Algebraic Structures Applied for Determining the Configurations in the n-Dimensional Orthogonal Pseudo-Polytopes Yaxal Arenas1, Ricardo Pérez-Aguila2

Departamento de Física y Matemáticas. Centro de Investigación en Tecnologías de Información y Automatización (CENTIA). Universidad de las Américas, Puebla (UDLAP) Ex-Hacienda Santa Catarina Mártir, México, 72820, Cholula, Puebla E-mails: [email protected], [email protected] 1

2

Abstract This article will describe some tools and algebraic structures that could support us in the task of obtaining in a more direct way the configurations in the n-Dimensional Orthogonal Pseudo-Polytopes. In order to speed up the determination of the topological equivalence between a pair of configurations, we describe a relation whose implementation compares any two configurations in a time which only depends of the number of hyper-octants in the space in which their hyper-boxes are embedded. We will show that our relation is in fact an equivalence relation which is ‘wider’ than equivalence relations based in geometrical transformations and therefore it provides an approximate solution to our problem. Index Terms: Algebraic Structures, Computational Geometry, Geometric and Topological Modeling. -

I. I

NTRODUCTION

A. The n-Dimensional Orthogonal Pseudo-Polytopes

Coxeter [6] defines an Euclidean polytope Πn as a finite region of the n-dimensional space enclosed by a finite number of (n-1)-dimensional hyperplanes. The finiteness of the region implies that the number Nn-1 of bounding hyperplanes satisfies the inequality Nn-1>n. The part of the polytope that lies on one of these hyperplanes is called a cell. Each cell of a Πn is a (n-1)-dimensional polytope Πn-1. The cells of a Πn-1 are Πn-2’s, and so on; we thus obtain a descending sequence of boundary elements Πn-3, Πn-4,…, Π1 (an edge), Π0 (a vertex). The n-dimensional Orthogonal

are Πn’s with all their Πn-1's, Πn-2's,…, Π1's oriented in n orthogonal directions. Finally, n-dimensional Orthogonal Pseudo-Polytopes (nD-OPP) are defined as nD orthogonal polytopes with non-manifold boundary [2]. Polytopes

B. Configurations for the nD-OPP’s A set of quasi-disjoint nD hyper-boxes determines a nD-OPP whose vertices must coincide with some of the hyper-boxes’ vertices [2]. We consider the hyper-boxes’ vertices as the origin of a nD local coordinate system, and they may belong to up to 2n hyper-boxes, one for each local hyper-octant. The nD-OPP’s vertices are determined

according to the presence or absence of each of these 2n surrounding hyper-boxes. In this work, an adjacency relation (or just adjacency) between two hyper-boxes refers to the intersection of those hyper-boxes [2]. The n possible adjacency relations between the 2n possible hyper-boxes can be of Π0 (vertex adjacency), Π1 (edge adjacency), …, Πn-2 ((n-2)D adjacency) or Πn-1 ((n-1)D adjacency). There is only one adjacency between any two hyper-boxes. There are

n

[8] possible combinations, which according to an 2 equivalence relation, can be grouped in equivalence classes also called configurations [9]. n n Let ƒ be the set of linear transformations in R generated by all possible compositions of reflections and rotations and their inverses, in any order, with repetition of these geometric transformations allowed. It is well known n that ƒ forms a group [6]. Let x and y be two combinations of n hyper-boxes (2 )

D . Then, we can define the equivalence relation R as x R y⇔(∃f∈ƒn)(f(x)=y). Under this equivalence relation, we have that the 2 possible combinations of hyper-boxes can be grouped into at least 2 [12] configurations. In the cases for the 1D-OPP’s, 2D-OPP’s, 3D-OPP’s and 4D-OPP’s we have 3, 6, 22 [1] and 402 [8] configurations respectively under equivalence relation R . II. THE PROBLEM OF DETERMINING THE CONFIGURATIONS FOR THE nD-OPP’s (n > 4) For the Euclidean n-Dimensional space we have 2n possible hyper-octants (4 quadrants for 2D space, 8 octants for 3D space, and 16 hyper-octants for 4D space). This number of hyper-octants has repercussion over the possible number of combinations of vertices described through the presence or absence of hyper-boxes each one in every hyper-octant. In 4D space we have 2 =65,536 possible combinations. Hill [8] determined that there are 402 configurations for 4D-OPP’s through the relation R . However, if we want to determine the configurations for 5D Orthogonal Pseudo-Polytopes (5D-OPP’s) through exhaustive searching, we would have to consider that there are 32 hyper-octants in 5D space, and for instance, to analyze 2 =4,294,967,296 combinations. Moreover, if the number of configurations is associated with the total number of combinations, it is evident that the first one is very less than the second one. For example, in 3D space we have 22 configurations for 256 possible combinations; this can be translated as that only the 8% of the combinations can perform the role of representatives [14]. See Table 1 for the application of this comparison over the configurations in 1D, 2D, 3D and 4D spaces. f

f

n

(2 )

