CN101170503A - An optimization method for multicast route ant group algorithm - Google Patents

Abstract

The invention discloses an optimization method of an ant colony algorithm of multicast routing, which includes that: solving network multicast routing by using the present ant colony optimization algorithm, and introducing a concept of iterative feedback value in the process of running the algorithm; calculating a feedback value according to the solving status of the algorithm and the distribution of information elements in every round of iteration, and dynamically adjust the value of control parameters, and thereby the ant colony algorithm in the invention is allowed to solve the multicast path in adaptive multi-iterations and to finally find out the multicast tree of the minimum cost. The invention overcomes the drawback that the performance of the prior ant colony algorithm relies too much on the initial value of parameters, effectively avoids that the ant colony algorithm is limited in the local optimization, and improves the solution of network multicast routing.

Description

A kind of optimization method of multicast route ant group algorithm

Technical field

The present invention relates to multicast (Multicast) route technology of network service, relate in particular to a kind of optimization method of multicast route ant group algorithm.

Background technology

Improving constantly of the increasingly extensive and broadband network services demand of using along with computer interconnected network, various business based on the Internet need service quality (QoS more and more, Quality of Service) assurance, and the bearing capacity of current raising existing network has just become to improve the important channel of QoS.Because it is higher to change or transform the cost of hardware in large quantities, and it is little to the influence that improves the existing network bearing capacity to change few parts, so people more and more pay close attention to the routing policy that can improve network resource utilization and network QoS, wherein, utilizing multicasting technology is a kind of very effective means that address the above problem at present.

Current Network Transmission pattern has three kinds, i.e. clean culture, broadcasting, multicast, and wherein, the multicast mode has development prospect most.In the multicast mode, a message transmits in network, and the control by router can guarantee that a plurality of destination nodes receive the message from source node.With respect to resource occupying big mode of unicast and the many broadcast modes of non-destination node expense, the multicast mode has resource and occupies the few characteristics of little, invalid expense.

Multicast routing problem is proved to be a nondeterministic polynomial, and (NP, NondeterministicPolynomial) difficult problem use heuritic approach, mostly as genetic algorithm, KMB algorithm, ant group algorithm etc. when existing method is found the solution multicast tree.Wherein, ant group algorithm is produced by the inspiration of occurring in nature ant foraging behavior, this algorithm simulation actual ant group seek the process of food.At occurring in nature, a shortest path between the ant group always can find from the nest to the food source.This is because ant can stay a kind of material that is called pheromone (Pheromone) on the path of its process in motion process.This material can be perceived by ant afterwards, and can volatilize gradually in time, and each ant is instructed the direction of motion of oneself according to the concentration of pheromone on the path, and tends to move towards the high direction of this material concentration.Therefore, if the ant of passing by on a certain path is many more, then Ji Lei pheromone is just many more, concentration is just big more, and the probability that this path was chosen by other ant in next time is just big more.Because within a certain period of time, short more path can be by many more ant visits, so along with the carrying out of said process, whole ant group finally can find from the ant cave to the shortest path the food.Pheromone is commonly called the routing information element in ant group algorithm.Ant group algorithm has utilized this characteristic of occurring in nature ant group to come shortest route problem is found the solution just.Ant group algorithm itself has implied the parallel distributed computing capability and based on the construction method of separating of probability, makes it can more easily evade network congestion, more effectively find the solution multicast routing problem.

Yet, ant group algorithm is the same with many other heuritic approaches, its performance often depends on the value of Control Parameter, since existing ant group algorithm in computational process seldom the value to Control Parameter adjust, therefore, if it is improper to exist initial value to be provided with, the defective that the performance of ant group algorithm is unsatisfactory.

Summary of the invention

In view of this, main purpose of the present invention is to provide a kind of optimization method of multicast route ant group algorithm, can effectively avoid finding the solution multicast path by the time may be absorbed in the problem of locally optimal solution, and can improve the speed of finding the solution the network multicast route.

