1 Introduction

In this paper, we present improved algorithms for the Steiner tree problem with the minimum number of Steiner points and bounded edge length (STP-MSPBEL) introduced by Lin and Xue [1]. Given a set of n terminal points P = {p₁, p₂, …, p_n} in the two-dimensional Euclidean space and a positive constant R, STP-MSPBEL asks for a tree spanning a superset of P such that each edge in the spanning tree has a length no more than R and the number of points other than those in P, called Steiner points [2], is minimized [1]. This problem has been applied to the relay node placement problem in wireless sensor networks [3, 4]. In monitoring sensor networks, connectivity among sensors is crucial for gathering information. Due to harsh environments like earthquakes, fires, or floods, a number of sensors can be simultaneously disabled. In this scenario, relay sensors are required to restore network connectivity. When the communication distance sets to R, STP-MSPBEL can be employed to recover the entire network at a low cost (with a minimal number of relay nodes). This problem also finds applications in wavelength-division multiplexing(WDM) optimal network design, VLSI design, and evolutionary/phylogenetic tree constructions in computational biology [1].

The classical Steiner tree problem, which aims to minimize the total length of the spanning tree, has been widely studied. This problem is NP-hard [5], and it is proven to be APX-hard, so it cannot be approximated within 96/95 in polynomial time unless p = np [6]. A number of approximation algorithms have been proposed [7–11]. The currently best-known approximation scheme achieved the approximation ratio of ln 4 + ϵ ≤ 1.39 using linear programming (LP) and iterative randomized rounding by Byrka et al. [7]. Later, Chen and Hsieh [8] proposed an efficient two-phase heuristic in greedy strategy with an approximation ratio of 1.4295. Recently, Traub and Zenklusen [10] proposed the purely combinatorial approach providing the approximation ratio of ln4+ ϵ, which is the same as the best-known approximation factor without solving an LP problem. Zhang et al. [11] presented a truthful-in-expectation mechanism that achieves the same approximation ratio. Berman et al. [12] presented a 1.25-approximation algorithm for the variation of the problem where a complete graph is given with edge lengths 1 and 2. Chen and Hsieh [10] analyzed the algorithm of Byrka et al. [7] and showed that it gives 1.162 + ϵ approximation for the problem on complete graphs with edge distances 1 and 2.

In STP-MSPBEL, Lin and Xue [1] proved that the problem is NP-hard and presented a polynomial-time approximation algorithm by exploiting a minimum spanning tree. Let len(e) be the length of an edge e. Given an edge e_i whose length is greater than the given bounded edge length R, this algorithm inserts Steiner points to break the edge e_i into small pieces of length at most R. Cheng et al. [3] called such edge as a steinerized edge. Given a tree, we can produce a steinerized tree for STP-MSPBEL using steinerized edges whenever the edge length exceeds the bounded edge length R. Lin and Xue [1] construct a steinerized minimum spanning tree from the minimum spanning tree of terminal points [13]. They showed that this algorithm has an approximation ratio of 5. Later, Chen et al. [13] proved that the steinerized minimum spanning tree indeed has an approximation ratio of exactly 4. They also presented a 3-approximation algorithm with O(n⁴) running time. At initialization, the algorithm adds edges between any pair of two terminal points if the distance between them is less than or equal to R. The algorithm examines all possible combinations of four terminal points from the entire set to determine if they are currently not connected and can be connected using a single Steiner point. If so, it inserts one Steiner point and four edges to connect those points and the Steiner point. After examining all subsets of four terminal points, the algorithm connects disconnected components using steinerized edges if the resulting graph is still disconnected. Cheng et al. [3] presented a 3-approximation algorithm with O(n³) running time. Instead of inspecting every subset of four terminal points, they investigate all subsets of three terminal points. Moreover, they proposed a randomized algorithm to give a 2.5 approximation with a probability of at least 0.5 in poly(n, q_P) time where q_P = max{q_a,b,c|a, b, c ∈ P} and q_a,b,c is the number of Stiner points used to steinerize the optimum Steiner tree on three terminal points {a, b, c} [3]. Senel and Younis [14] suggested an O(n⁴)-time algorithm to reduce Steiner points based on the Fermat point for all subsets of three terminal points. Later, they improved the algorithm with the discrete Fermat point in [15]; however, the time complexity of their algorithm to find the discrete Fermat point is not bounded because the algorithm should examine a large number of candidate points when the distance between input points is long. In summary, the best-known approximation ratio of deterministic algorithms for STP-MSPBEL has been 3 with running time O(n³).

