On the role of symmetry in solving maximum lifetime problem in two-dimensional sensor networks

We analyze continuous and discrete symmetries of the maximum lifetime problem in two dimensional sensor networks. We show how a symmetry of the network and invariance of the problem under a given transformation group G can be utilized to simplify its solution. We prove that for a G-invariant maximum lifetime problem there exists a G-invariant solution. Constraints which follow from the G-invariance allows us to reduce the problem and its solution to the subset of the sensor network. The subset we call an optimal fundamental region of network with respect to the action of the symmetry group G. We analyze in detail solutions of the maximum network lifetime problem invariant under a group of isometry transformations of a two dimensional Euclidean plane.


Introduction
Let us denote by S K N a sensor network build of N sensors and K data collectors.We split the set S K N into two subsets C K ∪ S N , the set of data collectors C K and the set of sensors S N .Elements p i of the network S K N we identify with points p i = (p 1 i , p 2 i ) of a two dimensional plane R 2 , where p i ∈ C K for i ∈ [1, K] and p i ∈ S N for i ∈ [K + 1, K + N ].Each sensor p i ∈ S N generates periodically the amount Q i of data and sends it, possibly via other sensors, to the data collectors.The data transmission cost energy matrix E i,j defines the energy required to send one unit of data between two elements p i , p j of the network S K N .The energy consumed by the p i sensor to send all of its data in a one cycle of the network lifetime is given by the formula E i (q, p) = K+N j=1,j =i q i,j (p)E i,j (p), (1) where p = (p 1 , ...., p K+N ) and q i,j (p) is the amount of data send by the p i sensor to the p j element of the network S K N .By definition the data collectors p i are elements of the network S K N which do not send any data, i.e., ∀p i ∈ C K ∀p j ∈ S K N q i,j (p) = 0. From the above assumption it follows that for the data collectors the energy given by ( 1) is equal to zero, E i (q, p) = 0, i ∈ [1, K].Because the sensors have limited power of energy, to extend the network lifetime there is a problem to find such transmission graph for the data transmitted in the network, for which the energy consumed by the most overloaded sensor would be minimal.Namely, if we assume that all sensors have the same initial power of energy E 0 and for a given data transmission graph the most overloaded sensor consumes in a one cycle E max i of energy, then [E 0 /E max i ] is the number of cycles until this sensor runs out of energy.The network lifetime we define as a number of cycles the data can be transmitted in the network until the first sensor runs out of energy, [1], [2].In the paper we will not discuss a particular solutions of the maximum lifetime problem and for our purposes we consider a simplified version of it.We assume that the only parameters which define the problem are the functions E i,j (p) and Q i (p).For example, we assume that the initial power energy of each sensor and the capacity of transmission channels between elements of the network S K N are sufficiently large that at least one solution of the problem exists.We also assume, that the sensors can send their data to any element of the network S K N and the data collectors can receive any amount of data without costs.Under the above assumptions the maximum lifetime problem for a sensor network S K N can be written in the form where the vector v 0 has the form (−1, ..., −1 and E T is a transposition of the data transmission cost energy matrix E. The undetermined numbers q (c) In the first equation of (2) we minimize the objective function of the maximum lifetime problem with respect to the q i,j variables.The second equation defines the energy consumed by each sensor to send all of its data in a one cycle of the network lifetime, and it is a matrix form of (1).The third equation in ( 2) is a data transmission flow conservation constrain, which states that the amount data Q i (p) generated by the p i sensor and the amount of data received from other sensors j q j,i must be equal to the amount of data j q i,j the p i sensor can send.The notation in (2) indicates that in general the functions E i,j (p), Q i (p) and the solutions q i,j (p) may depend not only on the coordinates of the points p i and p j but also on the coordinates of other elements of the network S K N and thus the functions may have a non-local character.In [3] it was identified five power-aware metrics for data transmission in mobile ad-hoc networks, which can be used to define a network lifetime.The above definition of network lifetime problem is equivalent to the minimization of the 'maximum node cost', defined by the fifth metric discussed in [3].
In the paper we discuss two types of symmetry of the maximum lifetime problem and impact of these symmetries on the solution of (2).We show, how a symmetry of the problem (2) and a symmetry of the set S K N can be used to simplify the solution of (2) and reduce it to some subset of S K N .The first type of symmetry is related with an invariance of the problem (2) under a continuous group of transformation G of the two dimensional plane R 2 onto itself in which the sensor network S N K is embedded.Under a transformation g ∈ G the elements p i of the network S K N are moved to another location g(p) = p ∈ S K N of the plane R 2 .We assume that the numbers of sensors N and data collectors K under these transformation remain unchanged.If the equations (2) are invariant under transformations group G, then there arises a question whether their solutions q i,j (p) for S N K and q i,j ( p ) for S N K coincide.We show in Sections 2, that it is indeed the case.We call this type of symmetry the space symmetry, because it exhibits the global properties of the functions E i,j (p), Q i (p), q i,j (p) and the whole problem in R 2 .For example, if the matrix elements E i,j (p) are functions of Euclidean distance between points p i and p j of S N K , for simplicity we assume that Q i (p) are constant functions, then the problem (2) is invariant under group of isometries of an Euclidean plane R 2 , [4].Invariance of E i,j (p) under isometry transformation means that the cost of data transmission between two elements of S K N do not depend neither on the direction of the transmission nor location of the network S K N in R 2 .This property is called an isotropy property of the problem (2) in R 2 .The second type of symmetry is related with an invariance of (2) under transformations group of the finite set S K N .In this paper we consider a bijective transformations of the set S K N onto itself.By definition, such transformations g are permutations, i.e. g(p i ) = p g(i) , and form a subgroup of symmetric group Π of the set S K N , [5].Because we do not want to mix the sensors and data collectors we assume that the group G acts separately on the sets C K and S N , which means that Thus, in the paper we consider transformation groups G of the network S K N which are subgroups of the symmetric group Π(C K ) ⊕ Π(S N ).In general, symmetries of the functions E i,j (p) and Q i (p) and the whole problem (2) do not have to be related with the shape of the set S K N .We can establish the relation by requiring that, for a given symmetry group G of the set S K N , we will consider only a G-invariant functions E i,j (p), Q i (p) and search for a G-invariant solution of (2).If there is the case, such problem (2) we call a problem with an internal symmetry group, because it is related with the shape of the set S K N .Symmetries quite naturally appeared in a several well known optimization and combinatorial problems, like partitioning or coloring problems.Examples of solving linear programming problems with symmetries can be found in [6].A review of results and techniques for solving a symmetric constraint programming problems can be found in [7].Existence of symmetries in a given optimization problem facilitates searching for a solution of it.Usually such problem splits into |G| identical parts, where |G| is an order of a symmetry group, and it is enough to solve the problem only for a one part.Solving given problem on a reduced part is called a 'symmetry breaking' procedure, because the reduced problem looses its symmetry, [6], [7].In this paper we utilize the technique of a 'symmetry breaking' to simplify solution of the G-invariant maximum lifetime problem in sensor networks.A 'symmetry breaking' of G-invariant problem ( 2) is performed by reduction of its solution to the optimal fundamental region F 0 ⊂ S K N of the symmetry group G.Because a selection of a fundamental region F for given symmetry group G is not unique, and not for all fundamental regions the problem (2) can be reduced, we show in the paper how to construct the optimal one and we investigate its properties.It is not always evident that for a given optimization problem with a symmetry group G there exists a G-invariant solution of it.The main result of the paper is a theorem, which states that for a considered G-invariant maximum lifetime problem there exists a symmetric (G-invariant) solution.In the paper we investigate in detail properties of the isometry invariant solutions of the problem in two dimensional sensor networks.It seems, that the presented paper is a first attempt of analyzing symmetries and utilize methods of solving optimization problems in sensor networks by means of their symmetry groups.

