Voronoi Diagrams with a Transportation Network on the Euclidean Plane

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1 Voronoi Diagrams with a Transportation Network on the Euclidean Plane Sang Won Bae Kyung-Yong Chwa Division of Computer Science, Department of EECS, Korea Advanced Institute of Science and Technology, Daejeon, Korea {swbae,kychwa}@jupiter.kaist.ac.kr Abstract This paper investigates geometric and algorithmic properties of the Voronoi diagram with a transportation network on the Euclidean plane. With a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm 2 + m 3 + nm log n) algorithm to compute the Voronoi diagram with a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm+m 2 +n log n) time with O(m(n+m)) space and the other O(nm log n + m 2 log m) time with O(n + m) space. Both algorithms output the diagram with linear size. Keywords: Transportation Network; Time Metric; Voronoi Diagram; Shortest Path. 1 Introduction With a transportation network, like streets in a city, subway or bus networks, or a highway over a nation, people have become capable to move faster. Finding a shortest (time) path using such transportation is very important due to the tendency of people trying to move with the shortest duration. In this situation, let us imagine a set of sites or service stations, and suppose that one should visit any of them; here, the Voronoi diagram problem arises. This paper considers geometric and algorithmic issues about the Voronoi diagram under such situation. The Voronoi diagram on the plane with a transportation network has been studied by several researchers. Abellanas et al.[1] discussed a single innite road network as a straight line lying on the Euclidean plane. Under their setting, they showed that the Voronoi diagram for n points can be computed in O(n log n) time. In addition, transportation networks on the L 1 plane have been dealt with; Abellanas et al.[2] considered isothetic and monotone networks and, more generally, Aichholzer, Aurenhammer, and Palop[3] introduced the city Voronoi diagram, the Voronoi diagram with an isothetic transportation network on the L 1 plane and presented an algorithm computing city Voronoi diagrams based on some relations among city Voronoi diagrams, straight skeletons, and abstract Voronoi diagrams. This work is supported by grant No.R from KOSEF. 1

2 The PhD dissertation of Palop[4] gives us a comprehensive survey for algorithmic problems on proximity and location under metrics induced by transportation networks on the plane. The author considered not only nearest Voronoi diagrams but also farthest Voronoi diagrams under such metrics. Also, transportation networks of wedge shape (two half lines with a common endpoint) or of a circle were taken into account. Actually, the plane with a transportation network can be regarded as a special and degenerate case of a weighted region. Mitchell and Papadimitriou[5] posed the weighted region problem and provided an interesting solution based on Snell's law of refraction. The weighted region problem is based on a partition of the plane into polygonal regions, each associated with an individual weight as the traveling speed inside the region. The authors provided an algorithm constructing a shortest path map under their assumption in O(m 8 L) time, where m is the complexity of the polygonal partition and L denotes the precision of the problem instance. All the previous work on transportation networks has assumed that every road in a transportation network has the same speed. But, as mentioned at the beginning, several kinds of transportation networks are established on our surroundings, hence transportation networks with diverse speeds need to be considered. This paper is the rst result dealing with transportation networks with multiple roads, possibly having arbitrary directions and diverse speeds, on the Euclidean plane. 1.1 The Model A transportation network consists of several roads as non-intersecting line segments on the plane, which may share a node, which is an endpoint, in common. Each road has its own speed. This structure can be generally represented as a planar straight-line graph G = (V, E) with speed v(e) on each road e E, where V is a set of nodes and E is a set of roads. In our model, one can enter into and exit from G at any point of roads or nodes. Along any road e E, one moves at xed speed v(e). Outside the network G, one moves at unit speed in any direction. If entry or exit points are restricted to nite xed points on the roads, a Voronoi diagram for additively weighted points is yielded; this problem can be computed in an optimal time.[3] We assume that v(e) > 1 for every road e E, since, if v(e) 1, the road e does not contribute any shortest paths. Given a transportation network G, the distance with the transportation network G, called d G, is measured as the length (duration) of the shortest (time) path using roads in G. In fact, d G induces a metric on R 2 as a shortest path metric. We call the metric induced on R 2 by d G the transportation metric with G. 1.2 The Results and the Organization We present two algorithms for constructing the Voronoi diagram V G (S) under the transportation metric with G for a set S of n sites. The rst algorithm runs in O(nm 2 log n + m 3 log m) time and uses O(m(n + m)) space, where m is the number of roads in G. In the second algorithm, we improve the running time to O(nm 2 + m 3 + nm log n) time. As a special case, we can reduce the complexity when the roads in the given transportation network have only a constant number of directions and speeds. In this case, we can compute the diagram in O(nm + m 2 + n log n) time with O(m(n + m)) space, or in O(nm log n + m 2 log m) time with linear space. These algorithms are presented in Section 5 and Section 6. In doing so, we introduce a generalized site, a needle. A needle generalizes a weighted line segment so that its weight is varied linearly. A Voronoi diagram for needles satisfying a certain 2

