It is well known that the Steiner minimal tree problem is one of the classical nonlinear combinatorial optimization problems. A visualization experiment approach succeeds in generating Steiner points automatically and showing the system shortest path, named Steiner minimum tree, physically and intuitively. However, it is difficult to form stabilized system shortest path when the number of given points is increased and irregularly distributed. Two algorithms, geometry algorithm and geometry-experiment algorithm (GEA), are constructed to solve system shortest path using the property of Delaunay diagram and basic philosophy of Geo-Steiner algorithm and matching up with the visualization experiment approach (VEA) when the given points increase. The approximate optimizing results are received by GEA and VEA for two examples. The validity of GEA was proved by solving practical problems in engineering, experiment, and comparative analysis. And the global shortest path can be obtained by GEA successfully with several actual calculations.

The Steiner minimal tree (SMT) problem can be stated as follows. Given a set

SMT, which is a well-known problem in combinatorial optimization, attracts considerable attention from theoretical to engineering point of view due to its applicability, which has been confirmed to be nondeterministic polynomial hard (NP-hard) in 1977, and scientists and researchers around the world have been devoting themselves to the resolution [

There are precise algorithm and heuristic algorithm for solving the system shortest path. The first precise algorithm for SMT problem is given by Melzak in 1961. Winter has given an opposite algorithm with the Melzak algorithm. The procedure Geo-Steiner given by Warme et al. had realized the algorithm previously given by Warme et al. [

The existing algorithms for SMT problem are mostly heuristic algorithms. The heuristic algorithm for SMT problem started from the early 1970s. The heuristics by Chang [

Fampa and Maculan presented a new mathematical programming formulation for SMT problem [

Although there are a large amount of methods for solving the SMT problem existing, most of them are heuristic algorithms and are easy to fall into partially optimum [

Here is the plan of our work. We first introduce the visualization experiment approach (Section

In this section, we will introduce the visualization experiment approach, since this method will be used in our new methods. First, we present the theoretical basis of visualization experiment.

Let

Some well-known properties should be given to obtain a solution for the SMT [

Given

A Steiner point has a degree equal to 3.

The edges emanating from a Steiner point have mutual angle equal to 120°.

All Steiner points lie in the convex hull of the given points.

The surfactant solution can easily form vesicle membrane. To achieve the steady state, in the effects of surface tension, the solution, perhaps the surface area of solution membrane, will also achieve smallest, but the surface area of sphere is of the smallest shape in all geometric solids which have the same volume. This is the solution physical property and principle of why the soap bubble always is sphere.

Many practical problems can be abstracted as SMT problem, such as the shortest path between cities, the distribution of grids, and the connections of network nodes. Since the cost can be reduced up to 13.4% by using SMT, the solution SMT is a hot spot issue, although SMT is an NP-hard problem. SMT can make a significant reduction in costs, so this problem has caused many mathematics and engineering researchers in the world to pay attention to it, and these researchers have established an international SteinLib test database [

Steiner minimal tree is often observed in our everyday life. For example, the multiple bubbles interact to form a Steiner tree [

The liquid surface tension can be defined as the surface energy of unit surface area. From the definition of the liquid surface tension, we hold that many systems in the nature tend to make its energy as low as possible, when certain external condition was satisfied. In other words, many systems in the nature tend to get the most stable state. So the liquid tends to get the shape with the minimum surface, so that the liquid has lowest surface energy [

In this paper, according to the properties of the liquid surface tension, we hold that the membrane has the smallest surface area, when the membrane formed by the liquid in the system is in the stable state. We design two parallel plates which can make the membrane surface with equal altitude. Then we can use the minimum surface area property to solve the minimum Steiner tree problem.

In the 19th century, scientists J.A.F. Plateau from Belgium observed and recorded the geometric shape of bubbles. He proposed the following rules [

Bubble membrane attaching to the wire frame or other closed structure has a smooth structure.

Bubbles connect in one or two ways, one of which is three surfaces connecting along with one smooth curve, and the other is the six-plane form curve, connecting at a vertex.

Bubble membrane connects at the same curve or at the same vertex and between the two surfaces it is equal; the angle should be 120° when three surfaces connect at a vertex; the angle approximately is 120° when six surfaces connect at a vertex.

VEA [

In the following part, we will introduce our visualization experiment device.

In the exploratory experiment, the parallel plates with fixed points are put in the surface active agent liquid slowly. After excluding the surface bubble, the parallel plates are taken out from the liquid slowly and measure the system shortest path (i.e., minimum Steiner tree) formed between the fixed points

The formula of active agent liquid: first put 2000 mL of distilled water into the vessel and add 20 g detergent with few additives and 8 g glycerin and mix all the materials. After 3 to 24 hours, the liquid will mix fully. Then the membrane traces can be kept more than 30 minutes.

