CN114001747B - Urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm - Google Patents

Urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm Download PDF

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CN114001747B
CN114001747B CN202111368321.3A CN202111368321A CN114001747B CN 114001747 B CN114001747 B CN 114001747B CN 202111368321 A CN202111368321 A CN 202111368321A CN 114001747 B CN114001747 B CN 114001747B
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intersection
source point
shortest path
kth
intersections
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CN114001747A (en
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丁建勋
殷慧娟
黄林煊
颜江楠
樊哲延
曾嘉涵
查菲菲
徐小明
龙建成
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Hefei University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/26—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 specially adapted for navigation in a road network
    • G01C21/34—Route searching; Route guidance
    • G01C21/3407—Route searching; Route guidance specially adapted for specific applications
    • G01C21/3415—Dynamic re-routing, e.g. recalculating the route when the user deviates from calculated route or after detecting real-time traffic data or accidents
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/20—Instruments for performing navigational calculations
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/26—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 specially adapted for navigation in a road network
    • G01C21/34—Route searching; Route guidance
    • G01C21/3446—Details of route searching algorithms, e.g. Dijkstra, A*, arc-flags, using precalculated routes

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Abstract

The invention discloses an urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm, which comprises the following steps: 1, acquiring a real-time urban road network diagram, and selecting a plurality of source point intersections; 2 if all the precursor intersections of an adjacent intersection of the intersection with the shortest path weight obtained by the previous iteration obtain the shortest path weight of the source point, updating the shortest path weight from the source point to the adjacent intersection, otherwise updating the upper bound of the shortest path weight of the adjacent intersection, and selecting the smallest updated shortest path weight; 3 backtracking the shortest path obtained in the iteration, and searching and updating the weight of the shortest path from other source points in the path to the intersection; and 4, ending the search and outputting when the shortest paths from all source points to all intersections in the intersection set are obtained, otherwise, continuing iteration. The method and the system can effectively improve the calculation efficiency of large-scale road network navigation planning and meet the travel demands of different travel points of multiple users at the same time.

Description

Urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm
Technical Field
The invention belongs to the field of vehicle navigation path optimization, and particularly relates to an urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm.
Background
With the continuous expansion of the urban road network scale, road network nodes are correspondingly increased, and the existing shortest path navigation system firstly utilizes a single-source shortest algorithm or a full-source shortest algorithm to solve the travel demands of users, so that the calculation workload is huge, and the real-time and efficient requirements of people on navigation software are hardly met. The common full-source shortest algorithm is used in a large-scale urban road network, particularly when road network nodes are dense, the range involved in calculation is wide, time consumption and resource occupation are correspondingly increased, and the method for exchanging the quantity of resources for efficiency and quality is a common coping scheme.
In solving the shortest solution of the vehicle navigation core problem, different students design different algorithms, and the single-source shortest algorithm is adopted one by one for the application of the single-source shortest algorithm, so that some calculated weights can be repeatedly calculated, and a large amount of calculation redundancy is caused; the whole source shortest path algorithm (from all points to all points) is also unnecessary to carry out the single source point shortest path algorithm, so that the calculated amount is increased, and the concept of real-time navigation is not adapted.
Meanwhile, the path of the shortest path is single in the same algorithm at present, the traditional Dijistra algorithm is used for gradually obtaining the shortest path only by searching leaf nodes, and the efficiency is low. The improvement of the relevant scholars on the basis of the method is still based on the point, the efficiency improvement effect is poor, and the parallel realization is not facilitated.
Disclosure of Invention
The invention aims to solve the defects of the prior art, and provides an urban road network multisource shortest path acquisition method based on shared calculation and dijkstra algorithm, so that the problem of optimizing a vehicle-mounted navigation path can be solved by obtaining the weight of the shortest path in multiple ways, the calculation efficiency of large-scale road network navigation planning is effectively improved, and the travel demands of different travel points of multiple users at the same time are met.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
the invention discloses an urban road network multi-source shortest path acquisition method based on shared calculation and dijkstra algorithm, which is characterized by comprising the following steps:
step 1: constructing an urban road network, defining parameters and initializing;
acquiring real-time road network data and obtaining an urban road network G= (V, A, W), wherein V represents an intersection set, and V= { V 1 ,v 2 ,v 3 ,…,v i ,…,v R }，v i Represents the i-th intersection, i=1, 2,3, …, R is the total number of intersections in the urban road network G, a represents the set of road segments between intersections, and a= { (v) i ,v j )|i,j＝1,2,3,…,R}，(v i ,v j ) Represents the i-th intersection v i To the jth intersection v j Directed road segments in between, W represents the set of weights for road segments between intersections, w= { ω i,j |i,j＝1,2,3,…,R}，ω i,j Is a directed road section (v) i ,v j ) If the weight of the ith intersection v i To the jth intersection v j Between which there is a directed road section (v i ,v j ) Then the j-th intersection v j For the ith intersection v i An i-th intersection v i For the j-th intersection v j Precursor intersection of (a), and omega i,j Not less than 0; if the ith intersection v i To the jth intersection v j There is no directed road section (v) i ,v j ) Let omega i,j = + infinity; defining the current iteration number as t;
step 2: acquiring a source point intersection set:
selecting K source point intersections and forming a source point intersection set V '= { V' k I k=1, 2, …, K }, where v' k Representing the kth source point intersection in the source point intersection set V ', and enabling the kth source point intersection V' k Sequence number s in intersection set V k I.e.