We propose an efficient approximation algorithm for STP-MSPBEL using the Voronoi diagram, a well-known geometric data structure in computational geometry. This paper extends our previous work [16]. Our algorithm runs in O(n log n) time with the worst-case approximation ratio of 3, which is the same as the best-known deterministic approximation ratio and remarkably reduces the running time compared to the previous 3-approximation algorithms [3, 13]. In the classical Steiner tree problem, it is known that the optimal Steiner tree for the given three points can be computed in constant time by using the Fermat point [17]; however, in STP-MSPBEL, there has been no exact algorithm even for three terminal points in constant time. After we presented the exact algorithm in our preliminary paper [16], Senel and Younis [18] proposed a simpler method to find the optimal Steiner tree for three terminal points; however, this paper addresses the counterexample that shows their method indeed cannot find the optimal solution. In this paper, we present the exact algorithm to find a Steiner tree for given three points in constant time, and we propose an improved algorithm for STP-MSPBEL by using the exact algorithm. The proposed algorithm ensures a worst-case approximation ratio of 3 in O(n log n) time while achieving improved performance since we utilize the optimal solution of three terminal points.

2 Preliminaries

We briefly introduce a Voronoi diagram and the Fermat point, mainly used by the proposed algorithms.

3 An efficient 3-approximation algorithm

We propose an O(n log n)-time approximation algorithm with an approximation ratio of at most three by exploiting the Voronoi diagram as described in Fig 1. More precisely, it takes O(n log n) time to determine the number of Steiner points. It takes O(n log n + S) to report the locations of Steiner points where S is the number of Steiner points in the optimal Steiner tree for STP-MSPBEL since the algorithm produces no more than three times the number of Steiner points in the optimal Steiner tree. Indeed, the number of Steiner points S is independent of the number of terminal points n. Similar to the previous works [3, 13], this paper does not consider the cost of making steinerized edges when analyzing algorithms. We first compute a minimum spanning tree and sort the edges in increasing order in Step 1. If the length of the edge is less than or equal to R, we insert the edge into the result tree in Step 2. If several connected components still remain after adding edges, it requires Steiner points to connect those components. In Step 3, we construct a Voronoi diagram of the terminal points as sites. At each Voronoi vertex, we investigate its three relevant sites, {p_a, p_b, p_c} ∈ P, to determine whether these points are in different connected components and whether the radius of the smallest enclosing circle of three sites is less than R. If so, we add one Steiner point at the center of the circle and insert three edges to connect relevant terminal points from the Steiner point. The center of the smallest enclosing circle for p_a, p_b, and p_c is located at the Voronoi vertex when the triangle Δp_ap_bp_c is not an obtuse triangle by property 1. If Δp_ap_bp_c is an obtuse triangle, the center of the smallest enclosing circle is located at the middle of the longest side of Δp_ap_bp_c. Note that if the Voronoi vertex is associated with more than three sites, the same approach can be applied using the same Voronoi vertex. Lastly, if two endpoints of the edges in the minimum spanning tree are in different connected components, we steinerize the edge and insert it into the result tree in Step 4.

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Fig 1. AlgA: An efficient 3-approximation algorithm.

https://doi.org/10.1371/journal.pone.0294353.g001

Now, we analyze the time complexity and performance of the proposed algorithm.

Lemma 1. AlgA takes O(n log n) time.

Proof. In Step 1, it takes O(n log n) time to compute the minimum spanning tree and to sort the edges of the minimum spanning tree. In Steps 2 and 4, we check at most n − 1 edges whether the edge connects two different connected components. Connected components can be efficiently maintained by disjoint-set forest, a well-known data structure in which connectivity query can be answered in O(log n) time, and two trees are merged in O(1) time [22]. Consequently, Steps 2 and 4 take O(n log n) time. The Voronoi diagram of n sites in the plane can be constructed in O(n log n) time [4]. Since the number of vertices in the Voronoi diagram is at most 2n − 5 in 2D, Step 3 also takes in O(n log n) time. Overall, AlgA takes O(n log n) time.

We show that the AlgA needs Steiner points no more than three times the number of Steiner points in the optimal solution in a similar way of [3, 13]. We briefly introduce the terminology used in the previous work. A full component of a Steiner tree is a subtree in which every Steiner point is a non-leaf node, and every input point is a leaf node. Given a Steiner tree T, C(T) denotes the number of Steiner points in T.

Lemma 2. [1]. There exists a shortest length optimal Steiner tree for STP-MSPBEL such that every Steiner point has degree at most five.

Lemma 3. [13]. Every steinerized minimum spanning tree has the minimum number of Steiner points among steinerized spanning trees.

Lemma 4. [13]. Let T* be a shortest optimal tree for STP-MSPBEL with property that every Steiner point has degree at most five. Let T_j be a full component of T*. Then the followings hold:

1. (1). The steinerized minimum spanning tree on terminals in T_j contains at most 3 ⋅ C(T_j) + 1 Steiner points.
2. (2). If T_j contains a Steiner point with degree at most four, then the steinerized minimum spanning tree on terminals in T_j contains at most 3 ⋅ C(T_j) Steiner points.
3. (3). If the steinerized minimum spanning tree on terminals in T_j contains an edge between two terminals, then it contains at most 3 ⋅ C(T_j) Steiner points.