(2

n− 2

)

f

16

f

32

Table 1. Percentages between the number of combinations and configurations for the nD-OPP’s. nD Space Combinations Configurations Percentage

1D 2D 3D 4D

4 16 256 65,536

3 6 22 402

75 % 37.5 % 8% 0.6 %

These situations lead us to conclude that the complexity imposed by the exhaustive searching makes difficult to determine the configurations for OPP’s in spaces of 5 dimensions and beyond [8]. In the following sections we will describe some tools that could support us in the task of obtaining the configurations in a more direct way. In section V, we will introduce an equivalence relation that will approximate the configurations obtained respect to equivalence relation Rf. Moreover, in section IV, we will introduce a scheme for representing combinations of hyperboxes which is based in binary strings. In the following section we will present some mathematical structures from

Such scheme, which in fact forms a vector space, will allow the extraction of topological characteristics of the combinations of hyperboxes. The new equivalence relation together with the binary representation will allow us to compare two combinations of hyper-boxes in a time which only depends of the number of hyper-octants in the space in which the hyper-boxes are embedded. Some of the equivalence classes that are produced by our relation are also equivalence classes under the relation Rf, however, there exist equivalence classes in Rf whose union composes an equivalence class under our relation. This last property will allow us to conclude that the partition, induced by the proposed equivalence relation, produces an approximation of the partition induced by relation Rf, but with the advantage of temporal complexity reduced considerably. which our scheme is supported.

DEVELOPMENT Lemma 3.1 [13]: The set G={0,1} under the AND (∧) III. THEORETICAL

operand forms a monoid.

Proof: The following properties are satisfied: • Closure: (∀a, b ∈ G)(a ∧ b ∈ G) • Associativity: (∀a, b, c ∈ G) (a ∧ (b ∧ c) = (a ∧ b) ∧ c) • Existence of the identity element: (∃ 1 ∈ G)(a ∧ 1 = 1 ∧ a = a, ∀a ∈ G) The monoid (G, ∧) is not a group because it not satisfies the existence of an inverse element for each element in G, i.e., (∃0 ∈ G)(∀a ∈ G)(a ∧ 0 = 0 ∧ a ≠ 1) Lemma 3.2 [13]: The set G={0,1} under the OR (∨) operand forms a monoid. Proof: The following properties are satisfied: • Closure: (∀a, b ∈ G)(a ∨ b ∈ G) • Associativity: (∀a, b, c ∈ G) (a ∨ (b ∨ c) = (a ∨ b) ∨ c) • Existence of the identity element: (∃0 ∈ G)(a ∨ 0 = 0 ∨ a = a, ∀a ∈ G) The monoid (G, ∨) is not a group because it not satisfies the existence of an inverse element for each element in G, i.e., (∃1 ∈ G)(∀a ∈ G)(a ∨ 1 = 1 ∨ a ≠ 0) Lemma 3.3 [13]: The set G={0,1} under the XOR (⊗) operand forms an Abelian group. Proof: The following properties are satisfied: • Closure: (∀a, b ∈ G)(a ⊗ b ∈ G) • Associativity: (∀a, b, c ∈ G)(a⊗(b⊗c) = (a⊗b)⊗c)

Existence of the identity element: (∃0 ∈ G)(a ⊗ 0 = 0 ⊗ a = a, ∀a ∈ G) • Existence of an inverse element for each element in G: (∀a ∈ G)(∃b ∈ G) (a ⊗ b = b ⊗ a = 0) • Commutativity (∀a, b ∈ G) (a ⊗ b = b ⊗ a) Theorem 3.1 [13]: (G, XOR, AND) form a ring. Proof: The following properties are satisfied: • By Lemma 3.3, (G, ⊗) is an Abelian group. • Right distributivity: (∀a,b,c∈G)(a∧(b⊗c)=a∧b⊗a∧c). • Left distributivity: (∀a,b,c∈G)((b⊗c)∧a=b∧a⊗b∧c). • Associativity of AND over G: (∀a, b, c ∈ G) (a ∧ (b ∧ c) = (a ∧ b) ∧ c). (G, XOR, OR) do not form a ring because right distributivity is not satisfied, i.e., (∃0, 1 ∈ G)(1∨ (0 ⊗ 0) ≠ 1 ∨ 0 ⊗ 0 ∨ 1). Theorem 3.2 [13]: The ring (G, XOR, AND) is a field. Proof: The following properties are satisfied: • Existence of the “multiplicative” identity element: (∃1 ≠ 0 ∈ G)(a ∧ 1 = a, ∀a ∈ G) • Existence of the “multiplicative” inverse element: (∀a ∈ G, a ≠ 0)(∃a-1 ∈ G)(a ∧ a-1 = 1) •