Figure BDA0003361675280000021
Figure BDA0003361675280000022
Representing the s < th > in intersection set V k K is more than 1 and less than or equal to R at each intersection;
step 3: initializing the weights from the source point intersection to each intersection of the t-th iteration:
defining a kth source point intersection V ' from the source point intersection set V ' at the t-th iteration ' k To the ith intersection V in intersection set V i The upper bound of the weight of the shortest path of (a) is
Figure BDA0003361675280000023
And initialize +.>
Figure BDA0003361675280000024
Is +' infinity;
defining a kth source point intersection V 'from the set of source point intersections V' k To the ith intersection V in intersection set V i The shortest path of (a) is weighted as P k (v i ) And initialize P k (v i ) Is +' infinity;
initializing t=1;
initializing the kth source point intersection v 'under the t-th iteration' k To the s < th) k Each intersection v sk The upper bound of the weight of the shortest path of (a) is
Figure BDA0003361675280000025
Then the kth source point intersection V ' in the source point intersection set V ' at the t-th iteration ' k To the s < th) k Each intersection
Figure BDA0003361675280000026
The shortest path of (a) is given the weight +.>
Figure BDA0003361675280000027
And will s k Crossing->
Figure BDA0003361675280000028
Added to the previous t-1 iterations at the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure BDA0003361675280000029
With the kth source point intersection v 'in the t-1 th iteration' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000210
Definition L k,i Representing the kth source point intersection V 'in the set of source point intersections V' k To the ith intersection V in intersection set V i Intersection set of shortest paths of (a) and initializing
Figure BDA00033616752800000211
Step 4: for the k source point intersection v 'in the t-1 iteration' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000031
Forward searches are performed at all intersections in (a) and the shortest path weights are updated:
step 4.1: initializing k=1;
step 4.2: if the k-th source point intersection v 'is used in the previous t-1 iterations' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000032
All intersections in the intersection set V are included, and the kth source point intersection V 'is used in the source point intersection set V' k Ending the shortest searching for the source point, and turning to the step 6, otherwise turning to the step 4.3;
step 4.3: initializing the kth iteration with the kth source point intersection v' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000033
Note that intersection v 'at kth source point in the t-1 th iteration' k Obtained as source pointsIntersection set of weights for shortest path
Figure BDA0003361675280000034
The number of intersections in (a) is M t-1 The method comprises the steps of carrying out a first treatment on the surface of the Taking out the intersection with m < th > of the intersection>
Figure BDA0003361675280000035
Let the mth intersection +.>
Figure BDA0003361675280000036
Sequence number +.>
Figure BDA0003361675280000037
I.e. < ->
Figure BDA0003361675280000038
Step 4.3.1: initializing m=1;
step 4.3.2: traversing the mth intersection
Figure BDA0003361675280000039
All adjacent intersections of (2) and selecting from them all intersections not belonging to the intersection set +.>
Figure BDA00033616752800000310
And add to the adjacency set +.>
Figure BDA00033616752800000311
Step 4.3.2.1: fetching contiguous sets
Figure BDA00033616752800000312
The q-th intersection of (a) is marked as the adjacency set +.>
Figure BDA00033616752800000313
In (q) th intersection->
Figure BDA00033616752800000314
Front of (2)The set of driving intersections is +.>
Figure BDA00033616752800000315
Let adjacency set +.>
Figure BDA00033616752800000316
In (q) th intersection->
Figure BDA00033616752800000317
Number r in intersection set V q I.e. +.>
Figure BDA00033616752800000318
Step 4.3.2.2: initializing q=1;
step 4.3.2.3: traversing precursor intersection sets
Figure BDA00033616752800000319
If all intersections in (a) are set of precursor intersections +.>
Figure BDA00033616752800000320
All intersections in (a) belong to the kth source point intersection v 'under the previous t-1 iterations' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000321
Step 4.4 is carried out, otherwise step 4.5 is carried out;
step 4.4: acquiring v 'from the kth source point intersection' k To all precursor intersections are obtained as the kth source point intersection v' k The shortest weight of the intersection, which is the weight of the shortest path of the source point:
step 4.4.1: precursor intersection set
Figure BDA00033616752800000322
The number of intersections in the method is B, and the precursor intersection set is taken out
Figure BDA0003361675280000041
B crossing of (a)>
Figure BDA0003361675280000042
Let its serial number in intersection set V be n b I.e. +.>
Figure BDA0003361675280000043
Step 4.4.2: initializing b=1;
step 4.4.3: if it is
Figure BDA0003361675280000044
Will->
Figure BDA0003361675280000045
Assign to->
Figure BDA0003361675280000046
And record the precursor intersection->
Figure BDA0003361675280000047
Sequence number n of intersection set V b Assignment to the (r) th in intersection set V q Crossing->
Figure BDA0003361675280000048
Number n of temporary precursor intersection b,min The method comprises the steps of carrying out a first treatment on the surface of the Wherein (1)>
Figure BDA0003361675280000049
Represents the kth source point intersection v' k To the nth of intersection set V b The right of shortest path of the intersection, < ->
Figure BDA00033616752800000410
Represents the nth of intersection set V b Crossing->
Figure BDA00033616752800000411
To the r q Crossing->
Figure BDA00033616752800000412
Is used for the weight of the (c),
Figure BDA00033616752800000413
represents the kth source point intersection v' k To the (r) th intersection set V q Crossing->
Figure BDA00033616752800000414
The weight of the shortest path of (a);
step 4.4.4: assigning b+1 to B, proceeding to step 4.4.3 until b=b, thereby outputting the kth source point intersection v' k To the (r) th intersection set V q Each intersection
Figure BDA00033616752800000415
Right of shortest path->
Figure BDA00033616752800000416
Sequence number n b,min And will be the r q Crossing->
Figure BDA00033616752800000417
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the nth of intersection set V b,min Crossing->
Figure BDA00033616752800000418
Intersection set of shortest path of +.>
Figure BDA00033616752800000419
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the (r) th intersection set V q Crossing->
Figure BDA00033616752800000420
Intersection set of shortest path of +.>
Figure BDA00033616752800000421
I.e.
