CN106651010B - Shortest path-based wire network dividing method - Google Patents

Abstract

The invention relates to a shortest path-based partitioning method on a non-directional connected network. The method is oriented to the actual problem to be solved, and the L2 Euclidean distance is replaced by the shortest path distance along the communication route between nodes on the line network when the Voronoi diagram on the line network is generated. The method is based on a shortest path algorithm on a multidirectional connected network, and after a group of randomly distributed seed points are given, a Voronoi unit calculated on the basis of the shortest path on the current network is calculated; and then, the Voronoi diagram CVT algorithm of the Euclidean space is popularized to the Voronoi diagram on the undirected connected net, and through iteration, the weighted length of the net corresponding to each seed point on the net is equal as much as possible, so that the optimized division of the net model is obtained. The method can effectively solve the problems of distribution of public infrastructure (gas stations, charging stations, sanitary facilities) and the like on urban roads, distribution of logistics storage nodes and the like.

Description

Shortest path-based wire network dividing method

Technical Field

The invention relates to a shortest path-based net dividing method.

Background

How to divide planes based on a point set on a given plane is an important mathematical problem, and the method has important practical value in important fields such as communication node optimization, logistics distribution, urban planning and the like. The division of planes by Voronoi diagrams is the most common form. The Voronoi diagram is a mathematically important geometric structure, and is widely applied as an important abstract model representation method in many fields such as physics, chemistry, biology, engineering and the like. A set of discrete points P on a given plane, where each point is called a seed point, corresponds to a partition unit. If the distance from any point in the division unit to the seed point is smaller than the distance from any point to other seed points, the plane is called as a Voronoi division, the corresponding division unit is called as a Voronoi unit (Voronoi Cell), and in 1850, Dirichlet uses 2-dimensional and 3-dimensional Voronoi diagrams in quadratic research. In 1908, Georgy Voronoy defined and studied the Voronoi diagram problem for general n-dimensional space. In order to find the optimal space division result, the Voronoi diagram can be calculated firstly, then the geometric barycenter of the Voronoi unit is continuously used as a new seed point, the Voronoi diagram is recalculated, and the areas of the Voronoi unit can be made equal as much as possible through continuous iteration. This is the center of gravity Voronoi subdivision (CVT for short). The wanwen parallel et al at hong kong university studied the CVT's fast algorithm and GPU-based acceleration, and wangxing et al studied the CVT problem on triangular meshes.

In practical applications, for example, when electric vehicle charging stations are distributed in a city, it is generally required that the area of the coverage area of each charging station is as equal as possible, which can be solved by using the CVT method of Voronoi diagram. However, in an actual city, charging stations can only be distributed on the sides of streets, and electric vehicles can only run on the streets, so that the plane cannot be divided by using the traditional method of calculating the Voronoi diagram by using the euclidean distance. But the actual road condition needs to be considered, and the plane division problem is converted into the division problem on the road network. Moreover, in consideration of practical situations such as congestion, the geometric area covered by each seed point is not required to be equal, but the total weighted length of the roads covered by each seed point is required to be equal. We therefore propose an optimized Voronoi partitioning problem on a connectionless mesh.

Disclosure of Invention

The method focuses on the optimal division problem on the non-directional connected network. First, description and definition of the net model are given. For practical problems to be solved, when a Voronoi diagram of a net model is generated, the method replaces the L2 Euclidean distance by the shortest path distance between two points on the net along the communication route between nodes on the net, so that the considered net space has no convex space property. The method is based on a shortest path algorithm on a multidirectional connected network, and after a group of randomly distributed seed points are given, a Voronoi unit calculated on the basis of the shortest path on the current network is calculated; and then, the Voronoi diagram CVT algorithm of the Euclidean space is popularized to the Voronoi diagram on the undirected connected net, and through iteration, the weighted length of the net corresponding to each seed point on the net is equal as much as possible, so that the optimized division of the net model is obtained. The method can effectively solve the problems of distribution of public infrastructure (gas stations, charging stations, sanitary facilities) and the like on urban roads, distribution of logistics storage nodes and the like.

Drawings

The invention is described in further detail below with reference to the following figures and detailed description:

FIG. 1 is a net model and Voronoi diagram definition on a net. Wherein (a) a wire mesh; (b) the Voronoi diagrams on the net, s1, s2, s3 respectively represent three seed points, and the corresponding colors identify Voronoi cells corresponding to the respective seed points.

FIG. 2 is a schematic diagram of the steps of a Voronoi cell computation method on a net.