We slightly modify the lemma in [13] to apply it to our algorithm.

Lemma 5. Let T* be an optimal tree for STP-MSPBEL and T_A be the tree produced by AlgA. Then C(T_A) ≤ 3 ⋅ C(T*).

Proof. Let T_S be the steinerized minimum spanning tree. Let k be the number of Steiner points inserted by AlgA in Step 3. Since each of those Steiner points combines three different connected components at once and every edge whose length is less than or equal to R in the minimum spanning tree is added in Step 2, it is obvious that (1) By Lemma 2, we assume that T* is the shortest length optimal Steiner tree such that every Steiner point has a degree at most five. We split T* into full components T_j(1 ≤ j ≤ N_f) where N_f is the number of full components. For each full component T_j, we construct corresponding steinerized minimum spanning trees . If any full component contains at least one Steiner point whose degree is at most four, the corresponding steinerized minimum spanning tree has not more than 3 times the number of Steiner points in the full components by Lemma 4 (2). Moreover, if the distance between any pair of two points in the full component is not more than R, the corresponding steinerized minimum spanning tree requires no more than 3 times the number of Steiner points in the full components by Lemma 4 (3). Now, we consider a spanning tree which consists of steinerized minimum spanning trees of each full component. By Lemma 4, (2) where g is the number of full components in which every Steiner point has degree 5 and the distance between any two points is greater than R. By Lemma 3, (3) From (1), (2) and (3), C(T_A) ≤ 3 ⋅ C(T*) + g − k. Therefore, the approximation ratio produced by AlgA is bounded by 3 if g ≤ k. To show that g ≤ k, we construct two forests F₁ and F₂. We put all input points into F₁ and F₂ each and then add edges whose length is less than or equal to R in the minimum spanning tree. Let p be the number of connected components in F₁ (= F₂) so far, which is the result of Step 2 of AlgA. We add k Steiner points and their induced edges to F₁ according to Step 3. The number of connected components in F₁ is equal to p − 2k.

Now, we examine g full components in which every Steiner point has a degree of 5. Such components are either only one Steiner point or multiple Steiner points. In the case of components with a single Steiner point in which 5 input points are in different connected components and enclosed by a circle with a radius of R, the input points belong to at most three connected components in F₁ after Step 3. We deal with three Voronoi vertices related 5 input points in Step 3. Let us consider one of Voronoi vertices and its relevant three points. If they are in all the different components, they are connected by a Steiner point. Otherwise, at least two of the points are already connected. Then we look into another Voronoi vertex associated with two input points unrelated to the former Voronoi vertex. Similarly, at least two of these points become a part of the same component. Accordingly, we connect two pairs of input points by adding two edges in F₂. Even though we merge them, the number of connected components in F₂ is still greater than or equal to that in F₁. Now, we consider the other case. If the full component has more than one Steiner point, we can always find two Steiner points whose four neighbor points are input points. These four points belong to at most three connected components in F₁ after Step 3. In a similar way to the previous case, at least three points are examined for connectivity due to the associated Voronoi vertex in Step 3. A Steiner point connects them, or two of them are already connected. Even if we add two edges into F₂ to connect a pair of input points at each of two Steiner points, the number of connected components in F₂ is still greater than or equal to that in F₁. In both cases, the number of connected components is equal to p − 2g in F₂ since we add two edges for every g full component. It satisfies that p − 2k ≤ p − 2g. Therefore, g ≤ k.

Lemmas 1 and 5 yield the following theorem.

Theorem 1. AlgA produces a Steiner tree in which the number of Steiner points is no more than 3 times the number of Steiner points in the optimal Steiner tree in O(n log n) time.

4 An exact algorithm for three input points

Given three terminal points p_a, p_b, and p_c, we present the first exact algorithm AlgB to determine the optimal number of Steiner points for STP-MSPBEL in constant time as well as we provide one of the optimal Stiner trees. An optimal tree connecting p_a, p_b, and p_c is formed as either a 3-star or a wedge, as shown in Fig 2. If an optimal Steiner tree is shaped as a wedge, we can easily find the optimal tree because the steinerized minimum spanning tree becomes optimal by Lemma 3. In the other case, p_a, p_b, and p_c are connected by a junction point p_J, which is a tree in the shape of a 3-star. In this case, it is important to seek the location of p_J at which is minimized. Since the optimal Steiner tree is formed as either a wedge, a 3-star, or both, it can be obtained by comparing the 3-star and the wedge. Since finding a steinerized minimum spanning tree is trivial for three points, the main part of AlgB aims to seek an optimal junction point.