Space symmetry of the maximum lifetime problem
We consider a group G of a one-to-one transformations of the two dimensional plane R 2 onto itself.Elements g of the group G transform points p of R 2 to some other points p = g(p) in R 2 .Because S K N ⊂ R 2 , the points of the network p i are transformed under g according to the equation p i = g(p i ).The transformed network g(S K N ) by an element g of G we denote by S K N .We assume, that the functions E i,j (p) and Q i (p) in ( 2) are invariant under transformations of the group G, which means that ∀g ∈ G we have where g(p) = (g(p 1 ), ...., g(p K+N )) and Proof.The invariance of q i,j (p) under transformation group G means that ∀ g ∈ G, the equations q i,j (g(p)) = q i,j (p) are satisfied.Since the functions Q i (p) are G-invariant, the feasible set given by the second equation in ( 2) is G-invariant The invariance of the feasible set under G means that the scopes of the parameters q and q in (2) for both networks S K N and S K N are the same.From the condition (5) and the invariance of a feasible set it follows that the function f (q, p), given by ( 3), is G-invariant with respect to the p variable ∀g ∈ G f (q, p) = f (q, g(p)).
The above equation means that f (q, p) is a constant function for any g ∈ G and the solution q i,j (p) of ( 2), as a minimal value of f (q, p), satisfies which is a G-invariance condition for q(p).
As an example of the application of the above proposition, let us consider a data transmission cost energy matrix E i,j (p) to be a function of the Euclidean distance between two elements p i , p j of the network S K N , i.e., The group which leaves invariant the metric ( 6) is a direct sum of two continuous abelian groups G = O 2 ⊕ T , the orthogonal group O 2 and the translation group ⊕T and E i,j (p) is of the form ( 7), then we know from the Proposition 1 that the solution of ( 2) is also G-invariant, which means that it must be a function of the distance d and the G-invariant functions Further analysis of the equations (2) allow us to deduce that the solution of (2) must be a linear function in Q i (p).We see on the above example, that simple analysis of symmetries of the functions E(p) and Q(p) and the structure of the equations (2) allows us to predict general form of its solution which greatly facilitates searching for it.