3 property, so called non-piercing, on the Euclidean plane can be computed in O(n log n) time and in O(n) space. Arguments and details about needles are presented in Section 3. And the algorithms that we present are based on several observations on transportation networks and on needles, which will be investigated in Section 4. Under transportation metrics, it is not so easy any more to plan a shortest path, even if a source and a destination are determined. In Section 7, we discuss how to plan a shortest path under the transportation metric with G. Given a set S of n sites and a transportation network G with m roads, we will show that a shortest path to a nearest site from a given point can be reported in O(log(m+n)+r) time by point location over the diagram obtained by the algorithms, where r is the complexity of the reported path. Also, a shortest path map for a xed source point under the transportation metric can be constructed in O(m 3 ) time, in general. Summary, conclusions, future works, and open issues are given in Section 8. 2 Basic Idea In this section, we describe our basic idea to solve the problem through an observation on the plane with a transportation metric. The transportation metric has some bad properties so that computing Voronoi diagrams under it is not so easy. For example, we can easily construct a cyclic bisector with Ω(m) segments between two points; a transportation network with cyclic roads and two sites, one of which is much closer to a road than the other, see Figure 1. This violates one of necessary conditions to apply the abstract Voronoi diagram, one of the most popular and useful tools to compute Voronoi diagrams (for details about abstract Voronoi diagrams, see Subsection 3.2 or Klein[6]). Thus, we should nd a different approach to solve the problem. Figure 1: A cyclic bisector. Black solid line segments are roads, two small circles are sites, and gray curved segments are the bisector between the two sites under the transportation metric. To nd such a different approach, we have to investigate the plane with a transportation network more carefully. Let us consider a simpler case, a single road transportation network. Given a single innite road on the Euclidean plane, we know how to nd a shortest path by Abellanas et al.[1] We draw the following observation from their results. Observation 1. To reach a destination as quickly as possible, in accessing a road e, 1. The entering or exiting angle with the road e should be π/2 ± α, where sin α = 1/v(e).[1] 2. If above is impossible, access the road e through the closer node of e. 3

4 Figure 2: Illustrations of t-neighborhoods when G has a single road. Observation 1 can be also shown by calculating and analyzing some equations, but it is believed that a simple observation or intuition is sufcient to show it. First, previous results tell us that, given an innitely long road, a shortest path is either a path that uses the road and enters or exits the road with angle π/2 ± α, or a direct path. This implies the rst rule of Observation 1. Assuming that we are given a road as a segment, if one cannot directly enter or exit the road with angle π/2 ± α, then he or she had better contact a node of the road; this implies the second rule of Observation 1. By Observation 1, we can nd the t-neighborhood of p under the transportation metric with G, dened as {x d G (x, p) < t}, for any point p on the plane and t 0. We shall denote it by N G (p, t). Figure 2 shows four illustrations for the boundary of N G (p, t) when G is a single road network. Note that N G (p, t) can be represented as a union of one large disk centered at p and two needleshaped regions along the road, each of which is a union of a set of disks whose radii are linearly decreased along the road. This provides an intuition to analyze the transportation metric in a different point of view. Our strategy is to consider such a needle-shaped region produced from p as an independent Voronoi site under the Euclidean metric. In Section 3, we dene such a Voronoi site, called a needle. Then, we can compute the distance with more roads by using needles on the roads, if we can compute the distance to a needle from any point under the Euclidean metric. We will show how to compute a set of needles from given sites S, which allows us to properly analyze the given transportation network G, in Section 4 and 5. 3 Needle: a Generalized Site In this section, we introduce a needle as a more generalized Voronoi site. Needles play a very important role in computing diagrams and in planning shortest paths under the transportation metric. In Subsection 3.1, we give a formal denition of needles and show how the distance from any point to a needle can be computed. In Subsection 3.2, we discuss how to compute Voronoi diagrams for needles on the Euclidean plane. 3.1 Denition of a Needle A needle is generalized from a line segment with an additive weight. A line segment with additive weight w can be viewed as a set of weighted points with weight w in common. Note that the distance to a weighted point p with weight w p from any point q is equal to d(p, q) + w p. A needle is also a weighted line segment but weights of its points are allowed to be varied. One necessary condition for a needle is that the weight on a needle should be assigned linearly. With this property, a needle is suitable to represent N G (p, t), as noted in Section 2. The following is a denition of a needle. 4

5 (a) t 1 < t < t 2 (b) t = t 2 (c) t > t 2 Figure 3: A needle and its t-neighborhoods. Denition 1. A needle p is an additively weighted line segment s(p), whose endpoints are p 1 and p 2, with weight function w p. The weight w p (x) is given linearly from t 1 to t 2, for each x s(p) and 0 t 1 t 2, such that w p (p 1 ) = t 1 and w p (p 2 ) = t 2. The needle p is represented by a 4-tuple, (p 1, p 2, t 1, t 2 ). We may use the terms p 1 (p), p 2 (p), t 1 (p), and t 2 (p), instead of p 1, p 2, t 1, and t 2, respectively. Figure 3 shows t-neighborhoods N(p, t) of a needle p under the Euclidean metric, when t 1 < t < t 2, t = t 2, and t > t 2, case by case. As t increases from 0, N(p, t) grows from a point p 1 (p) at t = t 1 (p) and reaches p 2 (p) at t = t 2 (p) along s(p). We also dene the speed of a needle p as v(p) = d(p 1 (p), p 2 (p))/(t 2 (p) t 1 (p)) and its direction as direction toward p 2 (p) from p 1 (p). Note that if v(p) 1, N(p, t) is of shape of a disk, and thus we can regard p as a weighted point p 1 (p) with weight t 1 (t). Throughout this paper, we will draw a needle as an arrow headed in its direction, as shown in Figure 3. We now discuss the distance from any point to a needle under the Euclidean metric. For any point x R 2 and any needle p, the distance is represented as follows: d(x, p) = min y s(p) {d(x, y) + w p(y)}, where w p (y) is the weight assigned to y. w p (y) can be computed by the denition of a needle as follows: w p (y) = d(p 1 (p), y) t 1 (p) + (t 2 (p) t 1 (p)) d(p 1 (p), p 2 (p)) = t 1 (p) + d(p 1 (p), y)/v(p), for all y s(p). We dene some terms associated with a needle p. Without loss of generality, we assume that p is horizontal and p 1 is to the left of p 2. Let l 1 denote the line whose slope is tan α and which meets p 1, and l 2 denote the line whose slope is tan(π/2 + α) and which meets p 2, where α = sin 1 (1/v(p)). We dene s + (p) to be a line segment associated with p whose endpoints are p 1 and the intersection point between l 1 and l 2. Symmetrically, we dene s (p); see Figure 4. Next, we let p + be the set of points above s(p) such that, for any point x p +, a line, which is perpendicular to s (p) and meets x, intersects s (p). In this manner, we also dene p, symmetrically. Note that p + and p decompose R 2 into four disjoint and connected regions as shown in Figure 4. Lemma 1. Let p be a needle represented by (p 1, p 2, t 1, t 2 ). Then, the distance from any point x R 2 to p is as follow. d(x, s (p)) + t 1 if x p + d(x, p) = d(x, s + (p)) + t 1 if x p min{d(x, p 1 ) + t 1, d(x, p 2 ) + t 2 } otherwise 5

