Theorem 7. Given a planar triangular subdivision T of complexity n with a translational flow and a positive constant, one can preprocess T in time O(( c skew n ) 2 ) log( c skew n )) using O(( c skew n ) 2 ) space such that given two query points x and t a (1 + )-approximate minimum cost path between x and t can be calculated in O(1/ 4 log( c skew n )) time. 4 Nearest neighbor query In this section we show how the structure we presented in the previous section easily can be modified for other problems. We consider the nearest neighbor(nn) and the k-nearest neighbor. We show how the data structure in Section 3 can be modified to answer (1 + )-nearest neighbor queries in anisotropic media. That is, given a planar triangular subdivision T, a set Z of m sites and a constant > 0, preprocess T and Z into a data structure such that given a query point q R 2, a (1 + ɛ)- approximately nearest site in Z is returned. 4.1 Preprocessing and query The preprocessing is done in almost the same way as in Section 3.1 but with the exception that the graph differs slightly than in Section 3.1 as follows. First omit the source point s and then for every site z i Z, we do the following. Triangulate the face r containing the z i into three (possibly degenerate) triangles with apex at z i. Then place Steiner points in geometric progression along the boundary of the subdivision (as described in Section 2.1), including the three new edges. Once the Steiner points are placed, we can construct a weighted directed graph G = (V, E ). Then for every non cite vertex in V compute the minimum cost path in G to every cite vertex, using Dijkstra s shortest path algorithm [10]. Note that this can be done by calling Dijkstra s m times. Then for each non cite vertex store in matrix M the cost of the minimum cost path among the computed paths. The query is done in almost the same way as in Section 3.1. Note that the approximate nearest cite is τ(x, p) + M[v p ], where M[v p ] is the minimum cost of the path from v p to any site. Using the same arguments as in the proof of Theorem 6 we can conclude: Theorem 8. Given a planar triangular subdivision T of complexity n with a translational flow, a point t R 2 and a positive constant, one can preprocess T in O(m ( c skew n log( c skew ) + m) log( c skew n log( c skew ) + m)) time using O(( c skew n ) log( c skew ) + m) space such that given a query point x R 2 a 19

20 (1 + )-approximate minimum cost path between s and t can be calculated in O(1/ 2 log( c skew n + m)) time. The above approach can also be extended to the case when one wants to report the approximate k nearest neighbors. 4.2 Preprocessing and query The preprocessing is almost the same as done with the nearest neighbor query. The only difference is that for each non cite vertex p in G we store in matrix M 2 the minimum cost path to all cites ordered according to their closeness from p. This step can be done in O(m ( c skew n log( c skew )+m) log( c skew n log( c skew )+m)) time using O(( m c skew n ) log( c skew ) + m) space. The query is done in almost the same way as in Section 3.1. The only difference is instead of computing the path minimum cost path from a good vertex to a destination point t, we compute the minimum cost path from each good vertex to each site. Finally, sort all the computed paths with respect to their increasing distance from q. Finally report the k approximate nearest neighbors. Using the same arguments as in the proof of Theorem 6 we can conclude: Theorem 9. Given a planar triangular subdivision T of complexity n with a translational flow, a point t R 2 and a positive constant, one can preprocess T in O(m ( c skew n log( c skew ) + m) log( c skew n log( c skew ) + m)) time using O(( c skew n ) log( c skew ) + m) space such that given a query point x R 2 a (1 + )-approximate minimum cost path between s and t can be calculated in O(1/ 2 m(log( c skew n ) + m)) time. 5 Concluding Remarks We considered the problem of computing a minimum cost path in a subdivision with a translational flow. In the basic case we presented an algorithm that slightly improves the running time of the algorithm by Reif and Sun [18]. Our main contribution is an effective data structure for the query version of the problem. There are many open problems remaining. For example, can one develop a more efficient data structure that has a smaller dependency on the triangulation? We believe our construction can be generalized to higher dimensions, but at what cost? References 1. L. Aleksandrov, H. N. Djidjev, H. Guo, A. Maheshwari, D. Nussbaum and J. R. Sack. Algorithms for approximate shortest path queries on weighted polyhedral surfaces. Discrete and Computational Geometry, 44(4): , L. Aleksandrov, M. Lanthier, A. Maheshwari and J.-R. Sack. An epsilon- Approximation for Weighted Shortest Paths on Polyhedral Surfaces. In Proceedings of the 6th Scandinavian Workshop on Algorithm Theory,

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