Optical network path and wavelength allocation method based on A* algorithm and pheromone scheduling

Abstract

The invention discloses an optical network path and wavelength allocation method based on an A * algorithm and pheromone scheduling, and relates to the technical field of optical networks, the fast directional search capability of the A * algorithm and the global adaptive optimization characteristic of an ant colony algorithm are fused, and the optical network path and wavelength allocation method based on the A * algorithm and the pheromone scheduling can be realized on the premise of ensuring the wavelength constraint. The service blocking rate is obviously reduced, and the resource utilization rate of the whole network is improved.

Description

Technical Field

[0001] This invention relates to the field of optical network technology, and more specifically to a method for optical network path and wavelength allocation based on the A* algorithm and pheromone scheduling. Background Technology

[0002] Currently, the mainstream path and wavelength allocation (RWA) schemes in optical networks mainly employ metaheuristic algorithms (such as ant colony optimization) or deterministic search algorithms (such as A* algorithm). Ant colony optimization achieves global adaptive optimization through distributed pheromone accumulation, which can cope with dynamic service loads; while A* algorithm uses heuristic functions to quickly determine a high-quality path with low physical cost, and has high search efficiency.

[0003] However, existing methods have significant limitations: ant colony optimization (ACO) suffers from blind searching in the early stages of iteration due to pheromone scarcity, and is prone to causing local congestion and resource fragmentation due to excessive pheromone concentration on a few paths; while A* algorithm, although capable of rapid pathfinding, lacks the ability to perceive and adaptively adjust the overall network resource status, making it difficult to achieve global resource balance in multi-service scenarios. Although some research has attempted to combine the advantages of both, there is still a lack of effective mechanisms to pre-detect and suppress resource bottlenecks in the processes of pheromone initialization, path guidance, and wavelength allocation, causing the algorithms to easily fall into local optima or suffer from low resource utilization efficiency under complex topologies and large-scale services.

[0004] Therefore, how to provide a method for optical network path and wavelength allocation that can integrate the fast directional search capability of the A* algorithm with the global adaptive optimization characteristics of the ant colony algorithm, thereby significantly reducing service blocking rate and improving the overall network resource utilization while ensuring wavelength constraints, is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] In view of the above problems, the present invention is proposed to provide an optical network path and wavelength allocation method based on A* algorithm and pheromone scheduling to overcome or at least partially solve the above problems.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for optical network path and wavelength allocation based on the A* algorithm and pheromone scheduling includes the following steps: S1: Obtain the source node and target node, and initialize the global pheromone matrix; S2: Use the A* algorithm to obtain an initial feasible path from the source node to the target node; wherein the initial feasible path satisfies topological connectivity constraints, maximum link delay constraints, and wavelength continuity constraints; S3: Increase the weight of each link in the initial feasible path and reuse the A* algorithm to obtain several candidate paths that are different from the initial feasible path; S4: The pheromone values ​​of each link in each path in the final candidate path set are amplified hierarchically to obtain an updated global pheromone matrix; wherein, the final candidate path set consists of the initial feasible path and several candidate paths obtained in S3; S5: Using the ant colony algorithm, a search is performed from the source node to the target node to obtain several feasible routing paths; wherein, the initial pheromone matrix of the ant colony algorithm is the updated global pheromone matrix; the several feasible routing paths all satisfy topological connectivity constraints, maximum link delay constraints, and wavelength continuity constraints; S6: Obtain the set of continuously available wavelengths corresponding to each feasible route path, and select the wavelength with the largest remaining availability or normalized remaining availability in the entire network from the set of continuously available wavelengths as the allocated wavelength for the feasible route path; wherein, if the set of continuously available wavelengths includes only one wavelength, reduce the pheromone value of each link on the feasible route path. S7: Perform an evaporation operation on the current pheromone matrix and add an increment to update the current pheromone matrix; S8: Repeat S5-S7 continuously until the iteration termination condition is met, and obtain the optimal path and optimal wavelength from the source node to the target node.