Internal symmetry of the sensor network
As an internal symmetry group G of a sensor network S K N we consider a one-to-one transformations of S K N onto itself.Such groups are subgroups of a symmetric group Π(S K N ).Because we cannot mix the sensors and the data collectors, the group G must be a direct sum of two subgroups Π(S N ) and Π(C K ), which transform sensors into sensors and data collectors into data collectors, see (4).For an action of g ∈ G on elements p i of the network we have the following transformation rules for E and Q In a matrix representation of G, the elements g ∈ G act on S K N as a linear transformations p i = j g i ,j p j , which induces the following transformations on the matrix E and the vector Q The next proposition shows that the problem ( 2) is covariant under action of the group Π(C K ) ⊕ Π(S N ).It means, that for any transformation (8) of E i,j and Q i , by an element g ∈ Π(C K )⊕Π(S N ), the solutions q and q of (2) are related by the transformation g.Proposition 2. Let q and q be solutions of the maximum lifetime problem (2) for (S K N , E, Q) and (S K N , E , Q ) sensor networks, where Proof.Since any permutation can be written as a product of transpositions, it is enough to check (9) for the transpositions The feasible set, defined by the second equation in ( 2) where h i (q, p) = K+N j=1 (q i,j − q j,i ) − Q i , is unchanged under the transpositions (10), because of the relations i .The transpositions (10) exchange the energies consumed by the r-th and r -th sensors g(E r ) = E r and remain unchanged energies of other sensors g(E i ) = E i , for i = r, r , where E i is given by (1).From this follows that the objective function of the maximum lifetime problem given by ( 3) is invariant under (10).The invariance of f (q, E, Q) under Π(C K ) ⊕ Π(S N ) means that ∀g ∈ G and for E = g(E), Q = g(Q), q = g(q) the functions are equal, f (q, E, Q) = f (q , E , Q ), and have the same minimal value with respect to the q variable min q f (q, E, Q) = min q f (q , E , Q ).
From the above equation it follows that, if q i ,j is a solution of (2) for E , Q then the matrix q i,j = g −1 (q i ,j ) is a solution of (2) for E , Q.
In the next section, based on the result of the Proposition 2 we show that for a G-invariant functions E i,j and Q i , where G is a subgroup of Π(C K ) ⊕ Π(S N ), there exists a G-invariant solution of (2).

