CN108924053B - Method for quickly searching multiple partially disjoint shortest paths for multipath routing - Google Patents

Method for quickly searching multiple partially disjoint shortest paths for multipath routing Download PDF

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CN108924053B
CN108924053B CN201810841121.7A CN201810841121A CN108924053B CN 108924053 B CN108924053 B CN 108924053B CN 201810841121 A CN201810841121 A CN 201810841121A CN 108924053 B CN108924053 B CN 108924053B
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path
shortest
paths
edge
point
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CN108924053A (en
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郭龙坤
邓芸芸
黄培煌
郝震东
陈建利
杨旸
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Fuzhou University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L45/00—Routing or path finding of packets in data switching networks
    • H04L45/12—Shortest path evaluation
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • H04L41/145—Network analysis or design involving simulating, designing, planning or modelling of a network
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L45/00—Routing or path finding of packets in data switching networks
    • H04L45/12—Shortest path evaluation
    • H04L45/128—Shortest path evaluation for finding disjoint paths

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Abstract

The inventionA quick searching method for multiple partially disjoint shortest paths of multi-path routing is disclosed, wherein a directed network is represented as a directed graph model; obtaining a shortest from the directed graph model
Figure DEST_PATH_IMAGE002
Route of travel
Figure DEST_PATH_IMAGE004
And make an order
Figure DEST_PATH_IMAGE006
(ii) a According to
Figure DEST_PATH_IMAGE008
All paths in (2) and establishing corresponding directed graphs
Figure DEST_PATH_IMAGE010
A traditional remainder of
Figure DEST_PATH_IMAGE012
(ii) a Based on traditional complementary graph
Figure 573284DEST_PATH_IMAGE012
Structural point decomposition residual graph
Figure DEST_PATH_IMAGE014
(ii) a Decomposing the residual graph from points
Figure 428108DEST_PATH_IMAGE014
Get a shortest one
Figure 667501DEST_PATH_IMAGE002
Route of travel
Figure DEST_PATH_IMAGE016
Along this path
Figure 393DEST_PATH_IMAGE016
To the path
Figure 752448DEST_PATH_IMAGE004
Carrying out augmentation; and decomposing to obtain a spiral optimal path. The method for quickly searching the plurality of partially disjoint shortest paths for the multi-path routing improves the efficiency and feasibility of searching the disjoint shortest paths in the network and can quickly find the partially disjoint shortest paths.

Description

Method for quickly searching multiple partially disjoint shortest paths for multipath routing
Technical Field
The invention relates to the field of network optimization, in particular to a method for quickly searching a plurality of partially disjoint shortest paths for multi-path routing.
Background
Network congestion is an inexperienced illness of a data transmission network, because a traditional data transmission network mainly uses a data transmission method based on a single shortest path, and because the traditional data transmission network selects an optimal link (with the maximum bandwidth/the lowest delay, and the like) for data transmission, the pressure of data transmission is more easily concentrated on those links and nodes with good performance, so that congestion is generated. Disjoint path routing is considered as a routing scheme that can completely address network congestion, and has better fault tolerance and network load balancing, but its requirement that nodes or links are completely disjoint is too strict and requires too many resources.
Disclosure of Invention
The invention aims to provide a method for quickly finding a plurality of partially disjoint shortest paths for multipath routing, so as to overcome the defects in the prior art.
In order to achieve the purpose, the technical scheme of the invention is as follows: a method for quickly finding a plurality of partially disjoint shortest paths for multi-path routing is implemented according to the following steps:
step S1: representing the directed network as a directed graph model G ═ V, E;
step S2: obtaining a shortest s-t path P from the directed graph model G*And make Ω ═ P*Omega represents a set of shortest s-t paths in the directed graph model G;
step S3: establishing a traditional residual graph corresponding to the directed graph G according to all paths in the omega
Figure BDA0001745225950000011
Step S4: based on the conventional residual graph
Figure BDA0001745225950000012
Structural point decomposition residual graph
Figure BDA0001745225950000013
Step S5: decomposing the residual map from the points
Figure BDA0001745225950000014
To obtain a shortest s-t path Q*Along this path Q*For path P*Performing augmentation (the augmentation process is described in detail in S5);
step S6: decomposing to obtain a spiral optimal path;
step S7: returning to said step S3 until all shortest S-t paths P*And finishing after the treatment is finished.