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Fig 2. Black dots represent input terminal points p_a, p_b, and p_c.

Empty dots represent Steiner points. (a) the steinerized minimum spanning tree requires 4 Steiner points (b) the steinerized 3-star tree using the Fermat point requires 4 Steiner points (c) the optimal Steiner tree for STP-MSPBEL requires 3 Steiner points. A junction point can be optimally placed in any of 3 shaded regions.

https://doi.org/10.1371/journal.pone.0294353.g002

The Fermat point p_F can be a good candidate as a junction point of p_a, p_b, and p_c because the Fermat point is defined as a point that minimizes |p_ap_F| + |p_bp_F| + |p_cp_F|. In STP-MSPBEL, unfortunately, p_F does not always become an optimal location for the junction point, as shown in Fig 2. Consider an equilateral triangle formed by p_a, p_b, and p_c where each length of a side is 2.25R. The steinerized minimum spanning tree and a steinerized 3-star tree using the Fermat point require four Steiner points to satisfy bounded edge length R. The optimal Steiner tree, however, needs three Steiner points, as shown in Fig 2(c).

Let T_F be a 3-star tree connecting input points p_a, p_b, and p_c using the Fermat point p_F as a junction point p_J.

Lemma 6. There is no steinerized 3-star tree T_3s such that C(T_3s) < C(T_F) − 2.

Prrof. It is obvious that and . Suppose that there is a steinerized 3-star tree T_3s such that C(T_3s) < C(T_F) − 2. Let p_J′ be a junction point of T_3s. To satisfy C(T_3s) < C(T_F) − 2, the following inequalities should hold: . Since T_F minimizes the sum of edge lengths, . It contradicts that . Thus, such T_3s cannot exist.

Let be a circle centered at a point p_x with a radius of y ⋅ R. We subdivide the plane by circles centered at each input point p_a, p_b, and p_c with various radii; we draw concentric circles , , and , as illustrated in Fig 4.

Observation 1. When the optimal Steiner tree of three points is formed as 3-star, the optimal location of p_J is inside the convex cell, which is an intersection area of , , and such that i* + j* + k* is minimized. Note that an optimal convex cell can be a single point if circles intersect at a point.

Proof. It is obvious that any point p_x in (or , ) requires at least q − 1 Steiner points to reach p_a (or p_b, p_c). When p_x is a junction point and , , and are the smallest circles containing p_x, we need i + j + k − 2 Steiner points including p_x to connect all input points. If p_x is not in the convex cell, we can always find the cell in which any point p_x′ can connect p_a, p_b, and p_c with fewer Steiner points. Suppose that the cell containing p_x has at least one concave arc on its boundary and w.l.o.g., we assume that this arc is the part of circle . Since p_x is outside of , any point p_x′ contained by , , and requires one fewer Steiner points than that of p_x. Therefore, an optimal junction point p_J should be inside of the convex region generated by , , and such that i* + j* + k* is minimum.

Based on Observation 1, it is easy to show that the optimal location of a junction point can be represented as a region rather than a single point. Therefore, we focus on searching for a convex region induced by the intersection of circles; more precisely, we search for an optimal region among convex cells in the plane subdivided by concentric circles.

Observation 2. An optimal convex cell may not be unique.

In the example of Fig 2(c), there are three cells in which we can optimally put a junction point. As the distance between terminal points increases, the number of convex regions requiring the same number of Steiner points can be highly many. Indeed, the number of optimal regions is not bounded by the input size. In this paper, we provide an algorithm to find at least one optimal convex cell to construct the Steiner tree, which requires the least number of Steiner points among all possible 3-star trees connecting p_a, p_b, and p_c.

Observation 3. The intersection of triangle Δp_ap_bp_c and optimal regions for the junction point of p_a, p_b, and p_c is not empty.

Proof. Let be a line through two points p and q. Suppose that an optimal convex cell is completely outside of Δp_ap_bp_c and w.l.o.g., this convex cell lies on the opposite side of from p_c. The boundary of the convex cell does not consist of since p_c is on the opposite side. If the convex cell consists of and , it should stab the line . Therefore, the optimal convex cells are either completely inside of Δp_ap_bp_c or on the edges of Δp_ap_bp_c.

We present AlgB, which provides an optimal Steiner tree for three input points as described in Fig 3. We consider a 3-star tree first and then compare it with the minimum spanning tree. The convex region formed by the intersection of three circles , , and is denoted by . Let be the intersection point of and that lies on the same side of as p_a or on , in other words, the intersection point closer to p_a.