[14]: Consider the set G = {0,1}. The set of vectors Bn, n ≥ 1, is defined as:

Definition 3.1

14243

B = G × ... × G ={( x1 ,..., x n

2n

Let

Definition 3.2:

x=(

2n

x ∈ G, i=1,…,2

):

n

i

}

x1 ,..., x2 n ) and y=( y1 ,..., y 2 n ) be

vectors in B . The vector addition in Bn is defined as: n

+:

B

n

× Bn (x , y )

a →

B

n

x+y Where x + y = ( x1 ⊗ y1 , ..., x2 n ⊗ y 2 n ) Definition 3.3:

Let

x = (

n x1 ,..., x2 n ) a vector in B and let n

a ∈ G. The scalar multiplication in B is defined as: n n B ⋅: → B (a, x) a⋅x Where a ⋅ x = a⋅( x1 ,..., x n ) = (a ∧ x1 , ..., a ∧ x n )

a

2

2

Theorem 3.3: The set Bn is a vector space over the field (G, XOR, AND). Proof: Let x=( x ,..., x y y ,..., y ) and z=( z ,..., z ) 1

be vectors in B

are satisfied:

n

2n

),

=(

2n

1

and let a, b ∈ G. The following properties

1) Closure of vector addition: By Definition 3.2, x + y = ( x1 ⊗ y1 Because xi , y i ∈ G, i = 1,…, n

n

n

2

, ...,

x2 n ⊗ y 2 n ).

⇒ x ⊗y i

i

∈G

∴(∀x, y ∈ B )(x + y ∈ B ) 2) Associativity of vector addition: x + (y + z) = ( x1 ,..., x2 n ) + ( y1 ⊗ z1 , ..., y 2 n ⊗ z 2n = (

x1 ⊗ ( y1 ⊗ z1 ), ..., x2 n ⊗ ( y 2 n ⊗ z 2n

= ((

2n

1

)

))

x1 ⊗ y1 )⊗ z1 , ..., ( x2 n ⊗ y 2 n )⊗ z 2n ) = (x + y) + z

∴(∀x, y, z ∈ Bn)(x + (y + z) = (x + y) + z)

{

3) Existence of zero vector in vector addition: Let 0 = ( 0,...0 ) ∈ B 2

n

⇒

n

x + 0 = ( x1 ⊗ 0, ..., x n ⊗ 0) = ( x1 ,..., x n ) = x and 2 2 0 + x = (0 ⊗ x1 , ..., 0 ⊗ x n 2

) = (

x1 ,..., x2 n ) = x

∴(∃0 ∈ B )(x + 0 = 0 + x = x, ∀x ∈ Bn) n

n

4) Existence of an inverse element for each element in B in vector addition: Let (-x) = x x + (-x) = ( x1 ⊗ x1 , ..., x n ⊗ x n )

⇒

x

= (- ) +

{

2

2

x = ( 0,...0 ) 2n

∴(∀x ∈ Bn)(∃(-x) ∈ Bn)(x + (-x) = (-x) + x = 0)

5) Commutativity of vector addition: x + y = ( x1 ⊗ y1 , ..., x n ⊗ y n 2

= (

)

2

1

y1 ⊗ x1 , ..., y 2 n ⊗ x2 n ) = y + x

Definition 4.1

∴(∀x, y ∈ Bn)(x + y = y + x)

6) Closure of scalar multiplication: By Definition 3.3, a ⋅ x = (a ∧ x1 , ..., a ∧ Because xi , a ∈ G, i = 1,…, 2

n

⇒a∧ x

i

x2 n )

∈G

n

n

∴(∀a ∈ G)(∀x ∈ B )(a ⋅ x ∈ B ) 7) Associativity of scalar multiplication: (a ∧ b) ⋅ x = ((a ∧ b) ∧ x1 ,..., (a ∧ b) ∧ x2 n ) = (a∧(b∧ x1 ),...,a∧(b∧ x n )) = a ⋅ (b∧ x1 ,...,b∧ x n ) 2

2

)) = a ⋅ (b ⋅ x) 2 ∴(∀a, b ∈ G)(∀x ∈ Bn)((a ∧ b) ⋅ x = a ⋅ (b ⋅ x)) 8) Distributivity of vector sums: a ⋅ (x + y) = a ⋅ ( x1 ⊗ y1 x2 ⊗ y 2 ) = a ⋅ (b ⋅ ( x1 ,..., x

n

, ...,

= (a∧( x1 ⊗ y1 ),...,a∧( x n ⊗ y 2

n

2n

2

n

2n

)=a⋅x+a⋅y

∴(∀a ∈ G)(∀x,y ∈ B )( a ⋅ (x + y) = a ⋅ x + a ⋅ y) 9) Distributivity of scalar sums: (a ⊗ b) ⋅ x = ((a ⊗ b) ∧ x1 ,..., (a ⊗ b) ∧ x2 ) n