In one embodiment of the present invention, in the step S1,
and representing the directed network as the directed graph model G (V, E), wherein V is a vertex in the directed graph, E is an edge in the directed graph, n is | V | represents the number of the vertex in the directed graph G, m is | E | represents the number of the edge in the directed graph G, a source point s and a destination node t are determined, and a weight function w (E) is defined, a cost function c (E) of the shared points, an upper bound delta of the shared points, and a path P (u, V) from the node u to the node V.
The goal of the method of finding partially disjoint shortest paths is to obtain a set of paths such that the total weight of the paths is minimal and the following 3 constraints are satisfied: 1) except for the source node and the destination node, the out degree of each node in the graph is equal to the in degree; 2) the number of sharing points of all paths is at most delta; 3) the variables 0 to 1: if the edge is included in the path, 1 is selected; otherwise, 0 is taken; i.e. to optimize the following mathematical model:
Figure BDA0001745225950000021
Figure BDA0001745225950000022
Figure BDA0001745225950000023
xe∈{0，1}e∈E
wherein the variables in the above formula: x is the number ofeRepresents a non-empty proper subset of the set {0,1}, w (e) represents the weight of the edge e, k represents the number of disjoint paths, s represents the shortest path P*T represents the shortest path P*End point of, delta+(v) Representing the set of edges, δ, from point v-(v) The set of edges representing entry point v, c (e) cost of edge e,
Figure BDA0001745225950000024
a point decomposition residual graph is shown.
In one embodiment of the present invention, in the step S2, a shortest S-t path P is found from the graph G by dijkstra method*And make Ω ═ P*Where Ω represents the set of shortest s-t paths in diagram G.
In an embodiment of the invention, in the step S3, the traditional remaining graph
Figure BDA0001745225950000025
Wherein the content of the first and second substances,
Figure BDA0001745225950000026
is formed by P*And the value of the weight is the inverse of the primary side, e ═ u, v ∈ P for each side*E 'are'Adding (v, u)
Figure BDA0001745225950000027
And the weight is w (e') -w (e).
In an embodiment of the present invention, in the step S4, the method further includes the following steps:
step S41: route P*Each internal point v, except the source point s and the destination point t, is decomposed into a point v1And v2；
Step S42: in the decomposition of the residual diagram
Figure BDA0001745225950000028
Two types of edges are added: edge e (v) with weight of 0 and cost of 11，v2) And a side e (v) with weight of 0 and cost of 02，v1)；
Step S43: replacing at least one point in the original edge
Figure BDA0001745225950000031
The edge of (2):
1) if it is
Figure BDA0001745225950000032
Then replace it with e (v)1，u2)；
2) If it is
Figure BDA0001745225950000033
And is
Figure BDA0001745225950000034
Then replace with e (x, u)1)；
3) If it is
Figure BDA0001745225950000035
And is
Figure BDA0001745225950000036
Then replace it with e (v)2，y)。
Also, the weights of all the replaced edges are equal to the weight of the primary edge, at a cost of 0.
That is, the shortest path P in the conventional remainder diagram*The points other than the starting point are divided into two points, i.e. assuming that the point u is at P*And u is not the starting point, divide u into u1And u2Adding two edges e (v) simultaneously1，v2) And e (v)2，v1)：
1) Each edge in graph G is further determined by whether it is in the shortest path P*The operation is as follows: if edge e (u, v) is in shortest path P*The method comprises the following steps: then the remaining map is decomposed at the points
Figure BDA0001745225950000037
Adding a new edge e (v)1，u2) Let the overhead c (e (v)) of this edge at the same time1，u2)0, whose weight is the weight of the negative edge e (u, v);
2) if edge e (x, u) is not in shortest path P*And if so, continuing to judge as follows: if point u is on the shortest path P*Except for the starting point s, t: then the remaining map is decomposed at the points
Figure BDA0001745225950000038
Adding a new edge e (x, u)1) Let the overhead c (e (x, u) of this edge at the same time1) 0) whose weight is the weight of the edge e (x, u);
3) if edge e (v, y) is not in shortest path P*And if so, continuing to judge as follows: if point v is on the shortest path P*Except for the starting point s, t: then the remaining map is decomposed at the points
Figure BDA0001745225950000039
Adding a new edge e (v)2Y), while letting the overhead c (e (v) of this edge be2Y)) is 0 and its weight is the weight of the edge e (v, y).