6 p + l 2 l 1 α s + (p) s(p) s (p) p α Figure 4: Plane decomposition around a needle p Proof. The t-neighborhood by the above equation is represented by N(s (p), t t 1 ) p + N(s + (p), t t 1 ) p (N(p 1, t t 1 ) N(p 2, t t 2 )) (R 2 \ (p + p )). One can easily show this is exactly the same as the t-neighborhood of a needle (p 1, p 2, t 1, t 2 ). By the above lemma, we can easily compute the distance from any point to a needle in constant time. 3.2 Voronoi Diagrams for Needles under the Euclidean Metric In this subsection, we compute the Voronoi diagram for a set of needles. In order to do so, we apply the abstract Voronoi diagram introduced by Klein.[6] In this model, a system (S, {J(p, q) p, q S, p q}) of bisecting curves for S is given, which is called admissible if the following conditions are fullled: 1. J(p, q) is homeomorphic to a line or empty, 2. the intersection of any two bisectors consists of nitely many components, 3. R(p, q) R(q, r) R(p, r), and 4. for any subset S S and p S, R(p, S ) is path-connected if it is nonempty, where R(p, q) = {x R 2 d(x, p) < d(x, q)}, R(p, S) = q S,p q R(p, q) and J(p, q) is the bisecting region between p and q. Needles, however, violate the rst and the fourth conditions, since needles generalize weighted line segments. Note that the bisector between two needles p and q is dened as J(p, q) = {x d(x, p) = d(x, q)}, which is computable in constant time by Lemma 1. We suggest a condition restricting needles so that the bisector system is admissible. We consider the following four cases for each needle p with respect to another needle q. 1. R(p, q) s(q) =. 2. R(p, q) s(q) and R(p, q) s(q) includes either p 1 (q) or p 2 (q), not both. 3. R(p, q) s(q) and R(p, q) s(q) includes both p 1 (q) and p 2 (q). 6

7 4. R(p, q) s(q) and R(p, q) s(q) includes neither p 1 (q) nor p 2 (q). We call two needles non-piercing if and only if, for each with respect to the other, the fourth case does not occur. Otherwise, we call them piercing. With only non-piercing needles, we are sufcient to solve our problem. We thus focus on a set of pairwise non-piercing needles. Note that the nonpiercing condition for needles generalizes the disjointness for line segments. Theorem 1. Let S be a set of pairwise non-piercing needles. Then, the system of bisecting curves for S, (S, {J(p, q) p, q S, p q}) is admissible. Proof. First, we show that J(p, q) is homeomorphic to a line. We can divide R 2 into p +, p and R 2 \ (p + p ), depending on the structures of shortest paths to p. These are four disjoint connected regions, we call them R p,1,, R p,4, see Figure 4. We can assign a function d p,i to each region R p,i such that d p,i (x) = d(x, p) for any point x R p,i and d p,i can further be interpreted as the distance to a certain weighted line segment or a certain point, which is placed out of the region R p,i. If x is on the boundary of R p,i and R p,j, d p,i (x) = d p,j (x), surely. Now, considering another needle q, R 2 can be divided into at most 16 regions {R p,i R q,j } 1 i,j 4. By the above argument, in each region we may have a bisecting curve, whose type may be a line segment, a parabolic arc or a hyperbolic arc. Further, such bisecting curves may share their endpoints only on the boundaries between regions. Hence, J(p, q) is a set of sequences of several quadratic curve segments. If p and q are non-piercing, the three regions, J(p, q), R(p, q), and R(q, p) are connected regions and have nonempty interiors, since a needle can be interpreted as a set of linearly weighted points, as described in Subsection 3.1. This implies J(p, q) is connected and acyclic, that is, homeomorphic to a line. Also, the above discussion implies nite connected intersections, since J(p, q) can be decomposed into at most 16 quadratic curve segment, each of which can be empty. For the second condition, if x R(p, q) R(q, r), then d(x, p) < d(x, q) < d(x, r), which implies that x R(p, r). Hence, R(p, q) R(q, r) R(p, r). Finally, we pick any subset S S. Since S is pairwise non-piercing, S is also pairwise nonpiercing. Thus, for any p S, R(p, S ) s(p), say U p, is connected or empty, and every p s(p) \ U p is dominated by S \ {p}. Hence, R(p, S ) is connected, further, path-connected, if it is nonempty. Constructing abstract Voronoi diagrams has been considered by several researchers, Klein[6], Mehlhorn et al.[7], Klein et al.[8], and Dehne and Klein [9]. All presented algorithms take an optimal time and space. Thus, we conclude the following corollary as a result of this section. Corollary 1. The Voronoi diagram for a set of n pairwise non-piercing needles on the Euclidean plane can be computed in O(n log n) time and O(n) space. In fact, Aichholzer, Aurenhammer, and Palop[3] introduced the concept similar to non-piercing needles. The authors discussed their concepts in the wavefront model and they dened a needle as a gure produced by the wavefront of a site while touching road segments. Note that, in the wavefront model, each site p sends out a wavefront and the interference pattern of these wavefronts contributes the diagram. In contrast with their approach, we dene a needle as an independent Voronoi site. 4 Relation between Transportation Networks and Needle Sets In this section, we discover a relation between a given transportation network and a set of needles. More precisely, we will show that there exists a set S of needles such that its Voronoi diagram V(S) 7