Π n−1 Π  n−2 Adjn (a, b) =  M Π  1  Π 0

n

= (a∧ x1 ⊗b∧ x1 ,...,a∧ x n ⊗b∧ x n ) 2

2

= (a∧ x1 ,...,a∧ x n )+(b∧ x1 ,...,b∧ x n ) = a ⋅ x + b ⋅ x

∴(∀a,b ∈ G)(∀x ∈ Bn)((a ⊗ b) ⋅ x = a ⋅ x + b ⋅ x)

10) Existence of the multiplicative identity element: Let 1∈G 1⋅x = (1∧ x1 ,..., 1∧ x n ) = ( x1 , ..., x

⇒

∴(1 ∈ G)(1 ⋅ x = x, ∀x ∈ B

∴B

n

IV. B

)

n

2

2n

R

EPRESENTATION FOR THE THE

nD-OPP's

5

C(a, b)

((n − 1) D adjacency) iff ((n − 2) D adjacency) iff

C(a, b) = n − 1 C(a, b) = n − 2

M

M

M

M

(edge adjacency) (vertex adjacency)

iff iff

C(a, b) = C(a, b) =

1 0

, )

For example, consider the vector representation of the

) =

x

4D combination c=(0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0,0). The adjacencies between its six hyper-boxes are shown in Table 2. Definition 4.3

C

4

(

is vector space over the field (G, ⊗, ∧). INARY

3

In other words: Adjn (a, b) = Π C a b

2

2

:

2

Definition 4.2 [14]: Let Adjn(a, b) be the function that computes the type of adjacency between two n-dimensional hyper-boxes referred through the binary digits that correspond to their respective hyper-octants. Then we will have:

))

= (a∧ x1 ⊗a∧ y1 ,...,a∧ x n ⊗a∧ y

scalar has a value equal to one then its referred hyper-octant is occupied by a hyper-box; otherwise, the hyper-octant is empty. Since we will have 2n scalars in the vector, the position of each scalar can be interpreted as a binary number with n digits ( 0L0 2 … 1L1 2). These n digits will be associated with each one of the nD space's main axes by considering the most significant bit as a reference to the X1 axis, the subsequent bit as a reference to the X2 axis, and so forth until we consider the least significant bit as a reference to the Xn axis. Moreover, if a bit is 0 then we will consider the positive part of the corresponding axis; otherwise, we will consider its negative part. Then, through the binary representation of the position of a bit in the combination's vector we can infer its corresponding hyper-octant. For example, if the 22-th bit in a 5D configuration's vector has a value equal to one, then we infer that there is a hyper-box in the hyper-octant x x x x x because 2210 = 101102. [3] Let be the number of bits that do not change from binary string 'a' respect to binary string 'b'. For example: C(110, 001) = 0 (no bit remains unchanged) C(110, 011) = 1 (bit 1 does not change) C(110, 111) = 2 (bits 0 and 1 do not change) By comparing the binary representations of the positions of two hyper-boxes in a n-dimensional combination we can infer the type of adjacency between them. If C(a, b) is equal to n-1, it implies that n-1 bits do not change and therefore these unchanged bits will refer to the positive or negative parts of n-1 main axes which define specifically a Πn-1 shared cell. If C(a,b)=n-2 then we have an (n-2)-D adjacency (a Πn-2 shared cell); and so forth until the cases when C(a,b)=1 (edge adjacency) and C(a,b)=0 (vertex adjacency).

ONFIGURATIONS IN

A nD-OPP's combination of hyper-boxes can be represented through a vector in the set Bn [3]. The positions of its scalars will indicate the nD space's hyper-octants. If a

[14]: Let

x

= ( x1 ,..., x n ) ∈ 2

B

. Γ(x) will de-

n

note the number of scalars in x such that xi = 1, 1 ≤ i ≤ 2n. For example, in 4D combination c we have that Γ(c)=6. n Definition 4.4 [14]: Let x∈B . We define Adj n Π ( x ) ,

{L

,

i

0 j and x =x =1. i

1L 23 n

n

2

j

k

Table 2. The adjacencies between the six hyper-boxes of a 4D combination (see text for details). Shared k-D cell (0 ≤ k < 4) Shared k-D cell (0 ≤ k < 4) Adj4(0001, 0011) = Π3 Adj4(0011, 1000) = Π1 Adj4(0001, 0101) = Π3 Adj4(0101, 0111) = Π3 Adj4(0001, 0110) = Π1 Adj4(0101, 1000) = Π1 Adj4(0001, 0111) = Π2 Adj4(0101, 0110) = Π2 Adj4(0001, 1000) = Π2 Adj4(0110, 0111) = Π3 Adj4(0011, 0101) = Π2 Adj4(0110, 1000) = Π1 Adj4(0011, 0110) = Π2 Adj4(0111, 1000) = Π0 Adj4(0011, 0111) = Π3

In 4D combination

Adj

Π1

4,

(c) = 4 , Adj

4,

Π2

c

we have that

(c) = 5 and Adj

4,

n

Π3

Adj

4,

Π0

(c ) = 1 ,

(c) = 5 .