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Fig 3. AlgB: An exact algorithm for three terminal points.

https://doi.org/10.1371/journal.pone.0294353.g003

In the first phase, we aim to construct an optimal 3-star tree which induces a Steiner tree requiring the minimum number of Steiner points under the bounded edge length R among all possible 3-star trees (refer to Step 1 and 2 of AlgB in Fig 3). By Observations 1 and 3, we can find the optimal location for the junction point by checking all convex cells in Δp_ap_bp_c. Because there could be a number of optimal regions in Δp_ap_bp_c by Observation 2, the algorithm focuses on finding one optimal convex cell. The first phase consists of three sub-parts symmetrically in which each input point is used as a reference point. W.l.o.g., we explain the sub-part on the basis of a point p_a. By Lemma 6, the number of Steiner points for an optimal 3-star tree is between i_F + j_F + k_F − 4 and i_F + j_F + k_F − 2 where , , and . In the case that the optimal number of Steiner points is equal to i_F + j_F + k_F − 2, the Fermat point becomes one of the optimal junction points. Otherwise, the algorithm searches for the convex cell such that i+ j+ k−2 is equal to i_F+ j_F+ k_F−3 or i_F+ j_F+ k_F−4. We first assume that the optimal 3-star tree requires i_F + j_F + k_F − 4 Steiner points. To figure out the existence of the target convex cell, we consider the series of particular convex cells which form where i + j + k = i_F + j_F + k_F − 2 and . When j and k are given, we check whether there exists δ_z satisfying that is not empty where i = i_F + j_F + k_F − j − k − 2. We build a distance function that signifies the distance between p_a and . If there exists any integer value satisfying , there exists a convex cell . Given i, j, and k, we can find the interval of δ_r to satisfy the following inequality: (4) If the intervals of δ_r satisfying f(i, j, k, δ_r) ≤ 0 include any integer value , is nominated for a candidate cell. If there is no convex cell satisfying i + j + k − 2 = i_F + j_F + k_F − 4, the algorithm seeks convex cells which require i_F + j_F + k_F − 3 Steiner points in the same manner.

For easy representation, we use the similarity transform by setting p_b = (0, 0) and p_c = (1, 0). The scale factor of the transform is . Accordingly, p_a is transformed to and bounded edge length R becomes . For given j and k, the coordinate of transformed is represented by (x_i, y_i) where and . Now, we rewrite the function f(i, j, k, δ_r) in (4) as follows. (5) where .

By substituting for δ_r, we can rewrite the function f′(i, j, k, δ_r) as follows: (6) where and t is the desired number of Steiner points. Note that t is initially set to i_F + j_F + k_F − 4 and later is set to i_F + j_F + k_F − 3. We can solve Eq (6) by the quartic formula since is of quartic form of for given i, j, and k. We calculate the valid interval of δ_r from satisfying inequality (7). If we find an integer value in the valid interval of δ_r, is set to a candidate cell. (7)

Once we set the desired number of Steiner points t, we should investigate with all possible combinations of j and k. In this paper, we reveal that only four pairs of j and k are enough to find the optimal cell for each of the two i values in each sub-part. In other words, instead of checking all convex cells, we solve Eq (6) at most 8 times in each sub-part with the parameters below. W.l.o.g., we assume that |p_bp_F| ≥ |p_cp_F|. If |p_bp_F| < |p_cp_F|, we regard p_b as p_c and p_c as p_b. In the case that the target number of Steiner points t is i_F + j_F + k_F − 4, we assign a set of parameters for Eq (6) to (i, j, k) ∈ {(i_F − 2, j_F, k_F), (i_F − 3, j_F, k_F + 1), (i_F − 4, j_F, k_F + 2), (i_F − 3, j_F + 1, k_F)}. If there exists any convex cell satisfying the condition, the convex cell becomes the optimal solution for the 3-star tree. Otherwise, we increase the i value of the above set of parameters by 1 and solve the equations again in order to examine whether there exists a 3-star tree requiring i_F + j_F + k_F − 3 Steiner points. If there is still no convex cell satisfying the condition, becomes the optimal solution for the 3-star tree.

In the second phase, we construct the minimum spanning tree T_MST of terminal points as described in Step 3. If the number of required Steiner points of T_MST is less than that of the optimal 3-star tree T_3S, we return steinerized T_MST as an optimal Steiner tree for the three terminal points. Otherwise, we return steinerized T_3S.

Now, we prove that the result tree of AlgB is an optimal Steiner tree for STP-MSPBEL given three terminal points. We begin by demonstrating that the steinerized 3-star tree, as generated through Step 2, demands the fewest Steiner points compared to all possible 3-star trees connecting terminal points. This is achieved by comparing the candidate convex cell produced by AlgB with all convex cells that are in the triangle Δp_ap_bp_c or intersect with the boundary of the triangle Δp_ap_bp_c. We partition the triangle Δp_ap_bp_c into seven distinct regions, delineated by , , and as illustrated in Fig 4. Subsequently, we investigate convex cells within each respective region.