[0007] Preferably, S3 specifically includes the following steps: S31: Multiply the initial cost weight of the selected link by a preset penalty multiplier to obtain the updated link weight matrix; wherein, the selected link is one of the links in the initial feasible path; the preset penalty multiplier is a constant greater than 1; S32: Execute the A* algorithm again under the updated link weight matrix to obtain a candidate path from the source node to the target node; wherein, the candidate path obtained in S32 satisfies the topology connectivity constraint, the maximum link delay constraint, and the wavelength continuity constraint. S33: Determine whether the candidate path obtained in S32 is not completely the same as any path in the current candidate path set. If so, add the candidate path to the current candidate path set and restore the cost weight of the selected link to the initial cost weight. S34: Repeat S31-S33 until all links in the initial feasible path are traversed, and several candidate paths different from the initial feasible path are obtained.

[0008] Preferably, S4 specifically includes the following steps: S41: Sort the paths in the final candidate path set in ascending order of total cost to obtain a new candidate path set { , , ,..., }; where, path , , ,..., The total cost increases; S42: Based on path , , ,..., The basic amplification factor and path of the allocation , , ,..., The proportion of idle wavelengths on each link is updated to determine the path. , , ,..., The pheromone values ​​of each link.

[0009] Preferably, S42 is implemented based on the following formula: ; ; ; In the formula, Representing a path The number of idle wavelengths on any link e; Representing a path The total number of wavelengths on any link e; This indicates the proportion of the number of idle wavelengths to the total number of wavelengths, i.e., the path. The proportion of idle wavelengths on any link e; Represented as a path The basic amplification factor for allocation; Indicates path The resource-aware amplification factor applied to any link e on the path; Representing a path The pheromone value of any link e before amplification; Representing a path The amplified pheromone value of any link e on the path; i represents the path. The starting node of any link e on the path; j represents the path. The termination node of any link e on the path; =1,2,...,K; > > >1; When k>3 0 or less A fixed value; This indicates the upper limit of pheromones.

[0010] Preferably, the remaining availability in S6 is obtained based on the following formula: ; ; In the formula, Represents the set of continuous available wavelengths corresponding to the selected path. The k-th wavelength in the spectrum; the selected path is one of the several feasible routing paths; Indicates the indicator function, when the link When the available space meets business requirements =1, otherwise =0; E represents the set consisting of all links in the entire network; Indicates wavelength Remaining availability across the entire network; This represents the set consisting of all links on the selected path; links For set Or any link in set E.

[0011] Preferably, the normalized residual availability in S6 is obtained based on the following formula: ; ; ; In the formula, Represents the set of continuous available wavelengths corresponding to the selected path. The k-th wavelength in the spectrum; the selected path is one of the several feasible routing paths; Indicates the indicator function, when the link When the available space meets business requirements =1, otherwise =0; E represents the set consisting of all links in the entire network; Indicates wavelength Remaining availability across the entire network; This represents the set consisting of all links on the selected path; Indicates wavelength Normalized remaining availability across the entire network; This indicates the total number of links in the entire network; links For set Or any link in set E.

[0012] Preferably, reducing the pheromone value of each link on the feasible routing path in S6 specifically includes the following steps: Multiply the pheromone value of each link on the feasible route path by the pheromone penalty factor. ;in, .

[0013] Preferably, step S7 updates the pheromone matrix based on the following formula: ; Among them, the right side of the formula This represents any link in the entire network at the start of the current iteration. The pheromone value; the left side of the formula Indicates the link The updated pheromone value is obtained after the superposition of evaporation and increment. The pheromone evaporation coefficient has a value range of [value range missing]. ; Indicates link The total increment of pheromone value contributed by all feasible routing paths in this iteration.

[0014] Preferred, Obtained based on the following formula: ; in, This represents the total number of feasible routing paths obtained in this iteration; if the link Belongs to the One feasible route ,but ,otherwise ; As a preset positive constant, For feasible routing paths The total cost.