[14]: Let x ∈ B . We define to Adj(x) as Adj(x) = (Γ(x), Adjn Π ( x) , ..., Adjn Π ( x) , Adjn Π ( x) ) n 1 1 0

Definition 4.5

,

−

,

,

Therefore, 4D combination c has Adj(c)=(6,5,5,4,1). The Algorithm 1 is based in definitions 4.4 and 4.5 in such way that it will determine, in na combination represented through a vector in the set B , the number of hyperboxes and adjacencies between them. Because the maximum number of hyper-boxes which are present in a combination depends of the number of hyper-octants of the space

in which they are embedded, we have that the execution time of Algorithm 1 is related precisely with this number. Let n h=2 be the number of hyper-octants in the n-dimensional n space. If we consider the combination with 2 hyper-boxes, that is, a hyper-box in each one of the hyper-octants, then 2 we will have that the execution time for this case is Θ(h ).

Such time is an upper-bound for the times related to remain n combinations with 0 to 2 -1 hyper-boxes. n Definition 4.6 [14]: Let x,y∈B . We will say that

Adj(x)=Adj(y) if and only if (Γ(x) = Γ(y)) ∧ ( Adjn Π ,

(

n−1

( x)

= Adj n Π ,

n −1

( y) )

∧ ... ∧

Adjn Π 0 ( x) = Adjn Π 0 ( y ) ) ,

,

V. THE EQUIVALENCE R ELATION Radj In this section we will describe our equivalence relation which is based in the definitions presented in previous section. Also, we will mention the way such relation will allow to improve the determination of the equivalence between two combinations of hyper-boxes. Let , be vectors in the set Bn. Let the binary relation Radj defined as x Radj y ⇔ (Adj(x) = Adj(y))

Definition 5.1:

x y

n

Theorem 5.1 [14]: The binary relation Radj in the set B is

an equivalence relation. n Proof: Let x,y,z ∈ B . There are satisfied the following

properties: Adj(x)=Adj(x) 1) Reflexivity: If x Radj x n ∴(∀x ∈ B )(x Radj x) Adj(y)=Adj(x) 2) Symmetry: If xRadjy Adj(x)=Adj(y) y Rf x ∴(∀x,y ∈ Bn)(x Radj y y Radj x) 3) Transitivity: If x Radj y ∧ y Radj z Adj(x)=Adj(y) ∧ Adj(y)=Adj(z) Adj(x)=Adj(z) x Radj z ∴(∀x,y,z∈Bn)(x Radj y ∧ y Radj z x Radj z) ∴ Relation Radj in Bn is an equivalence relation.

⇒ ⇒ ⇒

⇒

⇒

⇒

⇒

⇒

⇒

Algorithm 1. Computing the adjacencies between the hyper-boxes in a combination represented by a binary string. Input:

The number n of dimensions, n > 0. A binary string x with 2n bits which represents a combination of hyper-boxes.

Output: A vector adj which contains the number of hyper-boxes and the number of each type of adjacencies between them. Procedure Adj (BinaryString x, int n)

int h = 2n // h will be the number of hyper-octants in the n-dimensional space. /* Vector adj will have n available positions. Position zero will contain the number of hyper-boxes, position one will contain the number of vertex adjacencies, position two will contain the number of edge adjacencies, and so on. */ Initialize vector adj_x with all their n scalars equal to zero. int j = 0 while(j < h) if(x[j] = 1) // If scalar j is equal to one, then there exists a hyper-box. adj[0]++ // We update vector adj in position zero because we have found a new hyper-box. BinaryString a = getBinaryRepresentation(j) // We get the binary representation of position j. int k = j + 1 while(k < h) if(x[k] = 1) // If scalar k is equal to 1, then there exists a hyper-box which has an adjacency with hyper-box x[j]. BinaryString b = getBinaryRepresentation(k) // We get the binary representation of position k. /* We apply definition 3.3 in order to determine the type of adjacency between hyper-boxes x[j] and x[k] whose associated hyper-octants are represented though binary strings a and b. */ int adjacency = Adj(A, B) // Due to Adj(A,B)∈[0,n-1] we add the respective value of the adjacency in the position indicated by its output plus one. adj_x[adjacency + 1]++