The experiment for performance parameter: the smaller the distance between the parallel plates is, the longer the membranes are kept, more slowly the bubbles between the plates are excluded. Otherwise, the larger the distance between the parallel plates is, the shorter the membranes are kept and the faster the bubbles are excluded. The larger the diameters of the pillars are, the longer the membranes are kept, and the harder the membranes form. The smaller the diameters of the pillars are, the shorter the membranes are kept, and the more easily the membranes form.

The relative relationship between the plates and liquid: the liquid is still in the whole process. Get the membrane traces of global shortest path by putting test boards into the liquid and by taking test boards out of the liquid.

The process of the exploratory experiment.

Put the test boards into the liquid

Take the test boards out of the liquid

It is difficult to promote the exploratory experiment because the exploratory experiment has many limitations. The main reason is as follows.

It is difficult to introduce the engineering drawings to the experimental system because the test board contacts with liquid directly.

It is not easy to change the place and the number of the fixed points between the text boards, for the promotion will be blocked.

The entire operation is finished by hands, so human factors may reduce the accuracy of the experimental results.

To solve these problems, we assume that the stationary parts (the liquid) are changed to movable parts. Let the moving parts (the test plates) keep still. Then the above problems can be solved. The specific methods are the following

The liquid tank is composed of a double bottom with hollow parts. The placement of drawings is solved and the drawing is isolated with liquid.

Use magnet columns as fixed points. It is convenient to change to other engineering drawings.

Install the gradienter and the adjustable bolts to reduce the human factors to improve the accuracy of the test results.

Add the scale marks on the bottom of the tank to realize the measure of the visual paths. As a result, the movable cover board with the magnet columns in both sides and the bottom of the tank constitutes the instrument of the exploratory experiment. The membrane trace of the shortest path is formed through the variation of relative position. The visual instrument for the shortest path is shown in Figure

The shortest path experiment device.

Schematic diagram

Instrument image

When the number of given points is not too many, we can use VEA to get the shortest path. The shortest path of given points is obtained intuitively and simply through VEA, according to the physical properties of fluid. The step of using visualization experiment device is as follows:

configuration solution for experiment;

to insert drawing below the sink, select node in accordance with the drawing;

to pour the solution compounded into the channel, clearing up the bubble formed as the solution is poured into the channel;

open drainage switch and solution is discharged. The shortest path is formed at the same time;

when the membrane is stable, read the coordinates of given points and Steiner points, photograph and the experimental result chart is obtained;

at the end of the experiment, clean apparatus is used;

reorganize the empirical data.

Using VEA [

In the last section, we can use VEA to get the shortest path for the system which has not too many points. In this section, we will deal with the system with many points. We give two methods in this section. First, we introduce some definitions.

Suppose that

Simply speaking, each point of the given point set

The

The outer boundary of Delaunay diagram is a convex polygon, which is formed by connecting the convex sets in

The empty circumcircle property: the circumcircle of each triangular in the Delaunay diagram which is formed by points set

The property of maximizing the minimum angle: the minimum angle of each triangular in the Delaunay diagram is the biggest in the minimum angle of the triangular network formed by points set

The algorithm used in this paper for constructing Delaunay diagram is as follows.

Traverse all points and find the minimal horizontal coordinate and the minimal vertical coordinate (

Connect a pair of diagonal vertices of the rectangle. The rectangle is split into two triangles.

Put the points which coincide with the vertices and the boundary of rectangle in the rectangle. These points form the initial triangle network.

Add the remaining points in the corresponding triangle, using LOP algorithm to update the adjacent triangles from inside to outside.

Repeat Step

Algorithm process is shown in Figure

Initial construction of

Inner point insertion of

First, we introduce some definitions to be used later.

A topology is called Steiner topology, if the degree of each Steiner point is 3, and the degree of each given point is less than 3.

A topology is called full Steiner topology if the degree of each given point is 1 each the degree of each Steiner point is 3. The resulting Steiner tree is called full Steiner tree (full Steiner tree referred to as the FST).

Full Steiner topology satisfies the following conditions.

Degree condition: the degree of Steiner points must be 3, and the degree of the given points must be 1.

Angle condition: each angle between the three sides associated with the Steiner point is at 120 degrees.

The number of Steiner points: FST with

Melzak gives an algorithm for constructing a full Steiner tree from a full Steiner topology. This algorithm is the basis of constructing a minimal Steiner tree. We know that each nonfull topology can be decomposed into several small subfull topologies. The point sets of each subfull topology are the subsets of nonfull topology. Each side of nonfull topology only belongs to one full topology. For example, the five-point nonfull topology in Figure

Decomposition of Steiner topology.