Figure BDA00033616752800000422
Will be the r q Crossing->
Figure BDA00033616752800000423
Added to the intersection v 'at the kth source point' k Obtaining the set of shortest weights for the source point +.>
Figure BDA00033616752800000424
And the kth iteration with the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000425
I.e. < ->
Figure BDA00033616752800000426
Figure BDA00033616752800000427
Turning to step 4.6;
step 4.5: if it is
Figure BDA00033616752800000428
Will->
Figure BDA00033616752800000429
Assign to->
Figure BDA00033616752800000430
Otherwise, turning to step 4.6;
step 4.6: if Q is not equal to Q, q+1 is assigned to Q, the step 4.3.2.3 is shifted, otherwise, m+1 is assigned to M, and M > M is judged t-1 If so, executing the step 4.7; otherwise, turning to step 4.3.2;
step 4.7: selecting an intersection with the smallest upper bound of the weight of the shortest path from the intersection set V at the t-th iteration
Figure BDA00033616752800000431
And->
Figure BDA00033616752800000432
Not belonging to the kth source point intersection v 'under the previous t iterations' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000433
I.e. < ->
Figure BDA00033616752800000434
If->
Figure BDA00033616752800000435
Then the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure BDA00033616752800000436
Upper bound of the weight of the shortest path of +.>
Figure BDA00033616752800000437
Assigned to the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure BDA0003361675280000051
Right of shortest path->
Figure BDA0003361675280000052
And will be the ith min Crossing->
Figure BDA0003361675280000053
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the intersection set V +.>
Figure BDA0003361675280000054
Crossing->
Figure BDA0003361675280000055
Is a shortest path intersection set +.>
Figure BDA0003361675280000056
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the ith of intersection set V min Crossing->
Figure BDA0003361675280000057
Is a shortest path intersection set +.>
Figure BDA0003361675280000058
I.e. < ->
Figure BDA0003361675280000059
And will be the ith min Each intersection
Figure BDA00033616752800000510
Add to the collection->
Figure BDA00033616752800000511
And intersection set->
Figure BDA00033616752800000512
Step 5: backtracking the kth source point intersection v 'under the tth iteration' k To intersection set
Figure BDA00033616752800000513
The shortest path of the intersection of (a):
step 5.1: note that the kth source point intersection v 'in the t-th iteration' k Intersection set that gets the shortest path weight for source point
Figure BDA00033616752800000514
The number of intersections in (a) is M t Taking out the intersection with m < th > of the intersection>
Figure BDA00033616752800000515
And will cross the mthMouth->
Figure BDA00033616752800000516
Sequence number +.>
Figure BDA00033616752800000517
I.e. < ->
Figure BDA00033616752800000518
Step 5.2: initializing m=1;
step 5.3: record the k-th source point intersection v' k Into intersection set V
Figure BDA00033616752800000519
Crossing->
Figure BDA00033616752800000520
Is>
Figure BDA00033616752800000521
The number of the intermediate intersections in (a) is E m Marking the serial number rho of the epsilon-th intermediate intersection in the intersection set V ε ；
Step 5.3.1: initializing epsilon=1;
step 5.3.2: if it is
Figure BDA00033616752800000522
Then assume->
Figure BDA00033616752800000523
At the same time is the kth source point intersection V 'in the source point intersection set V' k′ I.e. +.>
Figure BDA00033616752800000524
Then take the shortest path point set +.>
Figure BDA00033616752800000525
In epsilon crossing->
Figure BDA00033616752800000526
To E th m +2 intersections->
Figure BDA00033616752800000527
All intersections as the kth 'source point intersection v' k′ To the intersection set V +.>
Figure BDA00033616752800000528
Crossing->
Figure BDA00033616752800000529
Is a shortest path intersection set +.>
Figure BDA00033616752800000530
And will->
Figure BDA00033616752800000531
Assign to->
Figure BDA00033616752800000532
If k' > k, and will be +.>
Figure BDA00033616752800000533
Crossing->
Figure BDA00033616752800000534
Added to the previous t-1 iterations to k 'th source point intersection v' k′ Set of weights to get shortest path for source point +.>
Figure BDA00033616752800000535
And the kth ' source point intersection v ' in the t-1 th iteration ' k′ Intersection set weighted for shortest path for source +.>
Figure BDA00033616752800000536
Otherwise, go->
Figure BDA00033616752800000537
Crossing->
Figure BDA00033616752800000538
Add to the previous t iterations to kth 'source point intersection v' k′ Set of weights to get shortest path for source point +.>
Figure BDA00033616752800000539
And the kth iteration with the kth 'source point intersection v' k′ Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000540
Step 5.3.3: judgment of ε=E m If so, turning to step 5.4, otherwise, assigning epsilon+1 to epsilon, and turning to step 5.3.2;
step 5.4: judging that m=m t If so, turning to the step 6, otherwise, assigning m+1 to m, and turning to the step 5.3;
step 6: judging whether k=k is satisfied, if so, turning to step 7, otherwise, assigning k+1 to K, and turning to step 4.2;
step 7: and if the shortest searching with all the source points as the source points in the source point intersection set V 'under the previous t iterations is finished, the shortest path from each source point intersection in the source point intersection set V' to any intersection in the intersection set V is obtained, otherwise, t+1 is assigned to t, and the step 4 is carried out.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention provides a partial source shortest path generation method between a single source shortest path algorithm and a full source shortest path algorithm, and provides an ingenious multi-source shortest path optimization method for solving the problem of complex calculation of a large-scale road network, wherein the shortest path weight of each source point can be calculated in a shared mode in the process of calculating the multi-source shortest path, and finally the shortest path weight is obtained in a multi-path mode through ideas such as shared calculation on the basis of dijkstra algorithm, so that the calculation efficiency is further improved, a large-scale road network diagram is processed more efficiently, resources are saved, the efficiency is ensured, the current situation of practical navigation application requirements is combined, unnecessary time loss is avoided, the calculation efficiency and the calculation quality are substantially improved, the calculation burden of navigation planning can be relieved, and the method is more practical.
2. In the existing full-source shortest path algorithm, the calculation efficiency is low when dealing with a complex large-scale map due to the intrinsic characteristics of the algorithm, and traversal search needs to be carried out on all intersection source points in the large-scale road network map, so that the efficiency is greatly reduced. The invention considers that only the source point expansion shortest path searching calculation with the user searching path requirement is selected, and omits the source point searching calculation without the user initiating the path searching, thereby carrying out a multi-source shortest path algorithm with higher efficiency under the condition of avoiding the parallel calculation of the full-source expansion path searching or the single-source shortest path algorithm, and providing a new solution for customizing the shortest path for a plurality of users at the same time for the navigation system.
3. Based on the theorem that the sub-path of a shortest path is always the shortest sub-path, when the shortest path from a source intersection to an intersection is obtained, the shortest paths from other source intersections possibly existing therein to the intersection can be traced back. The invention skillfully uses the point and records the weight of the shortest path traced back, avoids the independent calculation of each source point intersection, facilitates the utilization of the next iterative update and reduces the repeated calculation to cooperatively calculate, and forms a unique common calculation idea. This common computing idea complements the approach to obtain multi-source shortest paths, which reduces the computational effort as one of the new approaches to obtain the weights of the shortest paths.
4. The invention researches a directional traffic road network, namely, the road bidirectional traffic condition difference is considered in practical application, so that the road traffic road network is more accurately attached to practice. Meanwhile, the invention discovers that if all the precursor adjacent intersections of a certain intersection are searched for the shortest path taking the same source point intersection as a starting point, the weight of the shortest path from the source point intersection to the intersection can be directly calculated and obtained compared with the upper bound of the weight of the shortest path updated by continuing to perform dijkstra algorithm. With this, the present invention adds another approach to get the shortest path, which substantially increases the computational efficiency and quality. Meanwhile, the shortest path obtained by backtracking the common computing thought can be effectively utilized under certain conditions, so that the steps in the method can interact with each other, the work computing load is greatly reduced, and the shortest path searching of the large-scale road network graph is facilitated.
5. According to the method, under each iteration, after judging that the proper path for acquiring the shortest path is selected, the calculation of the two paths for acquiring the shortest path can be simultaneously carried out, and according to the characteristics, the method provides possibility for parallelization calculation under the multithreading condition, lays a foundation for fusion parallel multi-source search in the subsequent urban road network navigation planning work, is convenient to cooperate with graph dividing technology and parallel search, and further improves the speed for multi-user path planning.