8 under the Euclidean metric induces the diagram V G (S) for a set S of sites. Further, we will show that we can make the set S pairwise non-piercing through some procedures, in Subsection 4.1. Also, several lemmas that reveal computational and combinatorial properties about transportation networks and needles will be presented in Subsection 4.2. Discussions and results in this section will be helpful to develop algorithms in the next section. 4.1 The Reducibility A point p produces at most two new needles on a road e (see Figure 2). These needles are produced either at a node of e or at an intersection point of the road with a line whose angle with e is π/2 ± α and which meets p by Observation 1. One important fact is that the produced needles are also Voronoi sites and thus they possibly produce new needles on other roads. We will call such a newly produced needle a needle produced on a road from a (parent) needle. Note that a needle also produces in general at most two new needles on a road, as a point does. Suppose that a needle produces three needles on a road. Then, two of the three should be in the same direction and one of these two needles must be dominated by the original needle and/or other newly produced ones, since the speeds of the three needles are the same and their neighborhoods are convex under the Euclidean metric (see Figure 2 and 3). More formally, we dene σ G (p) to be a set of needles produced from a needle p on all roads in G. For a set X of needles, σ G (X) = p X σ G(p). In order to obtain all needles produced from a site p, we will apply σ G ( ), repeatedly. We let σg k (p) = σ G(σ k 1 G (p)) and σ0 G (p) = {(p, p, 0, 0)}. We then let Sp k denote k i=0 σi G (p) and S p denote Sp. Now, we are ready to prove the following theorem. Theorem 2. Given a transportation network G and a set of sites S, there exists a set S of needles such that V(S) induces V G (S), i.e., for any p S, there exists a site q S such that R(p, S) R G (q, S). Proof. First, we dene d k G (p, q) be the length of a shortest path from p to q where the path passes through at most k roads. Surely, d G (p, q) = d G (p, q). We claim that d k G (p, q) = d(p, Sk q ), which directly implies the theorem. There exists a needle q Sq k such that d(p, q) = d(p, Sq k ) and q σg l (q), for any point p and 0 l k. Let us consider a sequence of needles, q l = q, q l 1,, q 1, such that q i is produced from q i 1 and q i σg i (q). With these needles, we construct a path P from p to q passing through l roads. P starts from p and goes to the points y l on s(q l ) which obeys the rule of Observation 1. Next, P passes p 1 (q l ) along the road, on which q l is produced (see Figure 5(a)). Recursively, P passes y i and p 1 (q i ) next to p 1 (q i+1 ) and nally reaches q next to p 1 (q 1 ). Since each q i is produced on a road, P passes through l roads, which is at most k. Hence, d k G (p, q) d(p, Sk q ). p y l q l q l 1 y l 1 v i q i y i x i q i 1 v i 1 y i 1 x i 1 (a) (b) Figure 5: Proof of Theorem 2 8

9 Let us consider a path P of length d k G (p, q) from p to q and assume that P passes through l roads, for 0 l k. We then construct l needles q 1,, q l, inductively, such that q i σg i (q) and d(p, q l ) = d k G (p, q). P consists of l segments of roads e l, e l 1,, e 1 in order, each of which starts at y i and ends at x i on road e i. Also, v i denotes a node of e i in direction toward y i from x i (see Figure 5(b)). We construct q i as follows: q 1 = (x 1, v 1, d(x 1, q), d(x 1, q)+ 1 v(e 1 ) d(x 1, v 1 )) and q i = (x i, v i, d(x i, q i 1 ), d(x i, q i 1 )+ 1 v(e i ) d(x i, v i )). For convenience, let q 0 be (q, q, 0, 0). Then, q i σg i (q) since P is a shortest path that uses at most k roads and it obviously obeys the rule of Observation 1, locally. Surely, d(x 1, q 0 ) = d 0 G (x 1, q) = d(x 1, q). If d(x i, q i 1 ) = d i G (x i, q), a sub-path P of P from x i to q is a shortest path passing through i roads. We construct a path P of length d(x i+1, q i ) from x i+1 to q. P is obtained by adding the path x i+1 y i x i at the beginning of P. Then, d(x i+1, q i ) = d i+1 G (x i+1, q), since P passes through i + 1 roads and a sub-path of P. By induction, d(p, q l ) = d l G (p, q) = dk G (p, q), implying that dk G (p, q) d(p, Sk q ). Finally, we show that d k G (p, q) = d(p, Sk q ), and thus d G (p, q) = d(p, S q ). With taking S = p S S p, this claim implies that, for any p S, there exists a site q S such that R(p, S) R G (q, S). Theorem 2 shows the existence of a set of needles noted at the end of Section 2. But, S may not be pairwise non-piercing and further have innitely many needles by the construction from the proof of Theorem 2. Thus, we should reduce the number of needles and make them pairwise non-piercing in order to efciently compute V(S). For any p S, we call p effective if R(p, S), otherwise we call p ineffective. Lemma 2. There exists a set S of needles, which are effective and pairwise non-piercing, such that V(S ) = V(S). Proof. First, we remove all ineffective needles from S. We let S e S be the set of all effective needles in S. Trivially, V(S e ) = V(S). We will construct S from S e by making it pairwise nonpiercing without any change in its diagram. We can partition S e into S 0, S 1,, S m such that S 0 = {(p, p, 0, 0) p S} and all needles in S i lie on e i, for 1 i m, if all the roads in G are given as {e 1,, e m }. For each needle p S i, 1 i m, we can nd (at most) one needle q S i, the closest needle in the direction of p from p 1 (p). p stretches to a node of e i but the part over p 1 (q) does not contribute V(S e ), since both p and q are effective and their speeds are the same as v(e i ). We thus cut p by setting p 2 (p) to be p 1 (q) and t 2 (p) to be t 1 (p) + 1 v(e i ) d(p 1(p), p 1 (q)). We denote the set of these new needles from S i by Si and S 0 S1 S m by S. Since we cut only a useless part of every needle from S e, V(S ) = V(S e ). Now, we show that S is pairwise non-piercing. S 0 is clearly pairwise non-piercing, since it is a set of points. Si is also pairwise non-piercing, since all needles in S i lie on a line including a road e i and they are disjoint. Let us consider two needles in S, p and q, such that they are produced on two distinct roads. Note that one of p and q may be a point in S. We assume that p pierces q and q is produced on a road e i. First, q is not a needle produced from p, since p and q should be non-piercing if q is a needle produced from p. If a needle p produced on e i from p is effective, q cannot dominate any point on e i over p 1 (p ), and thus q is not pierced. Hence, no needles on e i from p are effective. This implies that q dominates two possible starting points of needles produced from p, and thus q dominates all points between the two points. Consequently, p does not pierce q, a contradiction. Therefore, S is pairwise non-piercing. 9