Fig. 3 is a schematic diagram of an application. The method comprises a Beijing urban area main traffic satellite map (upper left) and the connection condition of the main traffic roads within six rings (upper right, black square points are road intersections, and straight lines are the connection condition of a wire net), an initial charging station setting (middle), and a convergence result (lower) of the method after 35 iterations.

Detailed Description

And (3) defining a wire mesh model:

the wire mesh model represents a group of vertexes and the connection relation among the vertexes, and has wide application in the aspects of path planning and the like. We define net G ═ (V, E) over N-dimensional space, as shown in fig. 1 (a). Wherein the set of vertices is:

the set of edges is:

one continuous curve gamma of finite length_ijWith v_iAnd v_jEnd point and do not pass through

In the shortest path problem, there is also typically each edge (v) of the net_i,v_j) E gives weight w_i,j. From vertex v_iTo v_jThe sum of the weights of the edges passing through the path in sequence is called the weight of the path. From vertex v_iTo v_jThe path with the smallest weight is called vertex v_iTo v_jThe corresponding weight is called vertex v_iTo v_jThe shortest distance of (d)_i,j. If any two points in the vertex set have paths, the net is called connected, otherwise the net is not connected. If the edge is directional, i.e. (v)_i,v_j) And (v)_j,v_i) And if different directed edges are represented, the net is called a directed net, otherwise, the net is a non-directed net. For a multidirectional net, one can consider (v)_i,v_j) And (v)_j,v_i) Represent the same side and have w_i,j＝w_j,i。

Voronoi diagram on a wire mesh

The method considers the Voronoi diagram problem on the undirected connected net and defines a net space omega on an N-dimensional connected net G (V, E)_GComprises the following steps:

at curve gamma_ijTo (v)_i,v_j)∈E}. (3)

Note d_G(p₁,p₂) Is the space omega of the wire mesh_GTwo points of upper p₁,p₂Measure of distance between, w (p)₁,p₂) Is two points p on the same curve₁,p₂And (4) an inter-weight. Given omega_GSet of discrete points on

Point s_kCorresponding Voronoi cell

Is defined as:

balance

Is the space omega of the wire mesh_GA Voronoi division. FIG. 1(b) shows a Voronoi cell defined by three seed points on a net.

Center of gravity of wire mesh

For Voronoi partitioning on the wire mesh, we also want the sum of the "area" -weight of each Voronoi unit to be as equal as possible, for example in practical applications, we want to evenly set up charging stations so that the time required for each electric vehicle to reach its nearest charging station is as equal as possible; for another example, it is desirable to locate the garbage cans in the park so that each guest travels as equally as possible to the nearest garbage can.

To do this, a unidirectional connected net G ═ V, E, and net space Ω are given_GProbability density function p, for a given m seed points

Corresponding Voronoi subdivision

We define the "center of gravity" of each Voronoi cell as

Wherein E is_kAs a sub-net G_kIs set.

Calculation method for Voronoi diagram subdivision on undirected connected weighted line network

Given a non-directional connected net G ═ V, E, weight W, and net space Ω_GM seed points of

The Voronoi subdivision on the line network G is to calculate the Voronoi unit corresponding to each seed point

The key to calculating the Voronoi diagram on the undirected connected-wire network is to calculate the length of the shortest path between any two points in the network, i.e. to define the online network space

The distance between any two points. For this reason, we consider the seed point set first

Adding net G ═ V, E, and updating to obtain net G_S＝(V_S,E_S) And corresponding weight W_SThen calculate net G_SThe shortest path length between the upper vertices. Next, G can be calculated_S＝(V_S,E_S) The middle vertex belongs to the Voronoi unit and the division point of the edge. And finally forming the Voronoi subdivision on the wire mesh according to the dividing points. The specific method comprises the following steps:

(one) collecting the seed points

Adding net G ═ V, E, and updating to obtain net G_S＝(V_S,E_S) And corresponding weight W_S: each point S in S, assuming S is bounded by (v)_I,v_j) Then, there are:

E_S＝E_S∪{(s,v_i),(s,v_j)}\{(v_i,v_j)},W_S(s,v_i)＝w(s,v_i) (6)

W_S(s,v_j)＝w(s,v_j),W_S(v_i,v_j)＝+∞,W_S(s,s)＝0 (7)

and (II) acquiring a new shortest path. The method only considers the case of non-negative connected nets. For a non-negative-weight connected net, each vertex of the net is traversed, and the shortest paths from the vertex to the other vertices are calculated by adopting a Dijkstra algorithm to obtain the shortest path between any two vertices.