For achieving the above object, technical scheme of the present invention is achieved in that

A kind of optimization method of multicast route ant group algorithm, this method comprises:

A, each network node of initialization for all links between network node are provided with constraints, and are the pheromones tax initial value of every link;

B, initialization routing table search out the pheromones of destination node respective links afterwards, and pheromones is carried out local updating;

The valuation functions value of c, every possible path of iterative computation and this iterative feedback value, and select the optimal path of this iteration;

D, the Control Parameter of adjusting iterative feedback value renewal ant colony optimization algorithm and renewal global information element enter the next round iterative computation then;

E, judge whether to satisfy end condition, if satisfy end condition, then execution in step f; Otherwise, return step b;

F, select optimum multicast routed path of current iteration and output.

The described constraints of step a is specially:

The maximum of all the chain-circuit time delay sums between calculating from the source node to the destination node is as chain-circuit time delay constraints; And the minimum value that calculates all link expense sums, as the least cost binding occurrence of multicast routing tree.

The described pheromones for every link of step a is composed initial value and is specially: τ ₀=m/C ^NnWherein, m is the quantity of artificial ant, C ^NnThe serve as reasons length in path of structure.

The destination node of the described searching of step b satisfies condition: $j=\arg {\max }_{l\∈N_i^k}\left\{p_{\mathrm{ij}}^k\right\},$ $p_{\mathrm{ij}}^k=\frac{{\left[{\mathrm{\τ}}_{\mathrm{ij}}\right]}^{\mathrm{\α}}{\left[{\mathrm{\η}}_{\mathrm{ij}}\right]}^{\mathrm{\β}}}{{\mathrm{\Σ}}_{l\∈N_i^k}{\left[{\mathrm{\τ}}_{\mathrm{il}}\right]}^{\mathrm{\α}}{\left[{\mathrm{\η}}_{\mathrm{il}}\right]}^{\mathrm{\β}}};$

Wherein, $j\∈N_i^k;$ α, β are two Control Parameter pheromones and inspire the shared weight of the factor in Path selection, and set α=1; η _Ij(i, j) the inspiration factor on is got η for link _Ij=1/c _Ij* d _Ij

Step b is described to carry out local updating to pheromones and is specially: τ _Ij← (1-ρ) τ _Ij, wherein, ρ is the pheromones volatility coefficient of one of Control Parameter, and its span is [0.05,0.15].

The value of the valuation functions F (k) of the every paths of the described iterative computation of step c and this iterative feedback value M[t] be specially:

C1, according to valuation functions $F\left(k\right)=\underset{(i,j)\∈{\mathrm{route}}_k}{\mathrm{\Σ}}c(i,j)*\underset{(i,j)\∈{\mathrm{route}}_k}{\mathrm{\Σ}}d(i,j),$ Calculate the standard deviation sigma [t] of valuation functions;

C2, with the lasting algebraically N[t of current iteration optimal path in standard deviation sigma [t], the t time iteration] substitution iterative feedback value computing formula $M\left[t\right]=A\·\frac{N\left[t\right]}{\mathrm{\σ}\left[t\right]}$ In, obtain iterative feedback value M[t the t time].

The described adjustment iterative feedback of steps d value is upgraded the Control Parameter of ant colony optimization algorithm and is upgraded the global information element, comprising:

D1, revise to inspire the value of the weight beta of the factor, if this iterative feedback value M[t] greater than last time iterative feedback value M (t-1), then use β (t+1)=β (t)/0.95 to calculate the value of weight beta; Otherwise with the value of β (t+1)=0.95* β (t) calculating weight beta, wherein, the span of β is [0,5];

The value of d2, calculating and modification pheromones volatility coefficient ρ; If this iterative feedback value M[t] greater than last time iterative feedback value M (t-1), then use ρ (t+1)=ρ (t)/0.95 to calculate; Otherwise, calculate with ρ (t+1)=0.95* ρ (t), wherein, the span of ρ is [0.5,1.5];

After the value adjustment of d3, Control Parameter finishes, upgrade the global information element, the update rule of global information element is: ${\mathrm{\τ}}_{\mathrm{ij}}\←(1-\mathrm{\ρ}){\mathrm{\τ}}_{\mathrm{ij}}+\mathrm{\ρ\Δ}{\mathrm{\τ}}_{\mathrm{ij}}^{\mathrm{bs}},$ Wherein,  (i, j) ∈ T ^BsΔ τ _Ij ^Bs=B/F (best), F (best) are the valuation functions value of current iteration optimal path.

End condition described in the step e is: set maximum iteration time and/or find the solution multicast path by the precision of optimal solution.