G invariant solution of the maximum lifetime problem
The G-invariance of E i,j and Q i means that for any transformation (8) we have In the following theorem we construct a G-invariant solutions of (2).
Theorem 1.Let G be a symmetry group of the set S K N , which transforms sensors into sensors and data collectors into data collectors, E and Q be a G-invariant functions in (2), then there exists a G-invariant solution q of (2), i.e., ∀ g∈G gqg −1 = q. (11) Proof.We assume that for a given G-invariant matrix E and a vector Q there exists, not necessary a G-invariant, solution of ( 2).We denote it by q 0 .From the Proposition 2 we know that are solutions of (2) for the same E and Q.We show that any linear combination of q m q(λ 0 , ..., λ where λ m ≥ 0, m λ m > 0, M = |G| is also a solution of (2).We order the energies consumed by all senors from S N in a decreasing sequence where E i (q m ) is given by ( 1) and m ∈ [0, M − 1].For λ m ≥ 0, m λ m > 0, from (13) it follows that Because all q m are solutions of (2) for the same functions E, Q, then ∀ m E 1 (q m ) = E 1 (q 0 ), and ( 14) From the relation ( m λ m )E 1 (q 0 ) = E 1 ( m λ m q m ) it follows that any matrix of the form ( 12) is a solution of (2).Now, it is easy to show that for a symmetry group G of the problem (2) the matrix satisfies (11) and from ( 12)-( 14) it follows that it is a solution of (2).The G-invariance of (15) follows from the identity, ∀g n ∈ G where we used the fact that ∀ gn∈G g n G = G.
4 Reduction of maximum lifetime problem to the subset of S K N A G-invariant solution of the maximum lifetime problem (2) satisfy constrain (11), which can be used to reduce the number of variables q i,j and thus simplify the solution of (2).On other hand, we would like to relate the shape of the network S K N and its symmetry with the invariance of a solution q i,j of (2).This would allow us to determine a solution of (2) based on the shape of the network S K N .One can easily relate the shape of the network S K N with a symmetry of the data transmission cost energy matrix E i,j (p) by requiring that the matrix elements E i,j (p) are functions of the distance d(p i , p j ) between points of S K N .Transformations which preserve the distance (6) form a group of isometries of an Euclidean plane R 2 .Every isometry of a real Euclidean space is a composition of a translation and an orthogonal transformation, [4].For a finite set S K N in R 2 every isometry is an orthogonal transformation g ∈ O 2 , since there is no translation which transforms a finite set onto itself.In this section we consider as a symmetry group G of the sensor network S K N subgroups of the orthogonal group O 2 and assume that the functions E i,j and Q i are O 2 -invariant.The data transmission cost energy matrices E i,j , which are functions of the distance between elements of the sensor network are widely used in data transmission models in sensor networks.For example, in [2] it was consider a maximum lifetime problem for a one dimensional regular sensor network with the data transmission cost energy matrix of the form E i,j ∼ |i − j| α e γ|i−j| , α ≥ 2, γ ≥ 0. One may ask, what is the general form of the O 2 -invariant data transmission cost energy matrix E i,j .The matrix E = n λ n d an , which is a linear combination of powers of the matrix d, see (6), is trivially O 2 -invariant.The next simple lemma shows that O 2 -invariant E i,j matrix can be build of powers of the matrix elements of the metric d i,j .
Lemma 1.For the group O 2 of the orthogonal transformations of the Euclidean plane R 2 the matrix Proof.Since any orthogonal transformation g ∈ O 2 act as permutation on the set S K N and ∀g ∈ G d(p g(i) , p g(j) ) = d(p i , p j ), then for any a ∈ R we have the following implication from which follows the O 2 -invariance of E given by ( 16).
A linear action of the orthogonal group O 2 on the set S K N and the results of the Lemma 1 allows us to write a general form of the O 2 -invariant data transmission cost energy matrix E i,j where λ = (λ 1 , ...), ā = (a 1 , ...) and λ n , a n ∈ R. Most of the data transmission models in sensor networks utilize the data transmission cost energy functions given by ( 17).Because a finite subgroup of the orthogonal group O 2 is either a dihedral group D M or a rotation group R (M ) , [4], [5], we consider these groups as symmetries of the sensor network S K N and the functions E i,j and Q i .We show, that in both cases the problem (2) can be reduced to subset of S K N which we call an optimal fundamental region F (opt) 0 ⊂ S K N .For the dihedral symmetry group D M the optimal fundamental region can be easily determined.In case of the rotation group R (M ) the optimal fundamental region must be determined for a particular distribution of elements of S K N over the plane R 2 .Reduction of the maximum lifetime problem for sensor network (2) to the optimal fundamental region is proved under two assumptions.We assume, that two sensors can not exchange their data q i,j = 0 ⇒ q j,i = 0, p i , p j ∈ S N , which means that if the sensor p i sends some data to the sensor p j , then the sensor p j cannot send any data to p i .Second assumption is that we consider only a G-invariant E i,j matrices having the property which means that for such matrices the costs of data transmission grows whenever the distance between elements of the network S K N grows.