(III) calculating a Voronoi diagram:

given a net G ═ V, E, Voronoi seed point set

Computing point set S with respect to net space Ω_GA Voronoi division

I.e. the on-line network G_S＝(V_s,E_s) And wire mesh space thereof

Upper determination of V_sThe Voronoi cell to which each vertex belongs, and each edge E_sThe Voronoi cell to which the point on top belongs.

3.1 calculate V_sVoronoi cell of middle vertex

Definition of

Is a space of wire net

Mapping to subsets of the Voronoi seed Point set S, representing the wire mesh space

Each point in the set of seed points and the corresponding relation of the seed point set of the Voronoi unit.

Due to V_sThe points in (1) are divided into two parts, namely a vertex set V and a Voronoi seed point set S in the wire mesh G. For the

Vor(s) { s }; for the

The method comprises the following steps of (1) preparing,

3.2 calculation of E_sVoronoi cell of point on middle edge

If the shortest path from one end point of an edge to the seed point of a Voronoi cell to which the edge belongs passes through the edge, any point on the edge belongs to the Voronoi cell. Otherwise, the point on the edge may belong to a different Voronoi cell. So the following method is adopted to judge which Voronoi unit it belongs to:

if (v)_i,v_j)∈E_SMemory for recording

S_ij＝Vor(v_i)∩Vor(v_j)， (9)

d_iIs v is_iTo Vor (v)_i) Distance of the middle seed point, d_jIs v is_jTo Vor (v)_j) The distance between the seed points in the seed box,

(i) if | d_i-d_j|＝w_i,jThen to curve gamma_ijAny point p above (excluding the end point v)_i,v_j) Is provided with

Vor(p)＝S_ij(10)

(ii) If | d_i-d_j|≠w_i,jCurve gamma_ijHas a unique dividing point p_cSo that Vor (p)_c)＝Vor(v_i)∪Vor(v_j) And curve gamma_ijUpper p_cTo endpoint v_iThe weight of (A) is:

w(p_c,v_i)＝(w_i,j+d_j-d_i)/2。 (11)

the above two formulas are demonstrated as follows:

and (3) proving that: (ii) first proves (i). Without loss of generality, assume

d_j＝d_i+w_i,j。 (12)

To pair

Is obviously provided with

Then there are

Due to d_jIs a vertex v_jMinimum distance to all seed points, then s' ∈ Vor (v ∈ Vor)_j) I.e. by

So that there is a possibility that,

S_ij＝Vor(v_i)。 (15)

from (13), the sub-path properties of the shortest path are found from v_jThe shortest path to s' passes through the edge (v)_i,v_j). From the shortest-circuit property, for curve γ_ijAny point p above (excluding the end point v)_i,v_j) P belongs to the Voronoi cell in which s' is located, then

Consider that

And curve gamma_ijAny point p above (excluding the end point v)_i,v_j) We have

Also known from the shortest-circuit property

Also, from (12)

The compositions of (17) to (20) are,

known from (21), to

Is provided with

In addition, according to the closure property,

from (15), (16), (22) and (23), there is Vor (p) ═ S_ij. (i) Obtaining the syndrome.

And (ii) was confirmed. Without loss of generality, assume d_i<d_jThen must have

d_j<d_i+w_i,j。 (24)

Otherwise, if d_j>d_i+w_i,jThen we are v_jFinding a shorter path to the seed point, which is compared to the known d_jIs v_jThe shortest distance to the seed point contradicts. For curve gamma_ijAt any point p, as known from closeness,

as shown in fig. 2, consider

k ≠ l. Without loss of generality, assume

Then p to s_kMust first go directly through the end point v_iThen arrives at s_k. If not, p to s_kFirst via the end point v_jThen arrives at s_kIs provided with

And is composed of_jIs v_jThe shortest distance to the seed point is known,

as can be seen from (27) and (28) and the definition of the shortest path,

this contradicts (26). In this case, as is clear from the expressions (25) and (26),

known from (25) and (30) at this time

Vor(p)＝Vor(v_i)。 (31)

Due to curve gamma_ijIs a continuous curve, and the minimum distance from p to all the seed points is continuously changed in the process that the point p on the curve is continuously changed from one end point to the other end point. Also, as shown in (24), the minimum distance is monotonically increasing and then monotonically decreasing. Thus curve gamma_ijHas a unique dividing point p_cAnd at the division point has

Then, as shown in (31) and (32),

Vor(p_c)＝Vor(v_i)∪Vor(v_j)。 (33)

since at this time p_cTo s_kIs passing through the end point v_iIs then provided with

In the same way, there are

And due to

w_i,j＝w(p_c,v_i)+w(p_c,v_j)， (36)

Substituting (34) to (36) into (32) gives (11).