The optimization method of multicast route ant group algorithm provided by the present invention has the following advantages:

1) the present invention is by introducing the notion of iterative feedback value in ant group algorithm, and by calculating value of feedback according to multicast path again by the distribution of state of finding the solution and pheromones every the wheel in the iteration, thereby dynamically adjust the value of Control Parameter, efficiently solve the defective that the multicast routed path that obtains that ant group algorithm too relies on initial parameter value and caused usually is absorbed in local optimum.

2) optimization method of multicast route ant group algorithm of the present invention is revised Control Parameter according to the distribution dynamic of finding the solution state and pheromones in the algorithm running, improved the ant group and found the solution the speed of multicast routing problem, more can satisfy the requirement of network dynamic change.

Description of drawings

Fig. 1 is the process flowchart of multicast route ant colony optimization algorithm of the present invention.

Embodiment

Below in conjunction with the accompanying drawing and the embodiment of the invention method of the present invention is described in further detail.

In the present embodiment, computer communication network non-directed graph G (V, E) expression, wherein, V (v1, v2, v3 ..., vn) be the set of network node; E is the set of link, every link e (i, j) ∈ E have two parameters: link expense c (i, j), chain-circuit time delay d (i, j).In this non-directed graph, i, j represent network node, and the circuit between i and the j is called link, are referred to as the limit in non-directed graph.

In addition, represent source node with s, D (d1, d2 ..., the dn) set of expression destination node, the delay constraint of stipulating all destination nodes is a Δ.

The present invention find the solution multicast path by problem belong to the limited multicast routing algorithm problem of time delay, be specially: constructing a root is source node s, and cover the least cost multicast tree T of all destination nodes, require the time delay of each destination node in the tree will satisfy the delay constraint Δ simultaneously.

Wherein, the time delay d of destination node (i, j) be meant source node arrive the path p of destination node (time delay D of multicast tree (T) is meant the maximum of destination node time delay for s, the di) summation of last every limit time delay:

$D\left(T\right)=\max \underset{(i,j)\&Element;p(s,\mathrm{di})}{\mathrm{\&Sigma;}}d(i,j)<\mathrm{\&Delta;}---\left(1\right)$

Wherein, T is a multicast tree; I, j are network node; (i j) is chain-circuit time delay to d; Δ is the delay constraint threshold value of a setting.

And the value of the minimum cost of multicast routing tree constraint C (T) is:

$C\left(T\right)=\min \underset{(i,j)\&Element;T}{\mathrm{\&Sigma;}}c(i,j)---\left(2\right)$

Wherein, T is a multicast tree; I, j are network node; (i j) is the link expense to c.

Fig. 1 is the process flowchart of multicast route ant colony optimization algorithm of the present invention, and as shown in Figure 1, the processing procedure of this method comprises:

Step 101: the initialization network node provides the constraints of all links between network node and the pheromones initial value on each bar link.

Here, described constraints is time delay and cost value, and described time delay is D (T), is calculated by formula (1); Described cost value is C (T), is calculated by formula (2).The time delay of link and the factor of cost value mainly contain between the decision network node: link apart from the height of message transmission rate between length, internodal link connected mode and node etc. between node.

Described initialization network node also will be the plain initial value τ of the link configuration information between the adjacent nodes whole in the network ₀=m/C ^Nn, wherein m represents the quantity of ant, C ^NnBe length by the path of nearest neighbor (nearest-neighbor) heuristic structure.

Step 102: initialization routing table.

Described routing table is meant the network address table of throwing the net of one in the memory that is present in node device such as router.Suppose that routing table is route, during the initialization routing table, be provided with the only artificial ant of m, and to make the distance of distance sources node be 0, be i.e. s=0 from source node.

Described artificial ant is meant a bit of addressing, specific functional programs that has, and according to actual needs, the addressing computing that a plurality of such programs walk abreast simultaneously is set usually.

Step 103: make the pheromones of artificial ant searching target complete node, and the pheromones that finds is carried out local updating.

