for some I ∈ J (Pn−2). Without loss of generality, consider the position (i, j) such that M in(i, j, n − i, n − j) = t for all t ∈ [⌊n2⌋]and that eij∈ [rs − (t − s − 1) , r(s + 1) − (t − s − 1)]. Then we can write
rs − (t − s − 1) ≤ eij≤r(s + 1) − (t − s − 1) ⇔ 0 ≤ eij−rs + t − s − 1 ≤ r
⇔0 ≤ θE(i, j, k) ≤ r.
as required. To investigate the other two conditions (under the above assumptions) we consider the following regions in E:
Reg(1): i + j < n; Reg(2): i + j = n; Reg(3): i + j > n.
Note that by symmetries described in Table 1.7, a given configuration in Reg(2) or Reg(3) is just the anti-main diagonal reflection of the same configuration in Reg(1) and vice versa. Therefore, without loss of generality, it is sufficient to verify conditions (2) and (3) for all possible configurations for a given position (i, j) and its four adjacent neighbours in Reg(1). Also assume that (i, j, k) ∈ Pnis the associated position with (i, j) in Pn(I) ∈ Λ(n, r) in Reg(1). Then depending on the value of (i, j, k), the value of ωn(i, j) and the values of the entries adjacent to it, we can classify all the possible configurations into five disjoint categories each of which contains four (weak or strong) inequalities that need to be verified. As an illustration, the entries on the first shell of a given Pn(I) together with the regions Reg(1) − Reg(3) are shown in Figure 3.8. We begin with the configurations on the first shell that is, when t = 1. We investigate the configurations on the first shell separately since each entry in this shell belongs to the interval [0, r]. In the rest of this section, we will show the strict order relations between adjacent positions in each shell with “>>” or “<<” and each weak inequality with “>” or “<”, respectively.
Figure 3.8: The possible configurations on the first shell of Pn(I).
(a) Horizontal configurations: There are three possible configurations in this case shown in Figure3.9.
(a) (b)
(c)
Figure 3.9: Horizontal configurations.
Without loss of generality, consider the configuration shown in Figure 3.9a. For this configuration, we need to verify the following two sub-cases:
(1) We need to verify that:
θE(1, j, k) ≥ θE(1, j − 1, k4) ⇔ e1j≥e1,j−1
The last inequality e1j≥e1,j−1 is valid since by the definition of E, e1j−e1,j−1≥0 for j ∈ [n]. (2) Secondly, we need to verify that
θE(1, j + 1, k3) ≥θE(1, j, k) ⇔ e1,j+1≥e1,j
Again the last inequality e1,j+1≥ e1,j is valid since by the definition of E, e1,j+1−e1j ≥0 for j ∈ [n].
The verification of the inequalities associated with the configurations shown in 3.9b and 3.11c is very similar to the argument for the case (1) and (2).
(b) Right angled corner configurations: In this configuration, there are four possible scenarios that are shown in Figure3.10.
(a) The top left corner configuration. (b) The top right corner configuration.
(c) The bottom left corner configuration.
(d) The bottom right corner configuration.
Figure 3.10: Four possible right angled corner configurations for the first shell in Pn(I).
Without loss of generality, consider the configuration in Figure3.10a. Then we need to consider the following two subcases:
(1) We need to verify that
θE(1, 2, n − 3) ≥ θE(1, 1, n − 2) ⇔ e12≥e11
The last inequality e12≥e11clearly holds since by the definition of E, e12−e11≥0. (2) We need to verify that
θE(2, 1, n − 3) ≥ θE(1, 1, n − 2) ⇔ e21≥e11
By the same argument e21≥e11holds since by definition of E, e21−e11≥0.
Similar to the case for the horizontal case, the argument for the other three right angled corner configurations shown in Figures3.10b,3.11c and3.10dis very similar.
(c) Vertical configurations: There are three vertical configurations that are shown in Figure3.11.
(a) The vertical configuration on Reg(1). (b) The vertical configuration on Reg(3). (c) The vertical configuration on Reg(2).
Without loss of generality, consider the configuration in Figure 3.11a. Then we have the following two subcases:
(1) We need to verify that
θE(i, 1, k) ≥ θE(i − 1, 1, k6) ⇔ ei,1≥ei−1,1
Clearly the inequality ei,1≥ei−1,1holds since by the definition of E, ei,1−ei−1,1≥0 in Reg(1). (2) We need to verify that
θE(i + 1, 1, k5) ≥θE(i, 1, k) ⇔ ei+1,1≥ei,1
Clearly the inequality ei+1,1≥ei,1holds since by the definition of E, ei+1,1−ei,1≥0 in Reg(1). The verification for the other configurations in Figures3.11band3.11cis very similar.