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Fig 4. The optimal location for a junction point of a 3-star tree can be identified within the convex cells formed by concentric circles centered at each terminal point.

The triangle Δp_ap_bp_c is partitioned into 7 regions by , , and .

https://doi.org/10.1371/journal.pone.0294353.g004

First, we look into region 1, the intersection area of , , and , which indicates the smallest convex cell containing the Fermat point.

Lemma 7. No convex cell exists such that i + j + k < i* + j* + k*, i ≤ i_F, j ≤ j_F, and k ≤ k_F where is the candidate cell for the junction point of a 3-star tree as the result of AlgB.

Proof. In region 1, a convex cell satisfies that i + j + k ≥ i_F + j_F + k_F − 2, i_F − 2 ≤ i ≤ i_F, j_F−2 ≤ j ≤ j_F, and k_F − 2 ≤ k ≤ k_F by Lemma 6. In Step 2-1 of AlgB, convex cells are investigated by Inequality (7) with parameters (i, j, k) ∈ {(i_F − 2, j_F, k_F), (i_F − 1, j_F, k_F)}. Also, convex cells and are checked in Step 2-2 and 2-3 respectively. Consequently, all convex cells are investigated in by AlgB.

To prove Lemmas 8 and 9, we partition the plane into six wedges centered around the Fermat point p_F using the lines , , and . Note that each wedge has an interior angle of 60°. As illustrated in Fig 5, we denote W₁ as the wedge on the same side as p_b among two adjacent wedges of a line segment (p_a, p_F). Accordingly, we denote W₂, …, W₆ in clockwise order.

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Fig 5. 6 wedges (W₁, ⋯, W₆) are delineated by lines , , and (thick lines).

denotes an intersection point of circles and , positioned closer to p_a. Dotted curves represent an ellipse and a hyperbola with two focal points p_b and p_c through a point .

https://doi.org/10.1371/journal.pone.0294353.g005

Lemma 8. If and , no convex cell exists where δ_N < i/2, and .

Proof. Given a convex cell, we consider two cases: the first case is when one of vertices of the convex cell is closer to p_a and the second case is when the closest point to p_a is on the boundary. In the first case, we use to determine whether contains . In the case that , the intersection point of and the boundary of Δp_ap_bp_c is used for proof instead of .

We explain with the case that . We show that |p_ap_I| − 2R ≤ |p_ap_I+1| for any , and which implies that p_I+1 cannot be contained by unless p_I lies inside . As depiced in Fig 6, we denote l = |p_Ip_I+1| as the distance between p_I and p_I+1. If l ≥ 2R, (see Fig 6(b)). Since p_I ∈ W₁ ∪ W₂, . Thus, it should hold that . It implies that l < 2R. l + |p_ap_I+1| ≥ |p_ap_I| by triangle inequality. It holds that |p_ap_I+1| ≥ |p_ap_I| − l ≥ |p_ap_I| − 2R.

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Fig 6.

(a) The distance l between the intersection point of and , and . (b) For l ≥ 2R, the angle is strictly less than .

https://doi.org/10.1371/journal.pone.0294353.g006

Consequently, does not exist.

Let EP_b,c(p_x) be the ellipse through a point p_x with two focal points p_b and p_c (see Fig 5). Let HB_b,c(p_x) be the p_c-side curve of the hyperbola with two focal points p_b and p_c through a point p_x.

Lemma 9. If (or W₂) and , no convex cell (or ), , exists.

Proof. We consider a circle centered at p_a with a radius of and an ellipse as illustrated in Fig 7. Note that . When one of the intersection points of the ellipse and the circle lies inside of W₁, the other intersection point is located in W₂ ∪ W₃ ∪ W₄ since |p_bp_F| ≤ j ⋅ R. It implies that is outside of as well as . Thus, does not exist unless .

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Fig 7. When one of intersection points between EP_b,c(p_x) and where j ⋅ R > |p_bp_F| lies inside of W₁, the other point is inside of W₂ ∪ W₃ ∪ W₄ if it exists inside of the triangle.

https://doi.org/10.1371/journal.pone.0294353.g007

Now, we look into the cells in regions 2, 3, and 4. W.l.o.g., we show that there is no better convex cell compared to the candidate cell of the algorithm in region 2. We assume that |p_bp_F| ≥ |p_cp_F| for the proof of Lemmas 10 and 11.

Lemma 10. No convex cell exists such that i + j + k < i* + j* + k*, i ≤ i_F, j ≥ j_F, and k ≥ k_F where is the candidate cell as the result of AlgB.