end-of-if k++

end-of-while end-of-if j++

end-of-while return adj end-of-Procedure

Definition 5.2: The set [y]adj = {x ∈ Bn: x Radj y} will be the equivalence class of vectors x under relation Radj and vector y is their representative, i.e., y is a configuration. Because of execution time of algorithm 1 is Θ(h2), the execution time of an algorithm based in Definition 4.6 will be at most Θ(h2). Through the combination's binary representation and the equivalence relation Radj we have improved the procedure for comparing combinations of hyperboxes in terms of the time and memory complexity. Because a n-dimensional combination can be managed withn only 2n n bits instead of the 2 vertices for each one of the 2 possible hyper-boxes. Furthermore, a combination x of hyper-boxes could be evaluated with at most all the geometric transformations present in the set f n in order to determine if it is equivalent or not, under relation Rf, to a combination y. Now, we will introduce a third equivalence relation which is based in the isomorphism between weighted graphs. Such relationn will allow us to determine that the partition of the set B , induced by relation Radj, produces an approximation of the partition induced by relation Rf. We will refer to an undirected weighted graph G by its sets of vertices V; and edges E, by notation G=(V,E) [5]. Each one of its edges {u,v} will have associated a nonnegative number w({u,v}) called the weight of edge {u,v}. Definition 5.3: Two graphs G=(V,E) and G’=(V’,E’) are isomorphic if there exists a bijection f:V
V’ such that {u,v} is an edge of G ⇔ {f(u),f(v)} is an edge of G’ [5]. Definition 5.4: Let G=(V,E) and G’=(V’,E’) be two weighted graphs. G and G’ are isomorphic weighted graphs, denoted by G ≅ G’, if and only if G and G’ are isomorphic by bijection f:V V’ and (∀{u,v}∈E)(w({u,v})=w({f(u),f(v)})) Definition 5.5: Let G be a weighted graph. The weights sequence of vertex v ∈ V(G), denoted by w(v), is a list in decreasing order of the weights of the incident edges to v [4]. Definition 5.6 [14]: Let x = ( x1 ,..., x n ) ∈ Bn. The adjacencies




2

graph of x, denoted by G(x)=(V(x),E(x)) will be a weighted graph constructed in the following way: • V(x) = {i | x = 1, x ∈ x; i.e., the position of scalar xi equal to one in vector x} • E(x) = {{i, j} | i, j ∈ V(x), i ≠ j} • The weight w({i, j}) of each edge {i, j}∈ E(x) will be given by w({i, j}) = Adjn(i , j ). For example, G(c)=(V(c),E(c)) is described in Table 3. i

i

2

2

Table 3. Adjacencies graph of 4D combination c=(0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0,0)

Vertices V(c) = {1, 3, 5, 6, 7, 8} Edges E(c) = {{1, 3}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {3, 5}, {3, 6}, {3, 7}, Weights of edges

{3, 8}, {5, 6},{5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}

w({1,3})=Adj (0001,0011)=3 w({1,6})=Adj (0001,0110)=1 w({1,8})=Adj (0001,1000)=2 w({3,6})=Adj (0011,0110)=2 w({3,8})=Adj (0011,1000)=1 w({5,7})=Adj (0101,0111)=3 w({6,7})=Adj (0110,0111)=3 w({7,8})=Adj (0111,1000)=0 4

4

4 4 4

4 4 4

w({1,5})=Adj (0001,0101)=3 w({1,7})=Adj (0001,0111)=2 w({3,5})=Adj (0011,0101)=2 w({3,7})=Adj (0011,0111)=3 w({5,6})=Adj (0101,0110)=2 w({5,8})=Adj (0101,1000)=1 w({6,8})=Adj (0110,1000)=1 4 4

4 4 4

4 4

n

Theorem 5.2: (∃x, y ∈ B )(x Radj y ∧ G(x)

≅/

G(y)) n

Proof: We will show that there exist vectors x, y ∈ B such that Adj(x)=Adj(y) but their associated adjacencies graphs G(x) and G(y) are not isomorphic weighted graphs. 4 Let x=c (see previous example). Let y ∈ B defined by y=(1,0,1,1,0,1,0,1,1,0,0,0,0,0,0,0) In Table 4 are shown the corresponding weights sequences of vectors x and y. 4

Table 4. Weight sequences of two vectors in B (see text for details). V(x)={1, 3, 5, 6, 7, 8} V(y) = {0, 2, 3, 5, 7, 8}

w(0) = (3, 3, 2, 2, 1) w(2) = (3, 3, 2, 2, 1) w(3) = (3, 3, 2, 2, 1) w(5) = (3, 2, 2, 1, 1) w(7) = (3, 3, 2, 1, 0) w(8) = (3, 2, 1, 1, 0)

w(1) = (3, 3, 2, 2, 1) w(3) = (3, 3, 2, 2, 1) w(5) = (3, 3, 2, 2, 1) w(6) = (3, 2, 2, 1, 1) w(7) = (3, 3, 3, 2, 0) w(8) = (2, 1, 1, 1, 0)