For a nonfull Steiner topology, we can decompose it into several full Steiner topologies and then construct the full Steiner tree in each full Steiner topology by using Melzak method. Connecting all these full Steiner trees, we can get the full Steiner tree of the nonfull Steiner topology.

First, the Delaunay diagram [

Then the minimal spanning tree (MST) is constructed.

Subsequently, some subsets including 2, 3, or 4 points are constructed, for example,

(i) The subsets of 2 points are referred to as the sets only including 2 points, and the edge connecting these 2 points belongs to MST.

(ii) The subsets of 3 points are referred to as apexes of the triangle, and the 2 edges connecting these apexes belong to MST in Delaunay diagram.

(iii) The subsets of 4 points are referred to as apexes of convex quadrilateral. The apexes constitute two adjacent triangles in Delaunay diagram, and adjacent triangles have 3 edges belonging to MST.

The notion is used for dividing MST edges into some subsets including 2, 3, or 4 points.

When the given set is nonconvex, GEA description is as follows.

(1) After preparation, according to Definition

(2) Regarding subsets that only have one edge of MST, we can calculate its length and insert it into the rear of F in ascending.

If the subsets contain 3 points, please refer to FST Rule I; if the subsets contain 4 points, please refer to FST Rule II. Calculate

If the subset contains 2 points, the length is calculated directly, and it would be joined into the rear of queue

All the FSMT should be connected to be a tree according to the Kruskal algorithm, and the tree should be MST. In order to structure SMT, the queue of SMT is initialized to be null, taking out the subsets

The Rules for constructing FST are introduced as follows.

Picture for 3 points using Rule I.

Consider the equilateral triangle with the line segment

So

Picture for 4 points using Rule II.

So

Especially, when the given set is convex, we will adopt classical algorithm which was proposed by Melzak [

SMT of seven points in Melzak.

On the basis of Melzak algorithm [

The Delaunay diagram and the minimal spanning tree (MST) are constructed as in the Method

From the last section, we know that VEA can treat the system with points less than 10. Then, we can construct some subsets including

The subsets of

When the given set is nonconvex, GEA description is as follows.

This step can be done similarly as in Method

We can use VEA to construct FST for subsets containing

If the subsets contain 2 points, the length is calculated directly, and it would be joined into the rear of queue

All the FSMT should be connected to be a tree according to the Kruskal algorithm, and the tree should be MST. In order to structure SMT, the queue of SMT is initialized to be null, taking out the subsets

Optimize local line of the cities of Henan province, designing seventeen points for An Yang, Pu Yang, He Bin, Jiao Zuo, Xin Xiang, San Men Xia, Luo Yang, Zhen Zhou, Kai Feng, Shang Qiu, Xu Chang, Ping Ding Shan, Luo He, Zhou Kou, Nan Yang, Zhu Ma Dian, and Xin Yang as given points (Figure

Cities of Henan province.

Experiment Pictures.

Experiment picture of optimization of Part 1

Experiment picture of optimization of Part 2

When given points are increased, it is difficult to obtain SMT using experimental approach directly which requires higher operating techniques; meanwhile, the membrane is unstable, and the success ratio is lower.

GEA is used for shortest path planning for 65 given points as shown in Figure

Example of 64 points.

Experiment pictures.

Experiment picture of optimization of Part 1

Experiment picture of optimization of Part 2

Experiment picture of optimization of Part 3

Experiment picture of optimization of Part 4

Experiment picture of optimization of Part 5

Experiment picture of optimization of Part 6

A new geometry algorithm for SMT problem by referring visualization experiment is proposed in this paper. It is desired to make up for the insufficiency of experimental approach, that is, along with the increasing of given points. A new geometry algorithm for SMT problem by referring visualization experiment is proposed in this paper. It is desired to make up for the insufficiency of experimental approach, that is, along with the increasing of given points and irregular distribution; it takes more and more time to form membrane, and the objective is not steady; simultaneously the promotion of visualization approach is influenced. The example of the cities of Henan province has verified feasibility of GEA. When the given points are increased, GEA has obtained the shortest path when the membrane path is difficult to form through experimental approach, as shown in example of 65 points. By analyzing instances, contrasting compared with experiment results and verifying for many times, it is proved that GEA could find the location and number of Steiner points and could be used to solve global shortest path coordinating with visualization approach and to make up for the insufficiency of experimental approach.

The algorithm matched with visualization approach would be used in path planning not only for engineering problems, such as logistics warehouse system location, but also for the field of social services, such as transportation route planning. At the same time, the algorithm’ statement has broaden the thinking and provided a theoretical basis for the development of visualization approach.