Drawings
FIG. 1 is an overall flow chart of the present invention;
FIG. 2 is a flowchart of a multi-path acquisition shortest path in accordance with the present invention;
FIG. 3 is a schematic diagram of the main idea of the 4 th point (the right of the adjacent intersection can be directly calculated to obtain the shortest path) in the summary;
FIG. 4 is a simple city road network graph and weight table (with initial parameter values) of the present invention;
FIG. 5 shows the source point intersection v 'at iteration 3 of the present invention' 1 An upper bound table and a path diagram of the weight of the shortest path to each intersection;
FIG. 6 shows the source point intersection v 'at iteration 3 of the present invention' 2 An upper bound table and a path diagram of the weight of the shortest path to each intersection;
FIG. 7 is an upper bound table and path diagram of the weights of the shortest paths from each source point intersection to each intersection at iteration 6 of the present invention.
Detailed Description
In this embodiment, starting from the idea of solving a multi-source shortest path between a single source and a full source, an urban road network multi-source shortest path acquisition method based on a common calculation and dijkstra algorithm is provided, the specific flow steps are shown in fig. 1, the main idea framework is shown in fig. 2, and the main flows a, b, c, d in fig. 1 and 2 are in one-to-one correspondence, and the steps are as follows:
step 1: constructing an urban road network, defining parameters and initializing;
acquiring real-time road network data and obtaining an urban road network g= (V, a, W), taking the simple road network diagram in fig. 4 as an example, V represents an intersection set, and v= { V 1 ,v 2 ,v 3 ,…,v i ,…,v R }，v i Represents the i-th intersection, i=1, 2,3, …, R is the total number of intersections in the urban road network G, a represents the set of road segments between intersections, and a= { (v) i ,v j )|i,j＝1,2,3,…,R}，(v i ,v j ) Represents the i-th intersection v i To the jth intersection v j Directed road segments in between, W represents the set of weights for road segments between intersections, w= { ω i,j |i,j＝1,2,3,…,R}，ω i,j Is a directed road section (v) i ,v j ) If the weight of the ith intersection v i To the jth intersection v j Between which there is a directed road section (v i ,v j ) Then the j-th intersection v j For the ith intersection v i An i-th intersection v i For the j-th intersection v j Precursor intersection of (a), and omega i,j Not less than 0; if the ith intersection v i To the jth intersection v j There is no directed road section (v) i ,v j ) Let omega i,j = + infinity; defining the current iteration number as t;
step 2: acquiring a source point intersection set:
selecting K source point intersections and forming a source point intersection set V '= { V' k I k=1, 2, …, K }, where v' k Representing the kth source point intersection in the source point intersection set V ', and enabling the kth source point intersection V' k Sequence number s in intersection set V k I.e.
Figure BDA0003361675280000081
Figure BDA0003361675280000082
Representing the s < th > in intersection set V k K is more than 1 and less than or equal to R, as shown by the V' set in the figure 4 and the intersections corresponding to the black italic fonts in the simple road network diagrams in the figures 4 to 7;
step 3: initializing the weights from the source point intersection to each intersection of the t-th iteration:
defining a kth source point intersection V ' from the source point intersection set V ' at the t-th iteration ' k To the ith intersection V in intersection set V i The upper bound of the weight of the shortest path of (a) is
Figure BDA0003361675280000083
And initialize +.>
Figure BDA0003361675280000084
Is +' infinity;
defining a kth source point intersection V 'from the set of source point intersections V' k To the ith intersection V in intersection set V i The shortest path of (a) is weighted as P k (v i ) And initialize P k (v i ) Is +' infinity;
initializing t=1;
initializing the kth source point intersection v 'under the t-th iteration' k To the s < th) k Each intersection
Figure BDA0003361675280000085
The upper bound of the weight of the shortest path of (2) is +.>
Figure BDA0003361675280000086
Then the kth source point intersection V ' in the source point intersection set V ' at the t-th iteration ' k To the s < th) k Each intersection
Figure BDA0003361675280000087
The shortest path of (a) is given the weight +.>
Figure BDA0003361675280000088
And will s k Crossing->
Figure BDA0003361675280000089
Added to the previous t-1 iterations at the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000810
With the kth source point intersection v 'in the t-1 th iteration' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000811
As S in FIG. 4 3 ,S 5 ,ΔS 3 ,ΔS 5 Shown;
definition L k,i Representing the kth source point intersection V 'in the set of source point intersections V' k To the ith intersection V in intersection set V i Intersection set of shortest paths of (a) and initializing
Figure BDA00033616752800000812
Step 4: for the k source point intersection v 'in the t-1 iteration' k Intersection set that gets the shortest path weight for source point
Figure BDA00033616752800000813
Forward searches are performed at all intersections in (a) and the shortest path weights are updated:
step 4.1: initializing k=1;
step 4.2: if the k-th source point intersection v 'is used in the previous t-1 iterations' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000091
All intersections in the intersection set V are included, and the kth source point intersection V 'is used in the source point intersection set V' k The shortest search for the source point is ended and goes to step 6, S as illustrated in FIG. 7 3 Stop pairs at source point intersection V =v' k Searching for the shortest path of the source point, otherwise, turning to step 4.3;
step 4.3: initializing the kth iteration with the kth source point intersection v' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000092
Note that intersection v 'at kth source point in the t-1 th iteration' k Intersection set that gets the shortest path weight for source point
Figure BDA0003361675280000093
The number of intersections in (a) is M t-1 The method comprises the steps of carrying out a first treatment on the surface of the Taking out the intersection with m < th > of the intersection>
Figure BDA0003361675280000094
Let mth intersection->
Figure BDA0003361675280000095
Sequence number +.>
Figure BDA0003361675280000096
I.e. < ->
Figure BDA0003361675280000097
Step 4.3.1: initializing m=1;
step 4.3.2: traversing the mth intersection
Figure BDA0003361675280000098
All adjacent intersections of (2) and selecting from them all intersections not belonging to the intersection set +.>
Figure BDA0003361675280000099
And add to the adjacency set +.>
Figure BDA00033616752800000910
Step 4.3.2.1: fetching contiguous sets
Figure BDA00033616752800000911
The q-th intersection of (a) is marked as the adjacency set +.>
Figure BDA00033616752800000912
In (q) th intersection->
Figure BDA00033616752800000913
The precursor intersection set of (1) is +.>
Figure BDA00033616752800000914
Let adjacency set +.>
Figure BDA00033616752800000915
In (q) th intersection->
Figure BDA00033616752800000916
Number r in intersection set V q I.e. +.>
Figure BDA00033616752800000917
Step 4.3.2.2: initializing q=1;
step 4.3.2.3: traversing precursor intersection sets
Figure BDA00033616752800000918
If all intersections in (a) are set of precursor intersections +.>
Figure BDA00033616752800000919
All intersections in (a) belong to the kth source point intersection v 'under the previous t-1 iterations' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800000920
Step 4.4 is entered, otherwise step 4.5 is entered, v as shown in fig. 5 1 Is adjacent to intersection v of 2 Precursor intersection v of (2) 1 ，v 5 All have obtained the shortest path, i.e.>
Figure BDA00033616752800000921
And v 1 Another adjacent intersection v of (a) 6 Precursor intersection v of (2) 8 The shortest path is not obtained, so v 1 Go to step 4.4, v 6 Go to step 4.5.