⇒

y

Because Adj(x)=Adj(y)=(6, 5, 5, 4, 1) x Radj . However, the weights sequence of vertex 8 in G( ), w(8)=(2,1,1,1,0), does not coincide with any weights sequence of the vertices in G( ) (see table 4). ∴G(x) ≅ / G(y)

x

y y ∧ G( x ) ≅ / G(y)) adj Definition 5.7: Let x and y be vectors in the set Bn. Let R

⇒ (∃x, y ∈ B )(x R n

i

be the relation defined by x R y ⇔ G(x) ≅ G(y). n Theorem 5.3 [14]: The relation R defined over the set B is an equivalence relation. n Proof: Let x, y, z ∈ B . There are satisfied the following i

i

properties (‘o’ denotes the function composition operator): 1) Reflexivity: If x R x (∃ι:V(x) V(x),ι-identity mapping)(G(x) ≅ G(x)) ∴(∀x ∈ Bn)(x R x) 2) Symmetry: If x R y (∃ f:V(x) V(y), f-bijective)(G( ) ≅ G( )) Because f is biyective, (∃ f :V( ) V( ), f -bijective)(f o f = f o f = ι). R G( ) ≅ G( ) by bijection f ∴(∀ , ∈ Bn)( R R ) 3) Transitivity: If R ∧ R (∃ f:V( ) V( ), f-bijective)(G( ) ≅ G( )) ∧ V( ), g-bijective)(G( ) ≅ G( )) (∃ g:V( ) (∃ h:V( ) V( ), h-bijective)(h = g o f) G( ) ≅ G( ) by bijection h R ∴(∀ , , ∈ B )( R ∧ R R ) ∴ Relation R in Bn is an equivalence relation. The set [y]i = {x ∈ Bn: x Ri a} will be the equivalence class of vectors x under relation Ri and y is

⇒




i

i

⇒ ⇒ ⇒ ⇒ ⇒ ⇒


y


-1

i

x

x

y

-1

-1

-1

⇒y x x y⇒y x x y y z x
y x y y
z y z x
z x z ⇒x z xyz x y y z⇒x z

y xy

x

-1

i

i

i

i

i

i

n

i

i

i

i

Definition 5.8:

their representative, i.e., y is a configuration. n Theorem 5.3: (∀[x]i ⊂ B )(∃[y]adj)([x]i ⊆ [y]adj) n Proof: Let [x]i be any class in B under relation Ri. (∀z∈[x]i)(G(z) ≅ G(x)) (∀z∈ [x]i)(Adj(z) = Adj(x))

⇒ ⇒ ⇒ (∃[y] )(x∈[y] )⇒(∀z∈[x] )(Adj(z)=Adj(x)∧Adj(x)=Adj(y)) ⇒ (∀z ∈ [x] )(Adj(z) = Adj(y)) ⇒ (∀z ∈ [x] )(z ∈ [y] ) adj

adj

i

∴[x]i ⊆ [y]adj

i

i

adj

According to theorem 5.2, we determined the existence of combinations x and y such that Adj(x) = Adj(y), that is, both belong to the same equivalence class under relation Radj, however x∉[y]i and y∉[x]i. By theorem 5.2 we proved

the existence of equivalence classes under relation Ri such that they can be characterized as subsets of equivalences classes under Radj. Then, we can conclude that there exist n combinations x, y ∈ B such that ([x]i ⊆ [x]adj ∧ [y]i ⊆ [x]adj) where [x]i ∩ [y]i = ∅. Therefore, the equivalence classes induced by Radj can be seen as an approximation of the equivalence classes generated by relation Ri. We will conclude with the following definition:

Definition 5.9: We will say that the partition of the set Bn induced by equivalence relation Radj is coarser than the partition induced by equivalence relation Ri and we will denote it by n n Ri ≥ Radj ⇔ (∀[x]i ⊂ B )(∃[y]adj ⊂ B )([x]i ⊆ [y]adj)