When described artificial ant seeks destination node,, artificial ant k then is expressed as route if searching out s node _k(s).The artificial ant k that is positioned at node i is when next node j shifts, and node j should satisfy:

$j=\arg {\max }_{l\&Element;N_i^k}\left\{p_{\mathrm{ij}}^k\right\},$ $p_{\mathrm{ij}}^k=\frac{{\left[{\mathrm{\&tau;}}_{\mathrm{ij}}\right]}^{\mathrm{\&alpha;}}{\left[{\mathrm{\&eta;}}_{\mathrm{ij}}\right]}^{\mathrm{\&beta;}}}{{\mathrm{\&Sigma;}}_{l\&Element;N_i^k}{\left[{\mathrm{\&tau;}}_{\mathrm{il}}\right]}^{\mathrm{\&alpha;}}{\left[{\mathrm{\&eta;}}_{\mathrm{il}}\right]}^{\mathrm{\&beta;}}}$ , suppose $j=N_i^k;$

Wherein, $\arg {\max }_{l\&Element;N_i^k}\left\{p_{\mathrm{ij}}^k\right\}$ Be illustrated in node set N _i ^kIn, the p of node j _Ij ^kValue is maximum; $N_i^k=S-\mathrm{rout}{e}_k,$ S is the set of the node of all existence, route _kThe route set of representing the only artificial node that ant passes through of k; τ _IjBe limit (i, j) the pheromones value on; η _Ij(i, j) the inspiration factor on is got η for the limit _Ij=1/c _Ij* d _Ij, c _Ij, d _IjBe respectively limit (i, time delay j) and cost value; They and delay constraint Δ and expense restriction C (T) to be compared, just shift if satisfy constraints; Otherwise, just select other node, when constraints was not satisfied in all possible path, artificial ant just withdrawed from this searching process.

α, β are two Control Parameter, represent pheromones respectively and inspire the shared weight of the factor in Path selection.Suppose α=1, and β changes in the scope of [0,5], when every limit was selected, all the plain local updating rule of recalls information was upgraded the pheromones value on the limit of choosing.

The plain update rule of described local message is:

τ _ij←(1-ρ)τ _ij

Wherein, ρ is a Control Parameter, is called as the pheromones volatility coefficient, and ρ changes in the scope of [0.05,0.15].Artificial ant seeks in the process of destination node, and the behavior that node shifts is sustained, till all artificial ants are all found whole destination nodes.

Here, judge whether artificial ant finds the way of whole destination nodes to be: in multicast routing problem, a message transmits in network, can one of generation before transmitting in start of heading have the destination node of multicast request set D (d1, d2 ..., dn).When message transmits, in the information that artificial ant is carried, add destination node information, if the node of transferring in this set, is just passed to it to this node as destination node with message; If the node of transferring to does not belong to this set, then artificial ant continues to look for next node, up to finally finding whole destination nodes.

Step 104: the valuation functions value of the every paths of iterative computation and this iterative feedback value, and select the optimal path of this iteration.

Described valuation functions is $F\left(k\right)=\underset{(i,j)\&Element;{\mathrm{route}}_k}{\mathrm{\&Sigma;}}c(i,j)*\underset{(i,j)\&Element;{\mathrm{route}}_k}{\mathrm{\&Sigma;}}d(i,j),$ The described optimal path of selecting this iteration is meant and finds out the multicast path that makes described valuation functions value minimum.

Here, need calculate this iterative feedback value, adjust Control Parameter for next step and prepare.The computing formula of described iterative feedback value is:

$M\left[t\right]=A\&CenterDot;\frac{N\left[t\right]}{\mathrm{\&sigma;}\left[t\right]}$

Wherein, M[t] be the iterative feedback value of the t time iteration; N[t] be the lasting algebraically of current iteration optimal path in the t time iteration; σ [t] is the standard deviation of all path evaluation functional values in the t time iteration; A is a correction value.M[t] big more, adjust Control Parameter and impel the solution space of algorithm exploration away from the current iteration optimal solution, otherwise, then increase algorithm and near the iteration optimal solution, make up the new probability of separating.

Step 105: adjust the Control Parameter that the iterative feedback value is upgraded ant colony optimization algorithm, and upgrade the global information element.

The Control Parameter that described ant colony optimization algorithm relates to has three, is respectively: state transitions Control Parameter α, β and pheromones volatility coefficient ρ.