Step 4.4: acquiring v 'from the kth source point intersection' k To all precursor intersections are obtained as the kth source point intersection v' k The shortest-path weight of the intersection, i.e. the flow a in fig. 2, is the shortest-path weight of the source point, and the overall ideas and steps are as shown in fig. 3:
step 4.4.1: precursor intersection set
Figure BDA0003361675280000101
The number of intersections in the method is B, and the precursor intersection set is taken out
Figure BDA0003361675280000102
B crossing of (a)>
Figure BDA0003361675280000103
Let its serial number in intersection set V be n b I.e. +.>
Figure BDA0003361675280000104
Step 4.4.2: initializing b=1;
step 4.4.3: if it is
Figure BDA0003361675280000105
Will->
Figure BDA0003361675280000106
Assign to->
Figure BDA0003361675280000107
And record the precursor intersection->
Figure BDA0003361675280000108
Sequence number n of intersection set V b Assignment to the (r) th in intersection set V q Crossing->
Figure BDA0003361675280000109
Number n of temporary precursor intersection b,min The method comprises the steps of carrying out a first treatment on the surface of the Wherein (1)>
Figure BDA00033616752800001010
Represents the kth source point intersection v' k To the nth of intersection set V b The right of shortest path of the intersection, < ->
Figure BDA00033616752800001011
Represents the nth of intersection set V b Crossing->
Figure BDA00033616752800001012
To the r q Crossing->
Figure BDA00033616752800001013
Is used for the weight of the (c),
Figure BDA00033616752800001014
represents the kth source point intersection v' k To the (r) th intersection set V q Crossing->
Figure BDA00033616752800001015
The weight of the shortest path of (a);
step 4.4.4: assigning b+1 to B, proceeding to step 4.4.3 until b=b, thereby outputting the kth source point intersection v' k To the (r) th intersection set V q Each intersection
Figure BDA00033616752800001016
Right of shortest path->
Figure BDA00033616752800001017
Sequence number n b,min And is combined withWill be the r q Crossing->
Figure BDA00033616752800001018
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the nth of intersection set V b,min Crossing->
Figure BDA00033616752800001019
Intersection set of shortest path of +.>
Figure BDA00033616752800001020
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the (r) th intersection set V q Crossing->
Figure BDA00033616752800001021
Intersection set of shortest path of +.>
Figure BDA00033616752800001022
I.e.
Figure BDA00033616752800001023
Will be the r q Crossing->
Figure BDA00033616752800001024
Added to the intersection v 'at the kth source point' k Obtaining the set of shortest weights for the source point +.>
Figure BDA00033616752800001025
And the kth iteration with the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800001026
Go to step 4.6, v in FIG. 5 1 Is adjacent to intersection v of 2 For example, calculate the intersection v from the source point 3 Go through v respectively 2 Precursor intersection v of (2) 1 、v 5 Arrive v 2 Is the shortest path of (a)And selecting the smallest shortest path weight update as v 3 To v 2 The shortest path weight, P 3 (v 2 )＝min{P 3 (v 1 )+ω 1，2 ，P 3 (v 5 )+ω 5，2 } =15, and record that the precursor node is v at this time 5 I.e. lambda 3，2 ＝v 5 Will v 2 Added to L 3 (v 5 ) In (1) to obtain L 3 (v 2 )＝L 3 (v 5 )∪{v 2 }＝{v 3 ,v 4 ,v 5 ,v 2 Simultaneously v 2 Adding to set S 3 And->
Figure BDA00033616752800001027
In, S 3 ＝S 3 ∪{v 2 }，/>
Figure BDA00033616752800001028
Step 4.5: if it is
Figure BDA00033616752800001029
Will->
Figure BDA00033616752800001030
Assign to->
Figure BDA00033616752800001031
Otherwise go to step 4.6, v in FIG. 4 1 Is adjacent to intersection v 6 For example, P 3 (v 5 )+ω 5,6 ＝14＜T 3 3 (v 6 ) Thus updating T 3 3 (v 6 )＝P 3 (v 5 )+ω 5,6 =14, corresponding to scheme b in fig. 2;
step 4.6: if Q is not equal to Q, q+1 is assigned to Q, the step 4.3.2.3 is shifted, otherwise, m+1 is assigned to M, and M > M is judged t-1 If so, executing the step 4.7; otherwise, turning to step 4.3.2;
step 4.7: from the intersection at the t-th iterationThe intersection with the smallest upper bound of the weights of the shortest path in the set V
Figure BDA0003361675280000111
And->
Figure BDA0003361675280000112
Not belonging to the kth source point intersection v 'under the previous t iterations' k Intersection set with shortest path weight for source point +.>
Figure BDA0003361675280000113
I.e. < ->
Figure BDA0003361675280000114
If->
Figure BDA0003361675280000115
Then the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure BDA0003361675280000116
Upper bound of the weight of the shortest path of +.>
Figure BDA0003361675280000117
Assigned to the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure BDA0003361675280000118
Right of shortest path->
Figure BDA0003361675280000119
And will be the ith min Crossing->
Figure BDA00033616752800001110
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the intersection set V +.>
Figure BDA00033616752800001111
Crossing->
Figure BDA00033616752800001112
Is a shortest path intersection set +.>
Figure BDA00033616752800001113
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the ith of intersection set V min Crossing->
Figure BDA00033616752800001114
Is a shortest path intersection set +.>
Figure BDA00033616752800001115
I.e. < ->
Figure BDA00033616752800001116
And will be the ith min Each intersection
Figure BDA00033616752800001117
Add to the collection->
Figure BDA00033616752800001118
And intersection set->
Figure BDA00033616752800001119
Corresponding to the flow diagram in FIG. 2, according to FIG. 4 with respect to the source point v 3 The upper bound of the shortest path weight to each intersection is selected not to belong to S 3 Intersection v corresponding to the minimum upper limit value of (2) 6 ，