[0015] Preferably, the iteration termination condition is reaching the maximum number of iterations or the optimal path no longer changing.

[0016] As can be seen from the above technical solutions, compared with the prior art, the present invention discloses a method for optical network path and wavelength allocation based on A* algorithm and pheromone scheduling. It can integrate the fast directional search capability of A* algorithm with the global adaptive optimization characteristics of ant colony algorithm, thereby significantly reducing the service blocking rate and improving the overall network resource utilization while ensuring wavelength constraints. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0018] Figure 1 This is a flowchart illustrating an optical network path and wavelength allocation method based on the A* algorithm and pheromone scheduling, provided in an embodiment of the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] like Figure 1 As shown, this embodiment of the invention discloses a method for optical network path and wavelength allocation based on the A* algorithm and pheromone scheduling, including the following steps: S1: Obtain the source node and target node, and initialize the global pheromone matrix; It is understandable that S1 further includes: obtaining the current topology of the optical network, link status, and wavelength resource usage of each link.

[0021] S2: Use the A* algorithm to obtain an initial feasible path from the source node to the target node; wherein the initial feasible path satisfies topological connectivity constraints, maximum link delay constraints, and wavelength continuity constraints; It is understandable that the cost function of the A* algorithm can be adopted. ; in, This indicates the distance from the source node to the current node. The cumulative cost of a known path, combined with link distance and hop count; To start from the current node The heuristic cost estimation to the destination node can be based on the coordinates of the network topology, estimated through grid distance or logical topological distance between nodes. When expanding the neighboring nodes of the current node, only adjacent nodes that are topologically reachable and compatible with the current path in the wavelength dimension are retained, thus ensuring that the searched path satisfies the topological connectivity and wavelength continuity constraints.

[0022] S3: Increase the weight of each link in the initial feasible path and reuse the A* algorithm to obtain several candidate paths that are different from the initial feasible path; In one or more embodiments, S3 specifically includes the following steps: S31: Multiply the initial cost weight of the selected link by a preset penalty multiplier to obtain the updated link weight matrix; wherein, the selected link is one of the links in the initial feasible path; the preset penalty multiplier is a constant greater than 1; S32: Execute the A* algorithm again under the updated link weight matrix to obtain a candidate path from the source node to the target node; wherein, the candidate path obtained in S32 satisfies the topology connectivity constraint, the maximum link delay constraint, and the wavelength continuity constraint. S33: Determine whether the candidate path obtained in S32 is not completely the same as any path in the current candidate path set. If so, add the candidate path to the current candidate path set and restore the cost weight of the selected link to the initial cost weight. S34: Repeat S31-S33 until all links in the initial feasible path are traversed, and several candidate paths different from the initial feasible path are obtained.

[0023] S4: The pheromone values ​​of each link in each path in the final candidate path set are amplified hierarchically to obtain an updated global pheromone matrix; wherein, the final candidate path set consists of the initial feasible path and several candidate paths obtained in S3; In one or more embodiments, S4 specifically includes the following steps: S41: Sort the paths in the final candidate path set in ascending order of total cost to obtain a new candidate path set { , , ,..., }; where, path , , ,..., The total cost increases; What is understandable is: remembering the path The set consisting of the various links contained above is any link The cost weight is denoted as Then the path Total cost Defined as the sum of the cost weights of all links on that path, i.e. ; The link cost weight It is obtained by weighting one or more factors such as link length, link transmission delay, and link resource occupancy according to preset weights. In a specific implementation, it can be expressed as: ; in, These are preset weighting coefficients; , , They are defined as follows: 1) Distance Cost Defined as a link The physical length is the result after normalization to the maximum link length of the entire network: ; in, For link The physical length or the corresponding logical distance, It is the maximum value among all physical lengths of all links in the entire network or a preset normalization constant.

[0024] 2) Delay Cost Defined as a link The single-hop transmission delay is the result after normalization based on the maximum link delay of the entire network: ; in, For link Transmission delay, It is the maximum value of the transmission delay of all links in the entire network or a preset normalization constant.