Sensor network with a dihedral symmetry group
The dihedral group D M is a symmetry group of a regular polygon with M sides.It is a semidirect product R (M ) S (M ) of the rotation group R (M ) and the reflection group S (M ) .It consists of 2M elements, M reflections {S m } M −1 m=0 and M rotations {R m } M −1 m=0 .Because any rotation can be represented as a product of even reflections then the dihedral group can be generated by M reflections, [5].The first requirement means that there are no elements of the network S K N on the reflection lines X m .The second requirements states that the set S K N is a disjoint union of |D M | = 2M subsets F m , and S K N = m∈[0,2M −1] F m .The requirement, that there are no elements of the network S K N on the reflection lines X m , m ∈ [0, M − 1], can be written as which means that the stabilizer of any point p ∈ S K N is trivial.The selection of a fundamental region F ⊂ S K N is not unique.Among fundamental regions in S K N there is only one for which the D M -invariant problem (2) can be reduced.Let V 0 be a region between X-axis, X ≥ 0, and the reflection line X 1 ≥ 0. We assume, that the element S 0 of the dihedral group D M is a reflection along the X-axis and S 1 along the X 1 line.There exists only one fundamental region F 0 which is a subset of V 0 .From the set F 0 , by the following sequence of transformations , where S m are reflections of D M , we can generate |G| = 2M disjoint sets F m and represent the network S K N as a sum of them, S K N = 2M −1 m=0 F m .We show that the fundamental region F 0 is an optimal one, which means that there exists a D M -invariant solution q i,j of (2) for which q i,j = 0 ⇒ p i , p j ∈ F m and the matrix q i,j can be factorized into 2M identical sub-matrices, a one sub-matrix for each region F m .In other words, we show that the regions F m are closed for data transmission and inside each of them the data transmission paths are identical, so it is enough to find the solution of (2) in one of them, for example in F 0 .The reduction of the D M -invariant problem (2) to the subset of S K N is also possible when the data collectors are located on the reflection lines X m .Let us denote by ∂V 0 the border of the angle V 0 .The set ∂V 0 is a sum of two half-lines X 0 ≥ 0 and X 1 ≥ 0. By C 0 and C 1 we denote the set of data collectors which are located on the half-line X 0 ≥ 0 and X 1 ≥ 0 respectively.The set of data collectors C (X) which lie on all the reflections lines is the sum of the sets S m (C 0 ) and S m (C 1 ), i.e., C (X) = M −1 m=0 (S m (C 0 ) ∪ S m (C 1 )), and it is a D M -invariant.We add the elements of the set C (X) to the sensor network S K N and obtain a D M -invariant set S K N ∪ C (X) .The Figure 1 shows the set F 0 ∪ C 0 ∪ C 1 for D 4 group.In the fundamental region F 0 there are seven sensors and one data collector, three data collectors are located at the border of V 0 and lie in the sets C 0 and C 1 .The following proposition shows that the D M -invariant problem (2) over D M -invariant network S K N ∪ C (X) , with the data transmission cost energy matrix E i,j having the property (19), can be reduced to the region F 0 ∪ C 0 ∪ C 1 .
Proposition 3. Let D M be a symmetry group of the set S K N ∪ C (X) with trivial stabilizer for each p i ∈ S K N , F 0 ⊂ V 0 be a fundamental region in S K N for the group D M , C 0 , C 1 are the sets of data collectors which lie on the border of V 0 , then the solution of a D M -invariant problem ( 2) with E satisfying (19) can be reduced to Proof.From the Theorem 1 we know that for a D M -invariant problem (2) there exists a D Minvariant solution.We show that the D M -invariant solution q i,j of (2) can be factorized to 2M copies, and each copy is identical to the solution of (2) in the region F 0 ∪ C 0 ∪ C 1 .For a given reflection S m ∈ D M we write the set S K N ∪C (X) as sum of three subsets m , where C m is a set of data collectors which lie on the reflection line X m and (S K N ) (a) m , a = 1, 2 are sets of sensor network S K N elements which lie on both sides of X m .On the reflection line X m there is a set of data collectors isometric to the set C 0 ∪ C 1 for M odd, C 0 ∪ C 0 for M even and m even and C 1 ∪ C 1 for M even and m odd.The elements of (S K N ) m we denote by p a.i , a = 1, 2, where S m (p 1.i ) = p 2.i .The invariance of the solution q of (2) under transformation S m q(S m ) −1 = q can be written in the form    q 1.i,2.j= q 2.i,1.j, q 1.i,2.i= 0, q 1.i,1.j= q 2.i,2.j, for all p a.i ∈ (S K N ) m , a = 1, 2. The second equation in (20) follows from the requirement (18).From the geometric properties of the reflection symmetry we always have d(p 1.i , p 1.j ) ≤ d(p 1.i , p 2.j ), and because of the assumption (19), we get the set of inequalities E 1.i,1.j≤ E 1.i,2.jand E 2.i,2.j≤ E 2.i,1.j .From the above inequalities it follows that for a D M -invariant solution q of (2) for which q 1.i,2.j= q 2.i,1.j= 0, we can find a solution q for which q 1.i,2.j= q 1.j,2.i= 0 and q 1.i,1.j= q 2.i,2.j= q 1.i,2.j .This means that we can construct a D M -invariant solution q of (2) for which there is no data transmission across the reflection line X m and inside the sets (S K N ) m , (S K N ) m the data transmission is given by the same solution.From the definition of the maximum network lifetime problem (2) and the property (19) of the data transmission cost energy matrix E it follows that the sensors always send their data to the nearest data collector This means that the data collectors cannot receive data from senors which lie behind the reflection line X m .These properties are valid for any reflection S m , m ∈ [0, M − 1] of D M and from this it follows that the data is not sent across any reflection line X m , m ∈ [0, M − 1].Because of the symmetry, inside each of the 2M regions the solutions of (2) are identical and can be represented by a solution in F 0 ∪ C 0 ∪ C 1 .
In Figure 2, the dashed arrows indicate the optimal data transmission between sensors which lie on both sides of the reflection line X m .In the Proposition 3 the assumption that the set C (X) is D M -invariant can be omitted.In such case the network S K N ∪ C (X) and the solution of a D M -invariant problem (2) can be factorized into 2M parts, but in each part the solutions are different, due to the difference of the data collector sets on various reflection lines X m .