Generally, come adjustment information element and the ratio that inspires the weight of the factor in Path selection by the value of revising β earlier.The dynamic adjustment formula of β is:

$\mathrm{\&beta;}(t+1)=\left\{\begin{array}{cc}\mathrm{\&beta;}\left(t\right)/0.95& \mathrm{ifM}\left(t\right)>M(t-1)\\ 0.95*\mathrm{\&beta;}\left(t\right)& \mathrm{else}\end{array}\right.$

Wherein, M[t] be the iterative feedback value of the t time iteration, M (t-1) is the iterative feedback value of t-1 iteration; If this iterative feedback value was greater than last time, β (t+1)=β (t)/0.95 then; Otherwise, β (t+1)=0.95* β (t).As M[t] when increasing, pheromones distributes and becomes comparatively concentrated, and increase the value of β this moment, and the weight that inspires the factor is increased.The span of β is [0,5], adjusts value in the span of β automatically, can make the quality in the path of trying to achieve more stable.If β value gets certain fixed numeric values, though can be in certain be found the solution can be in the hope of separating preferably, can not guarantee can both to obtain for ten times, hundred times separating that quality is satisfied with.But adopt the mode of dynamically adjusting the β value, can make to find the solution and to obtain quality and separate preferably.

In addition, also can use the overall update rule of the local updating rule and/or the pheromones of pheromones simultaneously, these two kinds rules are used identical pheromones volatility coefficient ρ, and its formula is as follows:

$\mathrm{\&rho;}(t+1)=\left\{\begin{array}{cc}\mathrm{\&rho;}\left(t\right)/0.95& \mathrm{ifM}\left(t\right)>M(t-1)\\ 0.95*\mathrm{\&rho;}\left(t\right)& \mathrm{else}\end{array}\right.$

Wherein, M[t] be the iterative feedback value of the t time iteration, M (t-1) is the iterative feedback value of t-1 iteration; If this iterative feedback value was greater than last time, ρ (t+1)=ρ (t)/0.95 then; Otherwise, ρ (t+1)=0.95* ρ (t).As M[t] when increasing, increase the volatility coefficient of pheromones, make the possibility that makes up new routed path increase; As M[t] when reducing, reduce the value of ρ, make ant have big probability near the known iterative optimal path, to search for more outstanding path.If the value of ρ is a certain fixed value, if too high in the iterative process value, then algorithm will soon be restrained, and be absorbed in the situation of local optimum, and the ρ value is low excessively, just might make that pheromones can't be assembled on the high-quality path.Drawing here according to experiment, the ρ span is that [0.5,1.5] interior effect is better.

At last, after the value adjustment of Control Parameter finishes, carry out the renewal of global information element again.Can upgrade the global information element according to following formula:

${\mathrm{\&tau;}}_{\mathrm{ij}}\&LeftArrow;(1-\mathrm{\&rho;}){\mathrm{\&tau;}}_{\mathrm{ij}}+\mathrm{\&rho;\&Delta;}{\mathrm{\&tau;}}_{\mathrm{ij}}^{\mathrm{bs}},$ (i，j)∈T ^bs

Wherein: Δ τ _Ij ^Bs=B/F (best); B is a constant; F (best) is the valuation functions value of current iteration optimal path.

Step 106: judge whether ant colony optimization algorithm satisfies end condition, if satisfy, then execution in step 107; Otherwise, return step 102.

Described end condition is set according to real needs, can be set to maximum iteration time, also can be set to the precision of optimal solution.

Step 107: select optimum multicast routed path of current iteration and output.

The valuation functions value F (k) in all paths relatively selects the optimal path that the path that makes F (k) obtain minimum value is obtained as this iteration.

The above is preferred embodiment of the present invention only, is not to be used to limit protection scope of the present invention.

Claims (8)

1. the optimization method of a multicast route ant group algorithm is characterized in that, this method comprises:

A, each network node of initialization for all links between network node are provided with constraints, and are the pheromones tax initial value of every link;

B, initialization routing table search out the pheromones of destination node respective links afterwards, and pheromones is carried out local updating;

The valuation functions value of c, every possible path of iterative computation and this iterative feedback value, and select the optimal path of this iteration;

D, the Control Parameter of adjusting iterative feedback value renewal ant colony optimization algorithm and renewal global information element enter the next round iterative computation then;

E, judge whether to satisfy end condition, if satisfy end condition, then execution in step f; Otherwise, return step b;

F, select optimum multicast routed path of current iteration and output.