In an embodiment of the present invention, in the step S5, the decomposition residual map is extracted
Figure BDA00017452259500000310
By using Dijkstra methodObtaining a shortest s-t path, i.e. from said decomposed remainder
Figure BDA00017452259500000311
Obtaining a shortest s-t path with the cost not more than delta, marking the problem as a BRSP problem, obtaining the optimal solution of the BRSP problem in O (delta | E |) time by using a dynamic programming method, namely obtaining a shortest s-t path Q with the cost not more than delta, simultaneously obtaining the next shortest s-t path, and enabling P to be P*Let i ═ i +1 and Ω ═ P1 *，...，Pi *}。
In an embodiment of the present invention, in the step S6, note P*UQ is the optimal solution, representing path P*And all edges in path Q remove the set of edge pairs between the same vertices but in opposite directions; handle P*Decomposition of UQ into P1And P2And obtaining a spiral optimal path by two paths.
Compared with the prior art, the invention has the following beneficial effects: a method for quickly searching multiple partially disjoint shortest paths for multi-path routing converts the calculation of partially disjoint paths into the calculation of limited shortest paths by a new construction method of a graph structure (namely a point decomposition residual graph), and finds k-edge disjoint shortest paths allowing a limited number of sharing points by using an augmented path method, thereby balancing the degree of disjointness of transmission paths and occupied network resources. Meanwhile, computer simulation experiments are carried out on the proposed construction scheme of the partial disjoint shortest paths through related data on the virtual network and the real network, and the experiment results show that the method is more efficient, and the running time is reduced by three orders of magnitude. The efficiency and the feasibility of searching the disjointed shortest path in the network are improved, and partial disjointed shortest paths can be quickly found.
Drawings
Fig. 1 is a flow chart of a method for fast finding a plurality of partially disjoint shortest paths for multipath routing according to the present invention.
FIG. 2 is an original diagram for constructing a point decomposition residual diagram according to an embodiment of the present invention.
Fig. 3 is a residual graph of the original graph for constructing the point decomposition residual graph corresponding to the shortest path according to an embodiment of the present invention.
FIG. 4 is a point decomposition residual map for constructing a point decomposition residual map according to an embodiment of the present invention.
Detailed Description
The technical scheme of the invention is specifically explained below with reference to the accompanying drawings.
The invention relates to a method for quickly finding a plurality of partially disjoint shortest paths for multi-path routing, which finds k-edge disjoint shortest paths allowing a limited number of sharing points, thereby balancing the degree of the disjointness of transmission paths and occupied network resources. And (3) converting the calculation of the partial non-intersecting path into the calculation of the limited shortest path by constructing a point decomposition residual graph, so that the partial non-intersecting path can be searched by using a designed augmented path method, and a scheme for calculating the optimal solution of the spiral is obtained. The difficulty is mainly how to construct the point decomposition residual graph
Figure BDA0001745225950000041
Converting the computation of the partially disjoint path into a computation of a constrained shortest path. Thus, the device is provided with
Figure BDA0001745225950000042
Decomposed { P1，P2Is a spiral optimal solution for a δ V-2EDSP, where Q is the shortest s-t path bounded, in a point-decomposed residual graph
Figure BDA0001745225950000043
The upper cost bound in (1) is δ, i.e., there are δ sharing points at most. As used herein
Figure BDA0001745225950000044
It represents a path P*And all edges in path Q remove the set of edge pairs between the same vertices but in opposite directions. If delta V-kEDSP is found, the above point decomposition residual map-based augmentation path scheme is iteratively applied on delta V- (k-1) EDSP.