Sensor network with a rotation symmetry group
, where R m denotes a rotation by the angle α m = 2π M m.We assume that the rotations are around the point Proof.Let us assume that for R (M ) -invariant solution q of (2) the p m.i sensor sends to the sensor or to the data collector p n.j the amount q m.i,n.j of data, i = j.From (21) we know that the same amount of data is sent from the p (m+k).i sensor to the p (n+k).j sensor or the data collector, k ∈ [0, M − 1].As a result, each element of a sensor network from the j-th orbit receives the same amount of data q m.i,n.jfrom a one sensor from the i-th orbit.Due to the assumption that the data transmission cost energy matrix E i,j satisfies (19), the minimum energy of sending the amount of data q m.i,n.j by the p m.i sensor from the i-th orbit to the element p n.j ∈ S K N ∪ C (0) from the j-th orbit is achieved when the distance between p m.i and p n.j is minimal, i.e., it has the property ∀ p n .j∈S K N ∪C (0) d(p m.i , p n.j ) ≤ d(p m.i , p n .j).In Figure 3   Because the rotation group R (M ) is abelian the constrains R m qR −1 m = q, m ∈ [0, M − 1] which satisfy the R (M ) -invariant solution of (2) are not very restrictive.In general from the rotation invariance we cannot determine the optimal fundamental region F (opt) 0 , i.e., a region to which the problem (2) can be reduced.The next three propositions describe the size and location of the optimal fundamental region F (opt) 0 in S K N .Let us denote by V ± 0 two sub-regions of V 0 such that where the points of V − 0 lie on or between the X-axis, X ≥ 0 and the half-line p 2 = tan[ α 1 2 ]p 1 , p 1 ≥ 0. The region V + 0 lie above of V − 0 .The Figure 4 shows the location of the regions V ∓ 0 in V 0 .The regions V ± m can be obtained by rotating V ± 0 by elements of the group R (M ) , V ± m = R m (V ± 0 ).The next proposition shows, that the sensors which lie in V m can send their data to the data collectors or other sensors only when they lie in V m or in the neighboring regions V ± m∓1 .Proposition 6.Let q be a solution of R (M ) -invariant problem (2) for S K N ∪ C (0) network with E i,j satisfying (19), then the sensors p m.i from the region V m can send their data to the elements

Figure 1 :
Figure 1: Fundamental region F 0 for a dihedral group D 4 and nonempty sets C 0 and C 1 .

Figure 2 :
Figure 2: The dashed arrows indicate the optimal data transmission between two sensors.
the dashed arrows indicate the optimal data transmission between sensors p m,i from the i-th orbit and sensors or data collectors p m,j from the j-th orbit, m ∈ [0, M − 1].

Figure 3 :
Figure 3: The dashed arrows indicate the optimal data transmission between elements of S K N which lie on different orbits.

Figure 4 :
Figure 4: Location of the regions V ∓ 0 in V 0 .