Figure BDA00033616752800001120
At the same time v 6 Adding to set S 3 And->
Figure BDA00033616752800001121
In, S 3 ＝S 3 ∪{v 6 }，/>
Figure BDA00033616752800001122
Obtain v 3 To v 6 Is the shortest path L of (1) 3 (v 6 )；
Step 5: backtracking the kth source point intersection v 'under the tth iteration' k To intersection set
Figure BDA00033616752800001123
The shortest path of the intersection in (a), namely the step idea as in the flow d in fig. 1:
step 5.1: note that the kth source point intersection v 'in the t-th iteration' k Intersection set that gets the shortest path weight for source point
Figure BDA00033616752800001124
The number of intersections in (a) is M t Taking out the intersection with m < th > of the intersection>
Figure BDA00033616752800001125
And the mth intersection +.>
Figure BDA00033616752800001126
Sequence number +.>
Figure BDA00033616752800001127
I.e. < ->
Figure BDA00033616752800001128
Step 5.2: initializing m=1;
step 5.3: record the k-th source point intersection v' k Into intersection set V
Figure BDA00033616752800001129
Crossing->
Figure BDA00033616752800001130
Is>
Figure BDA00033616752800001131
The number of the intermediate intersections in (a) is E m Marking the serial number rho of the epsilon-th intermediate intersection in the intersection set V ε ；
Step 5.3.1: initializing epsilon=1;
step 5.3.2: if it is
Figure BDA0003361675280000121
Then assume->
Figure BDA0003361675280000122
At the same time is the kth source point intersection V 'in the source point intersection set V' k′ I.e. +.>
Figure BDA0003361675280000123
Then take the shortest path point set +.>
Figure BDA0003361675280000124
In epsilon crossing->
Figure BDA0003361675280000125
To E th m +2 intersections->
Figure BDA0003361675280000126
All intersections as the kth 'source point intersection v' k′ To the intersection set V +.>
Figure BDA0003361675280000127
Crossing->
Figure BDA0003361675280000128
Is a shortest path intersection set +.>
Figure BDA0003361675280000129
And will->
Figure BDA00033616752800001210
Assign to->
Figure BDA00033616752800001211
If k' > k, and will be +.>
Figure BDA00033616752800001212
Crossing->
Figure BDA00033616752800001213
Added to the previous t-1 iterations to k 'th source point intersection v' k′ Set of weights to get shortest path for source point +.>
Figure BDA00033616752800001214
And the kth ' source point intersection v ' in the t-1 th iteration ' k′ Intersection set weighted for shortest path for source +.>
Figure BDA00033616752800001215
I.e.
Figure BDA00033616752800001216
Figure BDA00033616752800001217
Otherwise, go->
Figure BDA00033616752800001218
Crossing->
Figure BDA00033616752800001219
Add to the previous t iterations to kth 'source point intersection v' k′ Set of weights to get shortest path for source point +.>
Figure BDA00033616752800001220
And the kth iteration with the kth 'source point intersection v' k′ Intersection set with shortest path weight for source point +.>
Figure BDA00033616752800001221
Order of the game
Figure BDA00033616752800001222
With v in FIG. 5 2 For example, L 3 (v 2 )＝{v 3 ,v 4 ,v 5 ,v 2 The 2 nd intermediate intersection (excluding v) 3 And v 2 )v 5 Belongs to the source intersection set V' and +.>
Figure BDA00033616752800001223
Then v will be taken 5 To v 2 All intersections between are added to L 5 (v 2 ) In (1) to obtain L 5 (v 2 )＝{v 5 ,v 2 }，P 5 (v 2 )＝P 3 (v 2 )-P 3 (v 5 ) At the same time v 2 Added to v 5 Is a set S of source points 5 And->
Figure BDA00033616752800001224
In, S 5 ＝S 5 ∪{v 2 }，/>
Figure BDA00033616752800001225
I.e. as shown in fig. 6;
step 5.3.3: judgment of ε=E m If so, turning to step 5.4, otherwise, assigning epsilon+1 to epsilon, and turning to step 5.3.2;
step 5.4: judging that m=m t If so, turning to the step 6, otherwise, assigning m+1 to m, and turning to the step 5.3;
step 6: judging whether k=k is satisfied, if so, turning to step 7, otherwise, assigning k+1 to K, and turning to step 4.2;
step 7: if the shortest path searching with all source points as source points in the source point intersection set V 'under the previous t iterations is finished, the shortest path from each source point intersection in the source point intersection set V' to any intersection in the intersection set V is obtained, if the shortest path searching from all source point intersections to each intersection is completed under the 6 th iteration in fig. 7, the shortest path values (as in table 1) from all source point intersections to each intersection and the corresponding shortest paths (as in table 2) are output; otherwise, t+1 is assigned to t, and the process goes to step 4.
The shortest path values from each source point to each intersection point obtained in this embodiment are shown in table 1:
TABLE 1 shortest Path values from Source points to intersections of the invention
Figure BDA0003361675280000131
The shortest paths from each source point to each intersection point obtained in this embodiment are shown in table 2:
table 2 shortest path acquisition results of the present invention
L v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9
v 3 v 3 ，v 4 ，v 1 v 3 ，v 4 ，v 5 ，v 2 v 3 v 3 ，v 4 v 3 ，v 4 ，v 5 v 3 ，v 4 ，v 5 ，v 6 v 3 ，v 4 ，v 7 v 3 ，v 4 ，v 7 ，v 8 v 3 ，v 4 ，v5，v 6 ，v 9
v 5 v 5 ，v 4 ，v 1 v 5 ，v 2 v 5 ，v 4 ，v 3 v 5 ，v 4 v 5 v 5 ，v 6 v 5 ，v 4 ，v 7 v 5 ，v 6 ，v 9 ，v 8 v 5 ，v 6 ，v 9
As can be seen from tables 1 and 2, at this time, all the intersections of the source point have been searched to obtain the shortest paths to all the intersections in the road network, and the calculation is finished, so that the efficiency is obviously improved.