The syndrome is two

To E_sAny one side (v) of_i,v_j) According to the method, whether the edge has the division point can be judged, and if the edge has the division point, the position of the division point is given by the formula (11). Note that all such sets of boundary segmentation points are C.

As can be seen from the definition of Vor and the definition of Voronoi cells on a net,

consider that

Sub-net G consisting of vertices and division points in cells_k＝(V_k,E_k) Then there is

This means that the net space

Voronoi division above

Corresponding to the wire net G_SA division of

And (IV) finally solving the gravity center Voronoi division of the given area by using an iteration mode in the Lloyd algorithm, and continuously and iteratively constructing the Voronoi division and updating the position of the seed point. Given the number m of seed points, the region Ω, and the probability density function ρ defined on Ω, the main process is as follows:

1) randomly selecting m initial seed points

2) According to the seed point

Computing a Voronoi subdivision for region Ω

3) Calculating each Voronoi cell in the step 2)

To obtain a new seed point set

4) If the new seed point set meets the given convergence criterion (typically determined using a method of whether the average distance of the new point set from the previous point set is less than a given threshold, which may be set to 0.1 times the unit length), the algorithm ends, otherwise, step 2 is performed.

This iterative approach has been demonstrated to have local convergence by Kieffer et al. The resulting Voronoi partitioning is the optimal partitioning of the wire mesh model for a given set of points as seed points.

Application of the method

As electric vehicles become a choice of more and more people, the demand for electric vehicle charging stations in urban traffic is increasing, and how to plan the locations of the charging stations so that users can find nearby charging stations as soon as possible and improve the utilization rate of each charging station becomes a problem to be considered by designers. Taking the main traffic road within six rings of Beijing as an example, the method is used for demonstrating how to determine the positions of 50 charging stations. The net generated by the road connection on the beijing map (1034 vertexes, 1801 edges, and the edge weight is the length of the corresponding road) and the final optimization result of placing 50 charging stations are shown in fig. 3. The application verifies that the method can effectively solve the problem of uniform distribution of the nodes on the road network.

Claims (4)

1. A shortest path-based partitioning method is characterized by comprising the following steps:

a) representing the data of the road scene as a multidirectional connected network, and abstracting the road congestion condition factors as the weight of edges in a network model;

b) giving a seed point set, adding the point set as a new node into a net model, and updating a vertex set, an edge set and the weight of the net;

c) obtaining a wire mesh Voronoi division based on a current point by a method of calculating a shortest path and edge division points;

d) calculating the gravity center of each current Voronoi unit, moving the seed point to the gravity center position, and calculating new Voronoi division;

e) repeating the step c and the step d until the position of the seed point set is converged;

wherein, the step of calculating the gravity center of each current Voronoi unit specifically comprises the following steps:

wherein, G ═ (V, E) is a net of unidirectional links in N-dimensional space, and V ═ V is a set of vertices₁,v₂,…,v_n},

Edge set

Ω_GIn order to connect the wire mesh space on the wire mesh G,

is the space omega of the wire mesh_GThe distance between the upper two points s and v is measured, and rho is the net space omega_GA probability density function of (a); g_S＝(V_S,E_S)，

And

respectively adding seed points into G

Then obtaining a new wire mesh model, and measuring wire mesh space and distance; e_kAs a sub-net G_kThe set of edges of (a) is,

as a sub-net G_kA corresponding wire mesh space;

the present point-based wire mesh Voronoi division specifically includes:

balance

Is the space omega of the wire mesh_GA Voronoi division.

2. A method according to claim 1, characterized in that: finally, the most balanced net partition based on the point set can be found, so that the sum of the distances from each point on the net to the final seed point position is minimum.

3. A method according to claim 1, characterized in that: only Voronoi units of the top points and the edge division points on the line network need to be calculated, and Voronoi division of the line network is formed according to the division points;

wherein the online fixed points comprise original fixed points and seed points.

4. The method according to claim 1, steps c and d, characterized in that: the steps of Voronoi division and gravity center calculation on the wire mesh are different from plane division, and redefinition and calculation are required based on the shortest path distance on the wire mesh model.

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