Further, in this embodiment, the specific steps are as follows:
step S1: representing the directed network as a directed graph G ═ V, E;
step S2: finding a shortest s-t path P from graph G*；
Step S3: establishing a traditional remainder of the corresponding graph G according to all paths in omega
Figure BDA0001745225950000051
Step S4: based on traditional complementary graph
Figure BDA0001745225950000052
Structural point decomposition residual graph
Figure BDA0001745225950000053
Step S5: from
Figure BDA0001745225950000054
Finds a shortest s-t path Q*Along this path to P*Performing augmentation (the augmentation process is described in detail in S5);
step S6: decomposing to obtain the optimal spiral path;
step S7: all shortest s-t paths P are judged*If the processing is finished, ending the processing, and outputting a plurality of partially disjoint shortest paths; otherwise, returning to the step S3
Further, referring to fig. 1, fig. 1 is a flow chart of a fast scheme for finding partially disjoint shortest paths that may be used for multi-path routing. The mathematical model of the method is described as follows:
in this embodiment, in step S1, a directed network is represented as a directed graph model G ═ V, E, where V is a vertex in the directed graph, E is an edge in the directed graph, n ═ V | represents the number of vertices in the directed graph G, and m ═ E | represents the number of pieces of the edge in the directed graph G, a source point S and a destination node t are determined, a weight function w (E) is defined, a cost function c (E) for sharing points, an upper bound δ for sharing points, and a path P (u, V) from the node u to the node V.
In this embodiment, the objective of the method for finding partially disjoint shortest paths is to obtain a set of paths, so that the total weight of the paths is the minimum, and the following 3 constraints are satisfied: 1) except for the source node and the destination node, the out degree of each node in the graph is equal to the in degree; 2) the number of sharing points of all paths is at most delta; 3) the variables 0 to 1: if the edge is included in the path, 1 is selected; otherwise, 0 is taken; i.e. to optimize the following mathematical model:
Figure BDA0001745225950000061
Figure BDA0001745225950000062
Figure BDA0001745225950000063
xe∈{0,1} e∈E
wherein x iseRepresents a non-empty proper subset of the set {0,1}, w (e) represents the weight of the edge e, k represents the number of disjoint paths, s represents the shortest path P*T represents the shortest path P*End point of, delta+(v) Representing the set of edges, δ, from point v-(v) The set of edges representing entry point v, c (e) cost of edge e,
Figure BDA0001745225950000064
a point decomposition residual graph is shown.
In this embodiment, in step S2, a shortest S-t path P is found from graph G using Dijkstra' S method*And make Ω ═ P*Where Ω represents the set of shortest s-t paths in diagram G.
In the present embodiment, in step S3, the shortest S-t path P found in the previous step*Creating a conventional remainder of the map G
Figure BDA0001745225950000065
Traditional complementary graph
Figure BDA0001745225950000066
Wherein
Figure BDA0001745225950000067
Is formed by P*And the weight is the inverse of the primary side, i.e. for each edge e ═ P ∈ v*Will be
Figure BDA0001745225950000068
And the weight is w (e') -w (e).
In the present embodiment, in step S4, based on the conventional remainder map
Figure BDA0001745225950000069
Structural point decomposition residual graph
Figure BDA00017452259500000610
Firstly, handle P*Each inner point (except s and t) v is decomposed into points v1And v2In a
Figure BDA00017452259500000611
Two types of edges are added: edge e (v) with weight of 0 and cost of 11,v2) And a side e (v) with weight of 0 and cost of 02,v1). Next, at least one point in the original image is replaced
Figure BDA00017452259500000612
The edge of (2):
1) if it is
Figure BDA00017452259500000613
Then replace it with e (v)1,u2)；
2) If it is
Figure BDA00017452259500000614
And is
Figure BDA00017452259500000615
Then replace with e (x, u)1)；
3) If it is
Figure BDA00017452259500000616
And is
Figure BDA00017452259500000617
Then replace it with e (v)2,y)。
Also, the weights of all the replaced edges are equal to the weight of the primary edge, at a cost of 0.