Claims (1)

1. A city road network multisource shortest path acquisition method based on shared calculation and dijkstra algorithm is characterized by comprising the following steps:
step 1: constructing an urban road network, defining parameters and initializing;
acquiring real-time road network data and obtaining an urban road network G= (V, A, W), wherein V represents an intersection set, and V= { V 1 ,v 2 ,v 3 ,…,v i ,…,v R }，v i Represents the i-th intersection, i=1, 2,3, …, R is the total number of intersections in the urban road network G, a represents the set of road segments between intersections, and a= { (v) i ,v j )|i,j＝1,2,3,…,R}，(v i ,v j ) Represents the i-th intersection v i To the jth intersection v j Directed road segments in between, W represents the set of weights for road segments between intersections, w= { ω i,j |i,j＝1,2,3,…,R}，ω i,j Is a directed road section (v) i ,v j ) If the weight of the ith intersection v i To the jth intersection v j Between which there is a directed road section (v i ,v j ) Then the j-th intersection v j For the ith intersection v i An i-th intersection v i For the j-th intersection v j Precursor intersection of (a), and omega i,j Not less than 0; if the ith intersection v i To the jth intersection v j There is no directed road section (v) i ,v j ) Let omega i,j = + infinity; defining the current iteration number as t;
step 2: acquiring a source point intersection set:
selecting K source point intersections and forming a source point intersection set V '= { V' k I k=1, 2, …, K }, where v' k Representing the kth source point intersection in the source point intersection set V ', and enabling the kth source point intersection V' k Sequence number s in intersection set V k I.e.
Figure FDA0003361675270000011
Figure FDA0003361675270000012
Representing the s < th > in intersection set V k K is more than 1 and less than or equal to R at each intersection;
step 3: initializing the weights from the source point intersection to each intersection of the t-th iteration:
defining a kth source point intersection V ' from the source point intersection set V ' at the t-th iteration ' k To the ith intersection V in intersection set V i The upper bound of the weight of the shortest path of (a) is
Figure FDA0003361675270000013
And initialize +.>
Figure FDA0003361675270000014
Is +' infinity;
defining a kth source point intersection V 'from the set of source point intersections V' k To the ith intersection V in intersection set V i The shortest path of (a) is weighted as P k (v i ) And initialize P k (v i ) Is +' infinity;
initializing t=1;
initializing the kth source point intersection v 'under the t-th iteration' k To the s < th) k Each intersection
Figure FDA0003361675270000015
The upper bound of the weight of the shortest path of (a) is
Figure FDA0003361675270000016
Then the kth source point intersection V ' in the source point intersection set V ' at the t-th iteration ' k To the s < th) k Crossing->
Figure FDA0003361675270000017
Is the shortest of (a)The right of the diameter is->
Figure FDA0003361675270000018
And will s k Crossing->
Figure FDA0003361675270000019
Added to the previous t-1 iterations at the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure FDA0003361675270000021
With the kth source point intersection v 'in the t-1 th iteration' k Intersection set with shortest path weight for source point +.>
Figure FDA0003361675270000022
Definition L k,i Representing the kth source point intersection V 'in the set of source point intersections V' k To the ith intersection V in intersection set V i Intersection set of shortest paths of (a) and initializing
Figure FDA0003361675270000023
Step 4: for the k source point intersection v 'in the t-1 iteration' k Intersection set that gets the shortest path weight for source point
Figure FDA0003361675270000024
Forward searches are performed at all intersections in (a) and the shortest path weights are updated:
step 4.1: initializing k=1;
step 4.2: if the k-th source point intersection v 'is used in the previous t-1 iterations' k Intersection set that gets the shortest path weight for source point
Figure FDA0003361675270000025
All intersections in intersection set V are included, then pair sourceWith the kth source point intersection V 'in the set of point intersections V' k Ending the shortest searching for the source point, and turning to the step 6, otherwise turning to the step 4.3;
step 4.3: initializing the kth iteration with the kth source point intersection v' k Intersection set that gets the shortest path weight for source point
Figure FDA0003361675270000026
Note that intersection v 'at kth source point in the t-1 th iteration' k Intersection set that gets the shortest path weight for source point
Figure FDA0003361675270000027
The number of intersections in (a) is M t-1 The method comprises the steps of carrying out a first treatment on the surface of the Taking out the intersection with m < th > of the intersection>
Figure FDA0003361675270000028
Let the mth intersection +.>
Figure FDA0003361675270000029
Sequence number +.>
Figure FDA00033616752700000210
I.e. < ->
Figure FDA00033616752700000211
Step 4.3.1: initializing m=1;
step 4.3.2: traversing the mth intersection
Figure FDA00033616752700000212
All adjacent intersections of (2) and selecting from them all intersections not belonging to the intersection set +.>
Figure FDA00033616752700000213
And add to the adjacency set +.>
Figure FDA00033616752700000214
Step 4.3.2.1: fetching contiguous sets
Figure FDA00033616752700000215
The q-th intersection of (a) is marked as the adjacency set +.>
Figure FDA00033616752700000216
In (q) th intersection->
Figure FDA00033616752700000217
The precursor intersection set of (1) is +.>
Figure FDA00033616752700000218
Let adjacency set +.>
Figure FDA00033616752700000219
In (q) th intersection->
Figure FDA00033616752700000220
Number r in intersection set V q I.e. +.>
Figure FDA00033616752700000221
Step 4.3.2.2: initializing q=1;
step 4.3.2.3: traversing precursor intersection sets
Figure FDA00033616752700000222
If all intersections in (a) are precursor intersections set
Figure FDA0003361675270000031
All intersections in (a) belong to the kth source point intersection v 'under the previous t-1 iterations' k Intersection set with shortest path weight for source point +.>
Figure FDA0003361675270000032
Step 4.4 is carried out, otherwise step 4.5 is carried out;
step 4.4: acquiring v 'from the kth source point intersection' k To all precursor intersections are obtained as the kth source point intersection v' k The shortest weight of the intersection, which is the weight of the shortest path of the source point:
step 4.4.1: precursor intersection set
Figure FDA0003361675270000033
The number of intersections in the method is B, and the precursor intersection set is taken out
Figure FDA0003361675270000034
B crossing of (a)>
Figure FDA0003361675270000035
Let its serial number in intersection set V be n b I.e. +.>
Figure FDA0003361675270000036
Step 4.4.2: initializing b=1;
step 4.4.3: if it is
Figure FDA0003361675270000037
Will->
Figure FDA0003361675270000038
Assign to->
Figure FDA0003361675270000039
And record the precursor intersection->
Figure FDA00033616752700000310
Sequence number n of intersection set V b Assignment to the (r) th in intersection set V q Crossing->
Figure FDA00033616752700000311
Number n of temporary precursor intersection b,min The method comprises the steps of carrying out a first treatment on the surface of the Wherein (1)>
Figure FDA00033616752700000312
Represents the kth source point intersection v' k To the nth of intersection set V b The right of shortest path of the intersection, < ->
Figure FDA00033616752700000313
Represents the nth of intersection set V b Crossing->
Figure FDA00033616752700000314
To the r q Crossing->
Figure FDA00033616752700000315
Weight of->
Figure FDA00033616752700000316
Represents the kth source point intersection v' k To the (r) th intersection set V q Crossing->
Figure FDA00033616752700000317
The weight of the shortest path of (a);
step 4.4.4: assigning b+1 to B, proceeding to step 4.4.3 until b=b, thereby outputting the kth source point intersection v' k To the (r) th intersection set V q Each intersection
Figure FDA00033616752700000318
Right of shortest path->
Figure FDA00033616752700000319
Sequence number n b,min And will be the r q Each crossMouth->
Figure FDA00033616752700000320
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the nth of intersection set V b,min Crossing->
Figure FDA00033616752700000321
Intersection set of shortest path of +.>
Figure FDA00033616752700000322
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the (r) th intersection set V q Crossing->
Figure FDA00033616752700000323
Intersection set of shortest path of +.>
Figure FDA00033616752700000324
I.e.