10 Theorem 2 and Lemma 2 say that the Voronoi diagram with a transportation network can be represented by that for an appropriate set of pairwise non-piercing needles under the Euclidean metric. Also, Lemma 2 provides a method to make a set of needles on roads to be pairwise non-piercing. More computational and quantitative issues about needles produced on the given network G are considered in the next subsection. Further, these theorem and lemma imply that a shortest path with a transportation network can be represented as a sequence of needles. More on shortest path planning under the transportation metric will be considered in Section Needles Produced on a Road from a Parent Needle We already know how a point produces at most two needles on a road by Observation 1. This subsection generalizes such an argument and we thus focus on how a needle produce needles on a road and on how many needles we have to handle to solve the problem. Lemma 3. If a needle p does not intersect a road e, needles produced on e from p are at most two of the following six needles; four needles produced from weighted points p 1 (p) with weight t 1 (p) or p 2 (p) with weight t 2 (p), and two needles starting from a node of e. Proof. Since a needle can be viewed as a set of weighted points, we consider all needles from such weighted points. And we assume e is horizontal without loss of generality. First, let S l s(p) be a set of points such that, for all x S l, there exists a needle q produced on e from x such that its direction is to the left. For any x S l, let q be a needle produced on e from x with weight w p (x) such that its direction is to the left. Then, we have t 1 (q) = w p (x) + d(x, p 1 (q)) and we minimize the time reaching the left node p of e, t 1 (q) + 1 v(e) d(p 1(q), p), over all x S l, where v(e) is the speed of e. x 2 S l x p x 1 x 1 p 1 (q) p 1 (q 1 ) e x h l α Figure 6: Minimizing the length to the left endpoint of e. Note that S l is a line segment whose endpoints are x 1 and x 2. Any x S l can be represented by #» x = x #» 1 + t #» d, for 0 t 1, where #» x is a vector with the same coordinates as x and #» d = x2 #» x #» 1. Then, we can say that w p (x) = w p (x 1 ) + w, d(x, p 1 (q)) = d(x, p 1 (q 1 )) + l and d(p 1 (q), p) = d(p 1 (q 1 ), p) + h, as in Figure 6. Furthermore, w, l and h are of the form ct, where c is a constant, 1 which implies that the equation is linear over 0 t 1. Hence, it is minimized when x is an endpoint of S l since p does not intersect e. Assuming that if at x = x 1 the equation is minimized, either x 1 is an endpoint of p or p 1 (q 1 ) is an endpoint of e. Consequently, the left-headed needle produced from p is a needle from an endpoint of p with its weight or a needle starting from a node of e. An analogous argument is applicable to the right-headed needle. We thus show this lemma. 1 Exactly, w = t(w p(x 2) w p(x 1)), l = t #» d sin θ #» d sec α and h = t #» d (cos θ #» d sin θ #» d tan α), where is the L 2 norm and θ #» d is the angle dened by #» d. 10

11 Lemma 4. Let p be a needle produced on a road e from a needle q. For another road e E, if p does not dominate any node of e or e, no needles produced on e from p are effective. Proof. Suppose that p does not dominate any node of e or e. Then, there exists a neighboring needle in S to the growing direction of p, as shown in Figure 7. However, since p does not dominate any node of e or e, the needles produced from p 2 (p) with weight t 2 (p) and the needle starting from a node of e are ineffective. Further, the needles from p 1 (p) with weight t 1 (p) are dominated by q, since they are closer to q than to p. By Lemma 3, we cover all candidates of effective needles produced on e from p. Thus, no needles produced on e from p are effective. q e p e Figure 7: No effective needles produced on e from p. These two lemmas reduce the number of necessary applications of σ G ( ). Thus, they will be very helpful to devise an algorithm to compute S in Subsection 5.1. Further, we can show the number of needles in S by Lemma 3 and 4. Lemma 5. S = O(m(n + m)), where n = S and m = E. Moreover, this upper bound for S is tight for the worst case. Proof. For a needle p, σ G (p) is at most 2m, if p is disjoint from all the roads. If a needle q σ G (p) does not dominate any node in V, then no needles in σ G (q) are effective by Lemma 4. In other words, we consider only needles dominating at least a node as an independent site. This implies that S = O(m(n + m)). Further, the tightness of the upper bound for S can be shown by a simple example, see Figure 8. The gure depicts a transportation network, a set of sites and a set S of produced needles; m/2 roads are fanned out and the remainders are placed horizontally as the gure. If we properly adjust the speed of each road, we can obtain m 2 (n + m 2 ) + n = O(m(n + m)) number of needles in S. 5 Algorithms In this section, we consider more algorithmic issues and present simple and efcient algorithms for computing the Voronoi diagram for sites S with a given transportation network G. First, we will give 11