That is, the shortest path P in the conventional remainder diagram*The points other than the starting point are divided into two points, i.e. assuming that the point u is at P*And u is not the starting point, divide u into u1And u2Adding two edges e (v) simultaneously1,v2) And e (v)2，v1) Respectively correspond to the following:
1) each edge in graph G is further determined by whether it is in the shortest path P*The operation is as follows: if edge e (u, v) is in shortest path P*The method comprises the following steps: then the remaining map is decomposed at the points
Figure BDA0001745225950000071
Adding a new edge e (v)1，u2) Let the overhead c (e (v)) of this edge at the same time1，u2)0, whose weight is the weight of the negative edge e (u, v);
2) if edge e (x, u) is not in shortest path P*And if so, continuing to judge as follows: if point u is on the shortest path P*Except for the starting point s, t: then the remaining map is decomposed at the points
Figure BDA0001745225950000072
Adding a new edge e (x, u)1) Let the overhead c (e (x, u) of this edge at the same time1) 0) whose weight is the weight of the edge e (x, u);
3) if edge e (v, y) is not in shortest path P*And if so, continuing to judge as follows: if point v is on the shortest path P*Except for the starting point s, t: then the point decomposition residueDrawing (A)
Figure BDA0001745225950000073
Adding a new edge e (v)2Y), while letting the overhead c (e (v) of this edge be2Y)) is 0 and its weight is the weight of the edge e (v, y).
In the present embodiment, in step S5, the slave
Figure BDA0001745225950000074
Using Dijkstra method to find a shortest s-t path, then using P pair along said shortest path*Augmentation was carried out as follows: from the configuration of step S3, a point decomposition residual map can be obtained, and solving the δ V-2EDSP problem can be converted into a point decomposition residual map
Figure BDA0001745225950000075
Find a shortest s-t path that costs no more than δ. The above problem is defined as BRSP problem, which is a special case of RSP problem, and the optimal solution of BRSP problem can be obtained in O (delta | E |) time by using dynamic programming method, i.e. a shortest st path Q with cost not more than delta, and P is made*Adding 1 to the subscript of (a), obtaining the next shortest s-t path, i is equal to i +1, and making
Figure BDA0001745225950000076
In the present embodiment, in step S6, P*UQ is just one optimal solution to the delta V-2EDSP problem, which represents path P*And all edges in path Q have the opposite set of edges removed. Finally, the P is put*Decomposition of UQ into P1And P2Two paths, namely a spiral optimal path, are the optimal solutions of the original problem.
In the present embodiment, in step S7, if an optimal solution to the δ V-kdepsp problem is found, the above-described augmented path scheme based on the point decomposition residual map, i.e., the process of step S3 to step S5, is iteratively applied to the δ V- (k-1) EDSP.
In order to further understand the technical solutions proposed by the present invention, the following is provided in conjunction with fig. 2 to 4Point decomposition residual graph of
Figure BDA0001745225950000081
Specific examples of the construction method are illustrated.
In this embodiment, first, a shortest s-t path P is found from the original G*Building a traditional remainder chart
Figure BDA0001745225950000082
Wherein
Figure BDA0001745225950000083
Is formed by P*And the weight is the inverse of the primary side, i.e. for each edge e ═ P ∈ v*Adding e ═ v, u
Figure BDA0001745225950000084
And the weight is w (e') -w (e).
Secondly, based on the traditional remaining graph
Figure BDA0001745225950000085
Structural point decomposition residual graph
Figure BDA0001745225950000086
Firstly, handle P*Each inner point (except s and t) v is decomposed into points v1And v2In a
Figure BDA0001745225950000087
Two types of edges are added: edge e (v) with weight of 0 and cost of 11,v2) And a side e (v) with weight of 0 and cost of 02,v1)。
Finally, at least one point in the original image is replaced
Figure BDA0001745225950000088
The edge of (2):
1) if it is
Figure BDA0001745225950000089
Then replace it with e (v)1,u2)；
2) If it is
Figure BDA00017452259500000810
And is
Figure BDA00017452259500000811
Then replace with e (x, u)1)；
3) If it is
Figure BDA00017452259500000812
And is
Figure BDA00017452259500000813
Then replace it with e (v)2Y). Also, the weights of all the replaced edges are equal to the weight of the primary edge, at a cost of 0.
The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.