Figure FDA00033616752700000325
Will be the r q Crossing->
Figure FDA00033616752700000326
Added to the intersection v 'at the kth source point' k Obtaining the set of shortest weights for the source point +.>
Figure FDA00033616752700000327
And the kth iteration with the kth source point intersection v' k Intersection set with shortest path weight for source point +.>
Figure FDA00033616752700000328
I.e. < ->
Figure FDA00033616752700000329
Figure FDA00033616752700000330
Turning to step 4.6;
step 4.5: if it is
Figure FDA00033616752700000331
Will->
Figure FDA00033616752700000332
Assign to->
Figure FDA00033616752700000333
Otherwise, turning to step 4.6;
step 4.6: if Q is not equal to Q, q+1 is assigned to Q, the step 4.3.2.3 is shifted, otherwise, m+1 is assigned to M, and M > M is judged t -1 If so, executing the step 4.7; otherwise, turning to step 4.3.2;
step 4.7: selecting an intersection with the smallest upper bound of the weight of the shortest path from the intersection set V at the t-th iteration
Figure FDA0003361675270000041
And->
Figure FDA0003361675270000042
Not belonging to the kth source point intersection v 'under the previous t iterations' k Intersection set with shortest path weight for source point +.>
Figure FDA0003361675270000043
I.e. < ->
Figure FDA0003361675270000044
If->
Figure FDA0003361675270000045
Then the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure FDA0003361675270000046
Upper bound of the weight of the shortest path of +.>
Figure FDA0003361675270000047
Assigned to the kth source point intersection v' k To the ith of intersection set V min Crossing->
Figure FDA0003361675270000048
Right of shortest path->
Figure FDA0003361675270000049
And will be the ith min Crossing->
Figure FDA00033616752700000410
Adding to the k-th source point intersection V 'in the source point intersection set V' k To the intersection set V +.>
Figure FDA00033616752700000411
Crossing->
Figure FDA00033616752700000412
Is a shortest path intersection set +.>
Figure FDA00033616752700000413
Thereby obtaining the kth source point intersection V 'in the source point intersection set V' k To the ith of intersection set V min Crossing->
Figure FDA00033616752700000414
Is a shortest path intersection set +.>
Figure FDA00033616752700000415
I.e. < ->
Figure FDA00033616752700000416
And will be the ith min Crossing->
Figure FDA00033616752700000417
Add to the collection->
Figure FDA00033616752700000418
And intersection set->
Figure FDA00033616752700000419
Step 5: backtracking the kth source point intersection v 'under the tth iteration' k To intersection set
Figure FDA00033616752700000420
The shortest path of the intersection of (a):
step 5.1: note that the kth source point intersection v 'in the t-th iteration' k Intersection set that gets the shortest path weight for source point
Figure FDA00033616752700000421
The number of intersections in (a) is M t Taking out the intersection with m < th > of the intersection>
Figure FDA00033616752700000422
And the mth intersection +.>
Figure FDA00033616752700000423
Sequence number +.>
Figure FDA00033616752700000424
I.e. < ->
Figure FDA00033616752700000425
Step 5.2: initializing m=1;
step 5.3: record the k-th source point intersection v' k Into intersection set V
Figure FDA00033616752700000426
Crossing->
Figure FDA00033616752700000427
Is set of shortest path points
Figure FDA00033616752700000428
The number of the intermediate intersections in (a) is E m Marking the serial number rho of the epsilon-th intermediate intersection in the intersection set V ε ；
Step 5.3.1: initializing epsilon=1;
step 5.3.2: if it is
Figure FDA00033616752700000429
Then assume->
Figure FDA00033616752700000430
At the same time is the kth source point intersection V 'in the source point intersection set V' k′ I.e. +.>
Figure FDA00033616752700000431
Then take the shortest path point set +.>
Figure FDA00033616752700000432
In epsilon crossing->
Figure FDA00033616752700000433
To E th m +2 intersections->
Figure FDA00033616752700000434
All intersections as the kth 'source point intersection v' k′ To the intersection set V +.>
Figure FDA00033616752700000435
Crossing->
Figure FDA00033616752700000436
Is a shortest path intersection set +.>
Figure FDA00033616752700000437
And will->
Figure FDA00033616752700000438
Assign to->
Figure FDA00033616752700000439
If k' > k, and will be +.>
Figure FDA00033616752700000440
Crossing->
Figure FDA00033616752700000441
Added to the previous t-1 iterations to k 'th source point intersection v' k′ Set of weights to get shortest path for source point +.>
Figure FDA00033616752700000442
And the kth ' source point intersection v ' in the t-1 th iteration ' k′ Intersection set weighted for shortest path for source +.>
Figure FDA00033616752700000443
Otherwise, go->
Figure FDA00033616752700000444
Crossing->
Figure FDA00033616752700000445
Add to the previous t iterations to kth 'source point intersection v' k′ Obtaining the shortest for the source pointSet of weights for diameter->
Figure FDA0003361675270000051
And the kth iteration with the kth 'source point intersection v' k′ Intersection set with shortest path weight for source point +.>
Figure FDA0003361675270000052
Step 5.3.3: judgment of ε=E m If so, turning to step 5.4, otherwise, assigning epsilon+1 to epsilon, and turning to step 5.3.2;
step 5.4: judging that m=m t If so, turning to the step 6, otherwise, assigning m+1 to m, and turning to the step 5.3;
step 6: judging whether k=k is satisfied, if so, turning to step 7, otherwise, assigning k+1 to K, and turning to step 4.2;
step 7: and if the shortest searching with all the source points as the source points in the source point intersection set V 'under the previous t iterations is finished, the shortest path from each source point intersection in the source point intersection set V' to any intersection in the intersection set V is obtained, otherwise, t+1 is assigned to t, and the step 4 is carried out.
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