12 Figure 8: A tight example for the upper bound of S. Circles with empty interior are sites and arrows are needles produced on the roads. One site is placed at the top of the gure and the others are placed below. an algorithm with time complexity O(nm 2 log n + m 3 log m) in Subsection 5.1. Next, an improved algorithm that takes O(nm 2 + m 3 + nm log n) time will be presented in Subsection An Algorithm As noted earlier, since S can include possibly innitely many needles, it is inadequate to compute S from S. The effectiveness of a needle is dened with respect to all other needles in S. However, the effectiveness of a needle p can be tested only with effective needles already emitting wavefronts at time t 1 (p). Thus, our algorithm simulates the wavefront model where the wavefronts sweep the plane. Recalling the wavefront model, each site starts sending out a wavefront at time 0 and the interference pattern between such wavefronts constitutes the Voronoi diagram. While simulating the wavefront model, at time t 1 (p), we test the effectiveness of p and we update some data structures for p, only if p is effective. In this way, we compute S directly, not from S. In order to do so, the algorithm handles two kinds of events, dened as certain situations in the wavefront model; one occurs when a needle produced on a road from another effective needle starts emitting a wavefront, called a birth event, and the other occurs when a wavefront sent out from a needle reaches a node, called a node event. From each event, we can ascertain when and where it occurs, the associated site (or needle), and so on. We will call an event effective if its associated needle dominates the point where the event occurs. To handle these events, we need two types of data structures: Q is an event queue implemented as a priority queue. The priority of an event e is its occurring time. Q supports inserting, deleting, and extracting-minimum in logarithmic time with linear space. T 1, T 2,, T m are balanced binary search trees, each associated with e i, where the road set E is given as {e 1, e 2,, e m }. Each T i stores needles on e i in order. The precedence is the position of the starting point of a needle and ties are broken by their directions. T i supports inserting and deleting of a needle in logarithmic time, and also a linear scan for needles currently 12

13 in T in linear time and space. The linear scan is possible by adding some links among the elements. For details about the above data structures, refer the book by Cormen et al.[10] Also, we discuss how an event is computed or created. A birth event b on a road e from a needle p is directly associated with a needle q produced on e from p. b occurs at time t 1 (q) at p 1 (q) on the road e. And, a node event a on a node v from a needle p occurs at time d(v, p) on v Computing S Now, we are ready to describe our rst algorithm. At the beginning of the algorithm, we compute the associating birth event with each needle in σ G (p) for all p S and insert it into Q. And, for each node v V, we compute the node event for the nearest site from v, insert it into Q, and set the variable event(v) to be the node event. Once the above initialization is done, we extract the upcoming event from Q and process it as follows repeatedly while Q is not empty. If a birth event b on e i occurs, we rst test the effectiveness of its associated needle p, and then insert it to T i. The effectiveness test can be done by checking at most two neighbor needles in T i. More precisely, we nd at most two neighbors of p by the dened order in T i, and test whether two neighbors dominate p 1 (p). (Note that this test is necessary but not sufcient, i.e., some ineffective needles may pass this test. This however does not increase the asymptotic number of needles we handle. We will discuss about this argument later.) If p passes the effectiveness test, we insert p into T i and compute node events from p. Note that event(v) is the currently earliest node event on v in Q. We let a be the node event on v newly computed from p. Then, only if a will occur earlier than event(v), we delete event(v) from Q, insert a into Q and set event(v) as a. When a node event a on a node v occurs, we compute σ G (p), where p is the needle associated with a. Then, from all needles in σ G (p), we compute the associating birth events and insert them into Q. After the event processing, all effective needles are stored in each T i. We then extract them with proper cutting, as described in the proof of Lemma 2. We denote S 1 i m T i by S a and the resulting set of needles after running the algorithm by S a. The following lemma guarantees the correctness of the algorithm. Lemma 6. Let S e be a set of effective needles in S. Then, S a is pairwise non-piercing and S e S a S. Proof. While running the algorithm, all the needles inserted into T i are obtained by applying σ G ( ) from any site p S, which implies that S a S. The algorithm lters ineffective needles efciently by Lemma 4 and by the effectiveness testing in processing a birth event, which is necessary. Hence, all needles in S \ S a are ineffective and S e S a. Note that all ineffective needles are totally dominated by other effective ones. To effective needles in S a, we can apply an analogue to the proof of Lemma 4. Thus, S a is pairwise non-piercing. Finally, we compute V(S a) by using Corollary 1 such that V(S a) = V(S ) = V(S) by Lemma 6 and 2. By Theorem 2, V(S) is a super diagram of V G (S) and thus we can easily obtain V G (S) from V(S) or V(S a). Figure 9 shows a brief description of the algorithm discussed above. 13

14 Input: A set S of sites and a transportation network G. Output: The Voronoi diagram V G (S). 1. Initialize Q and T i, 1 i m. 2. Compute all birth events from σ G (p) for all p S and insert them into Q. 3. For each node v V, compute the node event from the nearest site from v, insert it into Q, and set event(v) to be the node event. 4. While Q is not empty, 5. Extract the earliest event from Q. 6. Handle the event according to its type. 7. Let S a = S 1 i m T i. 8. Make S a pairwise non-piercing by Lemma 2. Let S a be the result. 9. Compute V(S a). 10. Extract V G (S) from V(S a). Figure 9: Description of the algorithm to compute V G (S) Analysis Our algorithm depends on the number of handled events and on the number of needles in S a. In fact, the number of events is O(m(n + m)), since the algorithm applies σ G ( ) only O(n + m) times and σ G (p) = O(m) for any needle p, and all the birth events are generated by a produced needle by σ G ( ). Note that the number of handled node events is exactly the number of nodes in V. Also, S a = O(m(n + m)) by Lemma 5 and 6. Finally, we conclude our main theorem. Theorem 3. Let G be a transportation network with m roads and S be a set of n point sites. The Voronoi diagram V G (S) for S under the transportation metric with G can be computed in O(nm 2 log n + m 3 log m) time and O(m(n + m)) space. Proof. Handling an event takes O(m log(n + m)) time. Thus, computing S a takes O(m 2 (n + m) log(n + m)) time which is the bottleneck of the algorithm. For any other steps of the algorithm, O(m(n + m) log(n + m)) time is sufcient. For the space complexity, the algorithm uses only linear space on the number of handled events and produced needles, which is O(m(n + m)). 5.2 An Improved Algorithm The algorithm in the previous subsection can be improved based on further observations on transportation networks. This subsection will show a new and important property of shortest paths under the transportation metric and an improved algorithm based on the new property Primitive Paths Here, we introduce a concept of a primitive path. 14