[0025] 3) Resource consumption cost Defined as a link Measurement of wavelength resource occupancy: ; in, Indicates link The ratio of the number of idle wavelengths to the total number of wavelengths, with a range of values ​​of [value missing]. ; The larger the value, the higher the link. The more abundant the wavelength resources, the higher the corresponding resource consumption cost. The smaller.

[0026] S42: Based on path , , ,..., The basic amplification factor and path of the allocation , , ,..., The proportion of idle wavelengths on each link is updated to determine the path. , , ,..., The pheromone values ​​of each link.

[0027] In one or more embodiments, S42 is implemented based on the following formula: ; ; ; In the formula, Representing a path The number of idle wavelengths on any link e; Representing a path The total number of wavelengths on any link e; This indicates the proportion of the number of idle wavelengths to the total number of wavelengths, i.e., the path. The proportion of idle wavelengths on any link e; Represented as a path The basic amplification factor for allocation; Indicates path The resource-aware amplification factor applied to any link e on the path; Representing a path The pheromone value of any link e before amplification; Representing a path The amplified pheromone value of any link e on the path; i represents the path. The starting node of any link e on the path; j represents the path. The termination node of any link e on the path; =1,2,...,K; > > >1; When k>3 0 or less A fixed value; This indicates the upper limit of pheromones.

[0028] S5: Using the ant colony algorithm, a search is performed from the source node to the target node to obtain several feasible routing paths; wherein, the initial pheromone matrix of the ant colony algorithm is the updated global pheromone matrix; the several feasible routing paths all satisfy topological connectivity constraints, maximum link delay constraints, and wavelength continuity constraints; It is understandable that each ant starts from the source node and selects the next hop neighbor node based on the transition probability during the iteration process.

[0029] Suppose the current ant is located at node i, and its set of possible neighbor nodes is: For any candidate neighbor node If the side Corresponding link If there exists at least one available wavelength compatible with the set of wavelengths already occupied by the current path in the wavelength dimension, then the probability of transitioning from node i to node j is defined as follows:

[0030] in, This represents the directed edge from node i to candidate neighbor node j recorded in the current iteration. The pheromone value on the surface; This represents a directed edge from node i to another candidate neighbor node k. The pheromone value on the surface For set The index of any candidate neighbor node in the denominator is used to enumerate all possible next-hop nodes. Represents a directed edge The heuristic function value can be determined based on the link. The path length and number of hops are determined by a combination of factors. Represents a directed edge The heuristic function value, and The definition method is the same, but it corresponds to the link characteristics from node i to candidate neighbor node k; , These are the importance coefficients of the pheromone and the heuristic function, respectively, used to adjust their relative weights in the transition probabilities. Under the taboo list constraint of the forbidden loop, the ant selects the next hop node according to the above transition probabilities and moves forward. When it reaches the destination node, the path traversed by the ant and the corresponding set of available wavelengths are recorded.

[0031] S6: Obtain the set of continuously available wavelengths corresponding to each feasible route path, and select the wavelength with the largest remaining availability or normalized remaining availability in the entire network from the set of continuously available wavelengths as the allocated wavelength for the feasible route path; wherein, if the set of continuously available wavelengths includes only one wavelength, reduce the pheromone value of each link on the feasible route path. In one or more embodiments, the remaining availability in S6 is obtained based on the following formula: ; ; In the formula, Represents the set of continuous available wavelengths corresponding to the selected path. The k-th wavelength in the spectrum; the selected path is one of the several feasible routing paths; Indicates the indicator function, when the link When the available space meets business requirements =1, otherwise =0; E represents the set consisting of all links in the entire network; Indicates wavelength Remaining availability across the entire network; This represents the set consisting of all links on the selected path; links For set Or any link in set E.