Proof. AlgB investigates , , , and by solving Inequality (7) in Step 2-1. We prove that there is no better cell in region 2 by comparing it with these four convex cells.

EP_b,c(p_F) is tangent to the circle centered at p_a with a radius of |p_ap_F| and tangent lines of EP_b,c(p_F) and HB_b,c(p_F) are orthogonal at a point p_F. Thus, becomes a tangent line of HB_b,c(p_F) at p_F and HB_b,c(p_F) lies in W₂ ⋃ W₃ ⋃ W₄. Since |p_bp_F| − |p_cp_F| < (j_F + 1 − k_F)R, is also in W₂ ⋃ W₃ ⋃ W₄. Thus, is in W₂. By Lemma 9, convex cells composed by and , require at least the same number of Steiner points as that of the convex cell consisting of and .

is in W₁ because j_FR − |p_bp_F| < R and the distance between two intersection points of W₂ and is less than . By Lemma 9, convex cells composed by and require Steiner points not less than that of the convex cell consisting of and .

Up to now, we have examined the convex cells such that i ≤ i_F and either j = j_F or k = k_F. Next, we consider a convex cell such that i ≤ i_F, j = j_F + δ1_N, k = k_F + δ2_N, and . When δ1_N ≥ δ2_N, we compare the convex cell with the convex cell CV_comp composed by and . By Lemma 8, is not better than CV_comp which is already checked above. Equally, we compare the convex cell with the convex cell CV_comp composed by and when δ1_N < δ2_N. By Lemma 8, is also not better than CV_comp. Consequently, there does not exist such that i + j + k < i* + j* + k*, i ≤ i_F, j ≥ j_F, and k ≥ k_F where is the candidate cell as the result of AlgB.

Lastly, we prove that there is no better convex cell in regions 5, 6, and 7 than the candidate cell. W.l.o.g., we show the case of region 5.

Lemma 11. No convex cell exists such that i + j + k < i* + j* + k*, i > i_F, j < j_F, and k < k_F where is the candidate cell as the result of AlgB.

Proof. HB_b,c(p_F) is tangent to at p_F as explained in Lemma 10. Since the algorithm investigates all convex cells of which one of vertices is on , we first look up convex cells such that i > i_F, j < j_F, k < k_F, and j − k < j_F − k_F − 2 and then such that i > i_F, j < j_F, k < k_F, and j − k > j_F − k_F + 1. If an ellipse with two focal points p_b and p_c intersects with at a single point, the point is in the same side of HB_b,c(p_F) as p_c. When the ellipse intersects with at two points, if one intersection point is on the same side of HB_b,c(p_F) as p_b, the other intersection point should be in the opposite side of HB_b,c(p_F). Therefore, if there exists a convex cell such that i + j + k < i* + j* + k*, i > i_F, j < j_F, k < k_F, and j − k < j_F − k_F − 2, there also exists a convex cell such that j′ + k′ is equal to j + k and j′ − k′ is either j_F − k_F − 1 or j_F − k_F − 2. Since the algorithm investigates and , cannot be better than the candidate cell of the algorithm.

Next, we inspect convex cells such that i > i_F, j < j_F, k < k_F, and j − k > j_F − k_F + 1. Since all intersection points of and are in W₄, any convex cell such that i > i_F, j < j_F, k < k_F, and j − k > j_F − k_F + 1 cannot be better than induced by by Lemma 9 with a slight modification.

Theorem 2. Given 3 input points, AlgB computes an optimal Steiner tree for STP-MSPBEL in constant time.

Proof. It is obvious that AlgB takes constant time. We can compute the Fermat point in constant time in Step 1. In Step 2, we solve Eq (6) at most 8 times for each sub-part. Finally, in Step 3, the minimum spanning tree of three points can also be constructed in constant time.

Now, we show that the result steinerized tree is optimal for STP-MSPBEL. There are two ways to connect three terminal points with minimum Steiner points: one is to connect three points with a junction point, and the other is to connect two pairs of input points directly. Through Step 2, the algorithm presents the optimal 3-star tree among all possible 3-star trees connecting input points. By Observations 1 and 3, at least one optimal convex cell for a junction point intersects with Δp_ap_bp_c. By Lemmas 9, 10, and 11, we have shown that the result convex cell of the algorithm is better than or equal to other convex cells in Δp_ap_bp_c. Therefore, an optimal 3-star Steiner tree can be constructed by putting the junction point in the result convex cell and connecting input points with the minimum number of Steiner points. Since the algorithm computes the minimum spanning tree in Step 3, the optimal Steiner tree for STP-MSPBEL is presented by comparing those two trees.