15 Denition 2. A path P under the transportation metric is called primitive, if the following are satised: 1. P contains no nodes in V in its interior. 2. P passes through at most one road. We call a primitive path with the shortest length a shortest primitive path. Lemma 7. Given a transportation network G on the Euclidean plane, for two points p, q R 2, there exists a shortest path P from p to q with G such that P is a sequence of shortest primitive paths whose endpoints are p, q, or nodes in V. Proof. Let P be a p-q path passing through a road e and next another road e. And let us assume that P contains neither a node of e nor a node of e. Then, Lemma 4 implies that there exists a path P no longer than P such that P passes through at least a node of e or e. This consequently implies that there exists a shortest path such that, if the path goes through roads e 1,, e k in order, it passes through at least one node of e i or e i+1 between every e i and e i+1 for 1 i k 1. Such a shortest path can be partitioned into a sequence of shortest primitive paths whose endpoints are p, q, or nodes in V. By Lemma 7, we can compute a shortest path for two given points, p and q. All shortest primitive paths whose endpoints are p, q, or nodes, can be generated since the number of such paths is exactly ( V + 2) 2 = O(m 2 ) and this can be represented as an edge-weighted complete graph whose vertex set is V {p, q} Modication Recall that a bottleneck of our algorithm is the birth event processing part. We handle O(m(n + m)) number of birth events and it takes O(m log(n + m)) time to process each event. More precisely, in processing a birth event b, we compute the node event on v from the needle p b associated with b and update event(v), for every node v V. However, updates of event(v) does not occur so much frequently, O(m) times for every birth event, indeed. We let G P = (V P, E P ) be a graph with weight w P over E P such that V P = V S and E P = {pv or vp p V S and v V }. We assign a weight w P (pq) to every pq E P as the length of a shortest primitive path from p to q. We denote by e(p, q) the road used by a shortest primitive path from p to q. Observe that no road may be used by shortest primitive paths and then e(p, q) is not dened. We modify the birth event processing part. Note that all the birth events are generated at the initialization step or at processing a node event. Let us consider a birth event b on road e i such that b is generated from p which is a node or a site in V S. We also assume that b is passed the effectiveness test. Then, for any node v V, v p, if e(p, v) e i, there should be a shortest primitive path from p to v that passes through not e i but another road. Thus, we do not need to update event(v) in that case. Hence, we can reduce the number of updates of node events. The following is a brief description of a birth event processing. If a birth event b on e i occurs, we rst test the effectiveness of its associated needle p, and then insert it to T i if the test is passed. And we assume that b is generated from p which is a node or a site in V S. For every node v V, if e(p, v) = e i, we compute the node event a on v from p and only if a will occur earlier than event(v), we delete event(v) from Q, insert a into Q, and set event(v) as a. Otherwise, if e(p, v) e i, we do nothing. 15

16 The above procedure takes O(m + l log(n + m)) time, where l is the number of updated node events. Since only one node event on a node is handled, the total number of updated node events is at most O(m(n + m)). Hence, with this new event processing procedure, the time complexity of the algorithm is reduced to O(m 2 (n + m) + m(n + m) log(n + m)) = O(nm 2 + m 3 + nm log n). Only remaining issues are how to compute the weight w P and how to evaluate e(v, w) for any vw E P. First, we can obtain w P (v, w) by comparing the lengths of primitive paths from v to w over all possible roads in E. It takes O(m) time for every edge vw E P. We further obtain e(v, w) while computing w P (v, w). If O(m(n + m)) storage is allowed to store e(v, w), we can evaluate e(v, w) in constant time. In summary, with O(m 2 (n + m)) preprocessing time and O(m(n + m)) space, we can compute the graph G P with the weight w P and we can evaluate e(v, w) in constant time for every vw E P. Consequently, we have a faster algorithm to compute the Voronoi diagram V G (S) for a set S of sites with a transportation network G. Theorem 4. Let G be a transportation network with m roads and S be a set of n point sites. The Voronoi diagram V G (S) for S under the transportation metric with G can be computed in O(nm 2 + m 3 + nm log n) time and O(m(n + m)) space. 6 Restriction on Transportation Networks In the previous section, we dealt with transportation networks as planar straight-line graphs with individual speeds and algorithms for computing the Voronoi diagram with such a transportation network. In this section, we consider a special case of transportation networks, which will lead to a reduction of running time and required storage in computing Voronoi diagrams. Here, we assume that there are at most k directions and speeds for the roads. We will obtain a better bound for S than that obtained in the previous section. Let d #» 1, d #» 2,, d #» k denote the k possible directions of the roads and v i be the speed associated with the direction d #» i for each i, 1 i k. Also, we denote by E i the set of roads in E whose directions are d #» i. Note that, in any E i, all the roads are parallel and their speeds are the same as v i. p e e q r Figure 10: Parallel roads with the same speed. r is always ineffective. Lemma 8. Given transportation network G with k possible directions and speeds of the roads, S = O(k(n + m)). Proof. In order to prove this lemma, it is sufcient to show that the number of effective needles in σ G (p) that are not starting from a node is at most O(k) for any needle p. 16