2. optimization method according to claim 1 is characterized in that, the described constraints of step a is specially:

The maximum of all the chain-circuit time delay sums between calculating from the source node to the destination node is as chain-circuit time delay constraints; And the minimum value that calculates all link expense sums, as the least cost binding occurrence of multicast routing tree.

3. optimization method according to claim 1 is characterized in that, the described pheromones for every link of step a is composed initial value and is specially: τ ₀=m/C ^NnWherein, m is the quantity of artificial ant, C ^NnThe serve as reasons length in path of structure.

4. optimization method according to claim 1 is characterized in that, the destination node of the described searching of step b satisfies condition: $j=\arg {\max }_{l\&Element;N_i^k}\left\{p_{\mathrm{ij}}^k\right\},$ $p_{\mathrm{ij}}^k=\frac{{\left[{\mathrm{\&tau;}}_{\mathrm{ij}}\right]}^{\mathrm{\&alpha;}}{\left[{\mathrm{\&eta;}}_{\mathrm{ij}}\right]}^{\mathrm{\&beta;}}}{{\mathrm{\&Sigma;}}_{l\&Element;N_i^k}{\left[{\mathrm{\&tau;}}_{\mathrm{il}}\right]}^{\mathrm{\&alpha;}}{\left[{\mathrm{\&eta;}}_{\mathrm{il}}\right]}^{\mathrm{\&beta;}}};$

Wherein, $j\&Element;N_i^k;$ α, β are two Control Parameter pheromones and inspire the shared weight of the factor in Path selection, and set α=1; η _Ij(i, j) the inspiration factor on is got η for link _Ij=1/c _Ij* d _Ij

5. optimization method according to claim 1 is characterized in that, step b is described to carry out local updating to pheromones and be specially: τ _Ij← (1-ρ) τ _Ij, wherein, ρ is the pheromones volatility coefficient of one of Control Parameter, and its span is [0.05,0.15].

6. optimization method according to claim 1 is characterized in that, the value of the valuation functions F (k) of the every paths of the described iterative computation of step c and this iterative feedback value M[t] be specially:

C1, according to valuation functions $F\left(k\right)=\underset{(i,j)\&Element;{\mathrm{route}}_k}{\mathrm{\&Sigma;}}c(i,j)*\underset{(i,j)\&Element;{\mathrm{route}}_k}{\mathrm{\&Sigma;}}d(i,j),$ Calculate the standard deviation sigma [t] of valuation functions;

C2, with the lasting algebraically N[t of current iteration optimal path in standard deviation sigma [t], the t time iteration] substitution iterative feedback value computing formula $M\left[t\right]=A\&CenterDot;\frac{N\left[t\right]}{\mathrm{\&sigma;}\left[t\right]}$ In, obtain iterative feedback value M[t the t time].

7. optimization method according to claim 1 is characterized in that, the described adjustment iterative feedback of steps d value is upgraded the Control Parameter of ant colony optimization algorithm and upgraded the global information element, comprising:

D1, revise to inspire the value of the weight beta of the factor, if this iterative feedback value M[t] greater than last time iterative feedback value M (t-1), then use β (t+1)=β (t)/0.95 to calculate the value of weight beta; Otherwise with the value of β (t+1)=0.95* β (t) calculating weight beta, wherein, the span of β is [0,5];

The value of d2, calculating and modification pheromones volatility coefficient ρ; If this iterative feedback value M[t] greater than last time iterative feedback value M (t-1), then use ρ (t+1)=ρ (t)/0.95 to calculate; Otherwise, calculate with ρ (t+1)=0.95* ρ (t), wherein, the span of ρ is [0.5,1.5];

After the value adjustment of d3, Control Parameter finishes, upgrade the global information element, the update rule of global information element is: ${\mathrm{\&tau;}}_{\mathrm{ij}}\&LeftArrow;(1-\mathrm{\&rho;}){\mathrm{\&tau;}}_{\mathrm{ij}}+\mathrm{\&rho;\&Delta;}{\mathrm{\&tau;}}_{\mathrm{ij}}^{\mathrm{bs}},$ Wherein,  (i, j) ∈ T ^BsΔ τ _Ij ^Bs=B/F (best), F (best) are the valuation functions value of current iteration optimal path.

8. optimization method according to claim 1 is characterized in that, the end condition described in the step e is: set maximum iteration time and/or find the solution multicast path by the precision of optimal solution.

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