[0032] In one or more embodiments, the normalized remaining availability in S6 is obtained based on the following formula: ; ; ; In the formula, Represents the set of continuous available wavelengths corresponding to the selected path. The k-th wavelength in the spectrum; the selected path is one of the several feasible routing paths; Indicates the indicator function, when the link When the available space meets business requirements =1, otherwise =0; E represents the set consisting of all links in the entire network; Indicates wavelength Remaining availability across the entire network; This represents the set consisting of all links on the selected path; Indicates wavelength Normalized remaining availability across the entire network; This indicates the total number of links in the entire network; links For set Or any link in set E.

[0033] In one or more embodiments, reducing the pheromone value of each link on the feasible routing path in step S6 specifically includes the following steps: Multiply the pheromone value of each link on the feasible route path by the pheromone penalty factor. ;in, .

[0034] It is understood that if the set of continuously available wavelengths includes only one wavelength, then the feasible routing path is considered a resource bottleneck path. Applying a pheromone penalty factor of less than 1 to the pheromone value of each link on the feasible routing path can reduce the probability that subsequent routing requests will continue to select the bottleneck path.

[0035] S7: Perform an evaporation operation on the current pheromone matrix and add an increment to update the current pheromone matrix; In one or more embodiments, step S7 updates the pheromone matrix based on the following formula: ; Among them, the right side of the formula This represents any link in the entire network at the start of the current iteration. The pheromone value; the left side of the formula Indicates the link The updated pheromone value is obtained after the superposition of evaporation and increment. The pheromone evaporation coefficient has a value range of [value range missing]. ; Indicates link The total increment of pheromone value contributed by all feasible routing paths in this iteration.

[0036] In one or more embodiments, Obtained based on the following formula: ; in, This represents the total number of feasible routing paths obtained in this iteration; if the link Belongs to the One feasible route ,but ,otherwise ; As a preset positive constant, For feasible routing paths The total cost.

[0037] S8: Repeat S5-S7 continuously until the iteration termination condition is met, and obtain the optimal path and optimal wavelength from the source node to the target node.

[0038] In one or more embodiments, the iteration termination condition is reaching the maximum number of iterations or the optimal path no longer changing.

[0039] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0040] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for optical network path and wavelength assignment based on A* algorithm and pheromone scheduling, characterized in that, The method comprises the following steps: S1: obtaining a source node and a target node, and initializing a global pheromone matrix; S2: obtaining an initial feasible path from the source node to the target node by using an A* algorithm; wherein the initial feasible path satisfies a topological connectivity constraint, a maximum link delay constraint and a wavelength continuity constraint; S3: increasing the weight of each link in the initial feasible path and repeatedly using the A* algorithm to obtain a plurality of candidate paths different from the initial feasible path; S4: hierarchically amplifying the pheromone values of each link of each path in a final candidate path set to obtain an updated global pheromone matrix; wherein the final candidate path set is composed of the initial feasible path and the plurality of candidate paths obtained in S3; S5: searching from the source node to the target node by using an ant colony algorithm to obtain a plurality of feasible routing paths; wherein the initial pheromone matrix of the ant colony algorithm is the updated global pheromone matrix; the plurality of feasible routing paths all satisfy the topological connectivity constraint, the maximum link delay constraint and the wavelength continuity constraint; S6: obtaining a set of continuous available wavelengths corresponding to each feasible routing path, and selecting a wavelength with the maximum residual available degree or normalized residual available degree in the set of continuous available wavelengths as the allocation wavelength of the feasible routing path in the whole network; wherein if the set of continuous available wavelengths only includes one wavelength, the pheromone values of each link on the feasible routing path are reduced; S7: performing a volatilization operation on the current pheromone matrix and superimposing an increment to update the current pheromone matrix; S8: constantly repeating S5-S7 until an iteration termination condition is met to obtain an optimal path and an optimal wavelength from the source node to the target node.

2. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, S3 specifically comprises the following steps: S31: multiplying the initial cost weight of a selected link by a preset penalty multiple to obtain an updated link weight matrix; wherein the selected link is one link in the initial feasible path; the preset penalty multiple is a constant greater than 1; S32: executing the A* algorithm again under the updated link weight matrix to obtain a candidate path from the source node to the target node; wherein the candidate path obtained in S32 satisfies the topological connectivity constraint, the maximum link delay constraint and the wavelength continuity constraint; S33: determining whether the candidate path obtained in S32 is completely different from any path in the current candidate path set; if yes, adding the candidate path to the current candidate path set and restoring the cost weight of the selected link to the initial cost weight; S34: repeating S31-S33 until all links in the initial feasible path are traversed to obtain a plurality of candidate paths different from the initial feasible path.

3. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method according to claim 1, wherein, S4 specifically comprises the following steps: S41: sort each path in the final candidate path set in ascending order of total cost to obtain a new candidate path set , , , }; wherein the total cost of path , , , increases; S42: update the information of each link on the path based on the base amplification factor assigned to the path and the free wavelength proportion of each link on the path , , , , , , , , , ​​​ 4. The A* algorithm and pheromone scheduling based optical network path and wavelength allocation method of claim 3, wherein, S42 is implemented based on the following formula: ； ； ； wherein denotes the number of free wavelengths on any link e of path ; denotes the total number of wavelengths on any link e of path ; denotes the proportion of free wavelengths to total wavelengths, i.e. the free wavelength proportion on any link e of path ; denotes the base amplification factor assigned to path ; denotes the resource-aware amplification factor applied to any link e of path ; denotes the pheromone value before amplification on any link e of path ; denotes the pheromone value after amplification on any link e of path ; denotes the start node of any link e of path ; denotes the end node of any link e of path = 1, 2,..., K; > > > 1; when k > 3, is 0 or a fixed value smaller than ; denotes the upper limit of pheromone.

5. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, The residual available degree in S6 is obtained based on the following formula: ； ； wherein, denotes the kth wavelength in the set of consecutive available wavelengths corresponding to the selected path; the selected path is one of the feasible routing paths; denotes an indicator function, = 1 when the link is free and satisfies the traffic demand, otherwise = 0; E denotes the set of all links in the network; denotes the remaining availability of the wavelength over the entire network; denotes the set of all links in the selected path; link is any link in the set or in the set E. ​​ 6. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, The normalized residual available degree in S6 is obtained based on the following formula: ； ； ； In the formula, Represents the set of continuous available wavelengths corresponding to the selected path. The k-th wavelength in the spectrum; the selected path is one of the several feasible routing paths; Indicates the indicator function, when the link When the available space meets business requirements =1, otherwise =0; E represents the set consisting of all links in the entire network; Indicates wavelength Remaining availability across the entire network; This represents the set consisting of all links on the selected path; Indicates wavelength Normalized remaining availability across the entire network; This indicates the total number of links in the entire network; links For set Or any link in set E.

7. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, The reduction of the pheromone values of each link on the feasible routing path in S6 specifically comprises the following steps: multiplying the pheromone value of each link on the feasible route path by a pheromone penalty factor ; wherein, .

8. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, Step S7 updates the pheromone matrix based on the following formula: ； Among them, the right side of the formula This represents any link in the entire network at the start of the current iteration. The pheromone value; the left side of the formula Indicates the link The updated pheromone value is obtained after the superposition of evaporation and increment. The pheromone evaporation coefficient has a range of values. ; Indicates link The total increment of pheromone value contributed by all feasible routing paths in this iteration.

9. The A* algorithm and pheromone scheduling based optical network path and wavelength allocation method of claim 8, wherein, Obtained based on the following equation: ； wherein, represents the total number of feasible routing paths obtained in the current iteration; if the link belongs to the th feasible routing path , then , otherwise ; is a preset positive constant, is the total cost of the feasible routing path .

10. The A* algorithm and pheromone scheduling based optical network path and wavelength assignment method of claim 1, wherein, The iteration termination condition is that a maximum number of iterations is reached or the optimal path no longer changes.

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