17 Without loss of generality, assume that #» d i is horizontal. For any needle p and two roads, e, e E i such that they are below p and e is closer than e to p, we let q be a needle produced from p on e and r be a needle produced on e. We also assume that q and r are not starting from a node. Then, we observe that N(r, t) N(q, t) for any positive t, since the two roads have the same speed of v i ; see Figure 10. This implies that r is not effective and the number of effective needles in σ G (p) that are not starting from a node is at most 4k. By Lemma 8, we can reduce the complexity of computing V G (S) with a minor modication from the algorithm presented in the previous section. The algorithm specied in Section 5 inserts O(m) birth events for each parent needle into Q. Recall that a birth event is created either at the initializing part or in handling a node event. We modify these parts to insert only O(k) birth events for each parent needle in O(k) time and we may need some preprocessing to do so. At the preprocessing step, we compute at most 4k roads for each p S V, which are candidates on which at most 4k effective needles are produced from p. We denote the set of such roads by E(p). First, we construct four maps M i,1,, M i,4 for each direction d #» i, 1 i k. Before describing how to construct them, we dene four directions d #» i,1,, d #» i,4 obtained by rotating d #» i by angles π/2 ± α i and π/2 ± α i, where sin α i = 1/v i. Note that these four directions induce shortest paths using any road in E i by Observation 1. We are now ready to construct the map M i,j, for 1 i k and 1 j 4. From every node incident to roads in E i, we draw along a ray with direction d #» i,j and stop drawing when reaching another road in E i, see Figure 11. With the roads in E i, we then obtain a polygonal subdivision of the plane with O( E i ) size and this is computable in O( E i log E i ) time by applying the plane sweep. Thus, we can construct such 4k maps in O(m log m) time and O(m) space. In addition, each region is labeled as the road to the direction d #» i,j. Note that there is one unlabeled region that bounds the other regions. d i,j di Figure 11: The map M i,j with E i and #» d i,j With the maps M i,j, we can compute E(p) for any point p R 2 by point location over each M i,j, in total 4k times. Let e be the road obtained by point location of p over M i,j. Then, by proof of Lemma 8, any needle produced from p on any road over e in direction d #» i,j is not effective. Hence, E(p) can be computed as a set of such roads obtained by point locations of p over the 4k maps. Finally, we have E(p) for every point p S V in O(k(m + n) log m) time and O(k(n + m)) space. As mentioned before, we now produce only O(k) birth events for each site at the initializing part and for each node event in O(k) time; This is done by checking only O(k) roads in E(p) instead of all roads in E, for any site or node p. In this way, the number of total birth events is reduced to O(k(n + m)) and also the complexity of the algorithm is. In Section 5, we proposed two algorithms. Based on each algorithm, Theorem 5 gives two algorithms and the analysis of them. 17

18 Theorem 5. Given transportation network G with k possible directions and speeds of the roads, V G (S) can be computed in O(knm+km 2 +kn log n) time with O(m(n+m)) space, or in O(nmk log n+ km 2 log m) time with O(k(n + m)) space. Proof. Here, we present two different algorithms based on the algorithms in Subsection 5.1 and 5.2. Based on the algorithm in Subsection 5.1. According to above discussion, we have O(k(n + m)) number of birth events and O(m) number of node events. By proof of Theorem 3, handling an event takes O(m log(n + m)) time, which implies that S a can be computed in O(km(n + m) log(n + m)) time. Also note that S a = O(k(n + m)) by Lemma 6 and 8. Hence, we can compute V G (S) under the assumption of this theorem in O(nmk log n + km 2 log m) time with O(k(n + m)) space. Based on the algorithm in Subsection 5.2. Using the maps M i,j, we can build the graph G P in O(km(n + m)) time. We handle O(k(n + m)) birth events and O(m) node events. By proof of Theorem 4, under the assumption of this theorem, we can compute V G (S) in O(km(n + m) + (M + k(n + m)) log(n + m)) time with O(m(n + m)) space, where M is the number of updates of node events during the modied algorithm. Actually, M = O(k(n + m)), since E(p) = O(k) and thus E(p) E(q) = O(k) for any p, q R 2. Hence, we have an O(knm + km 2 + kn log n) algorithm to compute V G (S) under the assumption of this theorem. Theorem 5 says that if we regard k as a constant (e.g., an isothetic network) the diagram can be computed in O(nm + m 2 + n log n) time or in O(nm log n + m 2 log n) time. We note that this restriction on a transportation network leads to a linear size of the resulting diagram. This result is analogous to the previous result by Aichholzer, Aurenhammer, and Palop[3] They also maintained a linear size of the diagram by restricting their transportation network as an isothetic network with the same speed for the roads on the L 1 plane. 7 Shortest Path Planning 7.1 Shortest Path to the Nearest Site By point location over Voronoi diagrams, we can easily nd the nearest site from any point in the plane. Under conventional metrics, for example, the Euclidean metric, the L 1 metric, and so forth, for any point p R 2, we can compute the nearest site in O(log n) time with the Voronoi diagram when n sites are given, and the complexity of shortest paths is constant. Thus, we can nd a shortest path to the nearest site, naturally and trivially. However, under transportation metrics, it becomes more complicated. The complexity of shortest paths is no longer constant but O(m) for the worst case, where m is the complexity of the given transportation network. In this subsection, we discuss how to construct a shortest path to the nearest site using diagrams obtained by our algorithms. With the Voronoi diagram V G (S) for a set S of sites under the transportation metric with G, we cannot plan a shortest path from a query point to the nearest site, although under conventional metrics we can. This is due to the fact that a region in V G (S) has insufcient information of the structures of shortest paths to the site associated with the region. However, fortunately, we have another diagram V(S ) obtained while running the algorithm. Note that the region R(p, S ) for each p S is a set of points from which the nearest site is p when R(p, S ) R G (p, S), and shortest paths to p have the same structures; details are shown in proof of Theorem 2. To construct a shortest path efciently, we maintain parent(p) for any p S in addition; parent(p) is dened as the parent needle in S, from which p is produced. The relation between p and parent(p) makes a forest structure that contains n rooted trees whose roots indicate sites in 18