VI. CONCLUSIONS AND FUTURE WORK It is essential to determine the configurations for the nD-OPP’s because they represent a finite subset which can be used to determine geometric and topologic properties for these nD-OPP’s. For example, in [2] there are used only the configurations to determine some properties for 4D-OPP’s. Moreover, two methodologies for determining the configurations for the nD-OPP's are described in [10]: Exhaustive searching and the 'Test-Box' Heuristic. Both procedures have in common that they determine whether two combinations of hyper-boxes x and y are equivalent (i.e. they belong to the same equivalence class) if there exists a function f∈ƒn such that f(x)=y. Both methodologies can be improved in terms of the time and memory complexity by considering the equivalence relation Radj. A formal treatment of this idea will be presented in [11]. Equivalence relation Radj provide us methods faster and more direct to obtain configurations for the nD-OPP’s of 5 dimensions and beyond. According to theorems 5.2 and 5.3, an equivalence class under relation Radj can be seen as the union of equivalence classes under relations Rf or Ri. In this sense, we propose, as future work, the design of new heuristics that allow the determination of those equivalence classes. By this way, given a class in Radj, we could obtain its corresponding classes in Ri or Rf. In another sense, according to section III, also in this paper we have experienced the possibilities to identify algebraic structures in two-valued algebras using, as operators, logic gates, finally achieving to identify some of the most important ones. Other operator, not considered in this work, that forms an Abelian group is the XNOR [13]. This result leads us to consider that ({0,1}, XNOR, OR) compose a field [13]. This field, along with the appropriate definitionsn of vector addition and scalar multiplication, in the set B , could allow us to define a new vector space. It would be interesting to define an homeomorphism between this vector space and the defined one in Theorem 3.3. This mapping between spaces could allow us to perform efficiently operations in one space that would be complex in the original space.

ACKNOWLEDGEMENTS We thank to Jorge Ojeda-Castañeda, PhD (Physics and Mathematics Department, UDLAP) and Antonio Aguilera, PhD (Computer Systems Engineering, UDLAP) for their valuable comments and suggestions. References

[1] Aguilera, A.. Orthogonal Polyhedra: Study and Application. Ph.D. Thesis. Universitat Politècnica de Catalunya, 1998. [2] Aguilera Ramírez A. & Pérez Aguila, R. Classifying the n-2 dimensional elements as Manifold or Non-Manifold for n-Dimensional Orthogonal Pseudo-Polytopes. Proc. of 12 International Conference on Electronics, Communications and Computers CONIELECOMP 2002, pp. 59-63. February 24 to 27, 2002. Acapulco, Guerrero, México. [3] Aguilera, A. & Pérez-Aguila, R. Representing and Computing Some Configuration Properties for the n-Dimensional Orthogonal Pseudo-Polytopes Proc. of 14th International Conference on Electronics, Communications, and Computers CONIELECOMP 2004, pp. 254-259. ISBN: 0-7695-2074-X. February 16 to 18, 2004. Veracruz, Veracruz, México. [4] Buckley, F. & Haraby, F. Distance in Graphs. AddisonWesley Publishing Company, 1990. [5] Cormen, T.H., Leiserson, C.E., Rivest, R.L. and Stein C. Introduction to Algorithms. Second Edition. MIT Press, 2001. [6] Coxeter,H.S.M: Regular Polytopes, Dover Publications, Inc., New York, 1963. [7] de Berg, M.; van Krevald, M.; Overmars, M. & Schwarzkopf, O. Computational Geometry, Algorithms and Aplications. Springer Verlag, 1997. [8] Hill, S. & Roberts J.C. Generating Surface Geometry in Higher Dimensions using Local Cell Tilers. Technical Report 4-98, University of Kent at Canterbury, Computing Laboratory, University of Kent, Canterbury, Kent CT2 7NF, March 1998. [9] Pérez Aguila, R. 4D Orthogonal Polytopes. B.Sc. Thesis. Universidad de las Américas, Puebla, 2001. [10] Pérez Aguila, R. Presenting the ‘Test-Box’ Heuristic for Determining the Configurations for the n-Dimensional Orthogonal Pseudo-Polytopes. Proc. of 13 International Conference on Electronics, Communications and Computers CONIELECOMP 2003, pp. 64-69. Feb. 24 to 26, 2003, Cholula, Puebla, México. [11] Pérez Aguila, R. & Aguilera, A. The ‘Test-Box’ Algorithm as a Novel Approach for Determining the Configurations for the n-Dimensional Orthogonal Pseudo-Polytopes. (Unpublished). [12] Ziegler, G.M. Lectures on Polytopes. Graduate Texts in Mathematics 152, Springer-Verlag, 1994. [13] Arenas, Yaxal; Ibarra, Héctor; Juárez, Montserrat; Parga, Francisco & Valenzuela, Sara. Mathematical Structures for Logic Operators. Submitted for Evaluation to “XVI Congreso Interuniversitario de Electrónica, Computación y Eléctrica CIECE 2006”. To be held April 5 to 7, 2006. Sonora, México. [14] Pérez-Aguila, Ricardo & Aguilera, Antonio. Mathematical Tools for Speeding Up the Determination of Configurations of the n-Dimensional Orthogonal Pseudo-Polytopes. Proceedings of the 2nd International Conference on Electrical and Electronics Engineering and XI Conference on Electrical Engineering ICEEE and CIE 2005, pp. 68-71. Published by the IEEE Computer Society. CD's ISBN: 0-7803-9230-2. September 7 to 9, 2005. México City, México. th

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