Claims (4)

1. A method for quickly finding a plurality of partially disjoint shortest paths for multi-path routing is implemented according to the following steps:
step S1: representing the directed network as a directed graph model G ═ V, E;
step S2: obtaining a shortest s-t path P from the directed graph model G*And make Ω ═ P*Omega represents a set of shortest s-t paths in the directed graph model G;
step S3: establishing a traditional residual graph corresponding to the directed graph G according to all paths in the omega
Figure FDA0002792865180000011
Step S4: based on the conventional residual graph
Figure FDA0002792865180000012
Structural point decompositionRemainder drawing
Figure FDA0002792865180000013
Step S5: decomposing the residual map from the points
Figure FDA0002792865180000014
To obtain a shortest s-t path Q*Along this path Q*For path P*Carrying out augmentation;
step S6: decomposing to obtain a spiral optimal path;
step S7: all shortest s-t paths P are judged*If the processing is finished, ending the processing, and outputting a plurality of partially disjoint shortest paths; otherwise, returning to the step S3;
in the step S1, in the above step,
representing the directed network as the directed graph model G ═ V, E, where V is a vertex in the directed graph, E is an edge in the directed graph, n ═ V | is the number of vertices in the directed graph G, m ═ E | is the number of edges in the directed graph G, a source point is s, a destination node is t, a weight function is w (E), a cost function of the number of shared points is c (E), an upper bound of the number of shared points is δ, and P (u, V) is a path from the node u to the node V;
the goal of the method of finding partially disjoint shortest paths is to obtain a set of paths such that the total weight of the paths is minimal and the following 3 constraints are satisfied: 1) except for the source node and the destination node, the out degree of each node in the graph is equal to the in degree; 2) the number of sharing points of all paths is at most delta; 3) the variables 0 to 1: if the edge is included in the path, 1 is selected; otherwise, 0 is taken; i.e. to optimize the following mathematical model:
Figure FDA0002792865180000021
Figure FDA0002792865180000022
Figure FDA0002792865180000023
xe∈{0,1} e∈E
wherein x iseRepresents a non-empty proper subset of the set {0,1}, w (e) represents the weight of the edge e, k represents the number of disjoint paths, s represents the shortest path P*T represents the shortest path P*End point of, delta+(v) Representing the set of edges, δ, from point v-(v) The set of edges representing entry point v, c (e) cost of edge e,
Figure FDA0002792865180000024
representing a point decomposition residual graph;
in the step S3, the conventional remainder
Figure FDA0002792865180000025
Wherein the content of the first and second substances,
Figure FDA0002792865180000026
is formed by P*And the value of the weight is the inverse of the primary side, e ═ u, v ∈ P for each side*Adding e ═ v, u
Figure FDA0002792865180000027
And the weight is w (e') -w (e);
in step S4, the method further includes the steps of:
step S41: route P*Each internal point v, except the source point s and the destination point t, is decomposed into a point v1And v2；
Step S42: in the decomposition of the residual diagram
Figure FDA0002792865180000028
Two types of edges are added: edge e (v) with weight of 0 and cost of 11,v2) And an edge e (with a weight of 0 and a cost of 0v2,v1)；
Step S43: replacing at least one point in the original edge
Figure FDA0002792865180000029
The edge of (2):
1) if it is
Figure FDA00027928651800000210
Then replace it with e (v)1,u2)；
2) If it is
Figure FDA00027928651800000211
And is
Figure FDA00027928651800000212
Then replace with e (x, u)1)；
3) If it is
Figure FDA00027928651800000213
And is
Figure FDA00027928651800000214
Then replace it with e (v)2,y)；
Also, the weights of all the replaced edges are equal to the weight of the primary edge, at a cost of 0.
2. The method of claim 1, wherein in step S2, a shortest S-t path P is obtained from the directed graph model G by dijkstra' S method*。
3. The method of claim 1, wherein in step S5, the decomposed remainder is extracted from the decomposed remainder
Figure FDA0002792865180000031
Wherein a shortest s-t path is obtained by using Dijkstra method, i.e. from said decomposed remainder
Figure FDA0002792865180000032
Obtaining a shortest s-t path with the cost not more than delta, marking the problem as a BRSP problem, obtaining the optimal solution of the BRSP problem in O (delta | E |) time by using a dynamic programming method, namely obtaining a shortest s-t path Q with the cost not more than delta, simultaneously obtaining the next shortest s-t path, and enabling P to be P*Subscript i +1 and Ω ═ P1 *,...,Pi *}。
4. The method for fast finding multiple partially disjoint shortest paths for multipath routing as set forth in claim 1, wherein in said step S6, note P*U.Q is the optimal solution, representing path P*And all edges in path Q remove the set of edge pairs between the same vertices but in opposite directions; handle P*Decomposition of U.Q into P1And P2And obtaining a spiral optimal path by two paths.
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