We argue that the optimal location for the junction point may not be found although investigating all intersection points among , , , , , and which implies that the method in [18] cannot provide the optimal Steiner tree for three points. The counterexample exists when |p_ap_F| is short, and both |p_bp_F| and |p_cp_F| are relatively long. As shown in Fig 8, we set R = 1, |p_ap_F| = 1.739, |p_bp_F| = 1100.95 and |p_cp_F| = 1000.3007. Then, the distance between p_a and is greater than 2; however,the distance between p_a and is less than 4. Thus, if the junction point is located at , it requires 2101 Stiner points which is one less than that using . If the junction point is located at any intersection points among , , , , , and , it requires at least 2102 Steiner points. In addition, the steinerized minimum spanning tree needs 2102 Steiner points.

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Fig 8. The location of a junction point for an optimal 3-star tree can be found outside of when R = 1, |p_ap_F| = 1.739, |p_bp_F| = 1100.95 and |p_cp_F| = 1000.3007.

It shows that none of the intersection points between , , , , , and might yield the optimal junction point location.

https://doi.org/10.1371/journal.pone.0294353.g008

5 A combined algorithm

In this section, we develop another 3-approximation algorithm by combining the exact algorithm AlgB and the approximation algorithm AlgA to further reduce the number of required Steiner points in O(n log n) time. As described in Fig 9, AlgC computes the minimum spanning tree (MST) of terminal points and sorts MST edges by the number of required Steiner points in ascending order. Next, AlgC constructs the Voronoi diagram of the set of terminal points as sites and calculates the number of Steiner points required at each Voronoi vertex to connect its relevant 3 terminal points by AlgB. Then, the algorithm sorts the Voronoi vertices by the number of required Steiner points. Let E_MST(i) be a set of MST edges requiring i Steiner points and V_VD(j) be a set of Voronoi vertices requiring j Steiner points. By looking up the sorted lists of MST edges and Voronoi vertices, we are able to construct a steinerized spanning tree. We process an edge case or a vertex case by the following rules:

1. Rule 1. E_MST(i) is processed before V_VD(j) where i < ⌈j/2⌉.
2. Rule 2. V_VD(j) is processed before E_MST(i) where j ≤ 2i.

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Fig 9. AlgC: A combined algorithm.

https://doi.org/10.1371/journal.pone.0294353.g009

In the edge case, if two endpoints are not in the same component, we steinerize the edge and add it to the resulting tree. In the vertex case, if three relevant terminal points of the Voronoi vertex are not in the same component, we add Steiner points and relevant edges into the resulting tree by AlgB. For example, at first, we add all MST edges in E_MST(0) by Rule 1, and then we look at the Voronoi vertices in V_VD(1) and V_VD(2) by Rule 2. Iteratively, we increase the result tree size by processing with E_MST(i), V_VD(2i + 1), and V_VD(2i + 2). The basic idea behind this strategy is that if three points are in different connected components after adding edges in E_MST(i), at least 2(i + 1) Steiner points are required to connect those three components by MST edges. Thus, it is beneficial to use the vertex case if V_VD(2i + 1) or V_VD(2i + 2) connects those three points. If three points are already in the same connected component before V_VD(j) are considered, the bottleneck of the path between any of two points is less than or equal to ⌈j/2⌉ − 1. Thus, Despite removing the two bottleneck edges, the result tree deteriorates when Steiner points are inserted based on V_VD(j).

Theorem 3. AlgC provides the Steiner tree T_C satisfying C(T_C) ≤ C(T_A) in O(n log n) time where T_A is the Steiner tree produced by AlgA.

Proof. It is straightforward that AlgC takes O(n log n) time. Since AlgB takes constant time, Step 1 and 2 of AlgC takes O(n log n) time. In Step 3, there are n − 1 MST edges and the O(n) Voronoi vertices in the sorted lists. Since each connectivity query takes O(log n) time, the overall time complexity is bounded by O(n log n). The performance of AlgC is better than or equal to that of AlgA. AlgC first processes with E(0) and V(1). The resulting tree (forest) is exactly the same as the result of AlgA after processing Step 3. In the context of AlgC, Steiner points are introduced through the vertex case exclusively if the spanning tree established by AlgB outperforms the tree formed by steinerized edges of the minimum spanning tree. Consequently, AlgC produces a spanning tree that requires an equal or fewer number of Steiner points than that of AlgA.

6 Conclusion

We present approximation algorithms for the Steiner tree problem with the minimum number of Steiner points and bounded edge length. Previously, the best-known deterministic approximation algorithm has O(n³) running time with an approximation ratio of 3. In this paper, we propose O(n log n)-time approximation algorithm with the same approximation ratio, significantly improving the time complexity. Additionally, to enhance performance, we introduce the first exact algorithm that can provide an optimal Steiner tree for any given three points in constant time. By using this exact algorithm, we develop another 3-approximation algorithm that runs in O(n log n) time with better performance in terms of the number of required Steiner points.