Now we investigate the five categories of configurations for positions in internal shells in Reg(1). For brevity, we pick one arbitrary configuration from each category to verify the validity of the associated inequalities with it since the argument for the rest is very similar by symmetry.
cat(1): The first category contains 16 configurations (in all regions) each of which consists of four strict inequalities (either>>or<<regarding the position (i, j, k) and its four adjacent positions (i, j −1, k1), (i−1, j, k2), (i, j +1, k3)and (i+1, j, k4). Without loss of generality, consider the configuration shown in Figure (3.12) in Reg(1).
Figure 3.12: The first category containing one configuration with four strict relations.
In this configuration, we need to verify the following inequalities:
θE(i, j, k) >> θE(i, j − 1, k1) (3.23) θE(i, j, k) >> θE(i − 1, j, k2) (3.24) θE(i, j, k) >> θE(i, j + 1, k3) (3.25) θE(i, j, k) >> θE(i + 1, j, k4) (3.26)
M in(i, j − 1) = tij−1 , ei,j−1∈ [rsi,j−1− (tij−si,j−1−2) , r(si,j−1+1) − (tij−si,j−1−2)] ; M in(i − 1, j) = tij−1 , ei−1,j∈ [rsi−1,j− (tij−si−1,j−2) , r(si−1,j+1) − (tij−si−1,j−2)] ; M in(i, j + 1) = tij , ei,j+1∈ [rsi,j+1− (tij−si,j+1−1) , r(si,j+1+1) − (tij−si,j+1−1)] ; M in(i + 1, j) = tij , ei+1,j∈ [rsi+1,j− (tij−si+1,j−1) , r(si+1,j+1) − (tij−si+1,j−1)] .
(3.27) Now by definition of Φ, for (3.23) we need to verify that
θE(i, j, k)>>θE(i, j − 1, k1) ⇔eij−rsij+tij−sij−1>>ei,j−1−rsi,j−1+ti,j−1−si,j−1−2 ⇔eij− (r + 1)(sij−si,j−1) + (tij−ti,j−1) +1>>ei,j−1
(3.28)
To verify the validity of the last inequality in (3.28), we note that by the definition of E (3.15), eij−ei,j−1≥0 (or equivalently eij≥ei,j−1)). Moreover, by our assumptions tij=1 + ti,j−1implies that sij≤si,j−1(and thus −(r +1)(sij−si,j−1) ≥0). Therefore, we can write eij− (r +1)(sij−si,j−1) ≥ei,j−1 and subsequently eij− (r + 1)(sij−si,j−1) +1>>ei,j−1 as required.
The verification for inequality (3.24) is very similar since it is the anti-main diagonal reflection of the given configuration with respect to the position (i, j, k).
For inequality (3.25), we need to verify that
θE(i, j, k)>>θE(i, j + 1, k3) ⇔eij−rsij+tij−sij−1>>ei,j+1−rsi,j+1+ti,j+1−si,j+1−1 ⇔eij− (r + 1)(sij−si,j+1)>>ei−1,j
(3.29)
To verify the last inequality in (3.29), first we note that tij =ti,j+1 implies that si,j+1≥1 + sij (and thus −(r + 1)(sij +1) ≥ −(r + 1)(si,j+1)). Also by the definition of E (3.15), ei,j+1−eij ≤ r (or equivalently eij≥ei,j+1−r). Then we can write
eij− (r + 1)sij− (r + 1) ≥ ei,j+1− (r + 1)si,j+1−r ⇔ eij− (r + 1) (sij−si,j+1) −1 ≥ ei,j+1
⇔ eij− (r + 1) (sij−si,j+1)>>ei,j+1
The verification for the inequality in (3.26) is very similar to the case (3.25) since it is the anti-main diagonal reflection of the given configuration with respect to position (i, j, k).
cat(2): This category consists of 56 configurations (in all regions) each of which contains three strict inequalities (either>>or<<) and one weak inequality (either>or <) regarding the position (i, j, k)
and its four adjacent positions (i, j − 1, k1), (i − 1, j, k2), (i, j + 1, k3)and (i + 1, j, k4). Without loss of generality, consider the configuration in Figure3.13in Reg(1).
Figure 3.13: An arbitrary configuration from second category that consists of three strict inequalities and one weak inequality, respectively.
For this configuration, we need to verify the following inequalities:
θE(i, j, k) >> θE(i, j − 1, k1) (3.30) θE(i, j, k) >> θE(i − 1, j, k2) (3.31)
θE(i, j, k) < θE(i, j + 1, k3) (3.32)
θE(i, j, k) >> θE(i + 1, j, k4) (3.33)
To verify them, let us assume the same assumptions described in (3.27) regarding the shells that these positions belong to. Then the argument for (3.30), (3.31) and (3.33) is exactly the same as the ones we had for (3.23), (3.24) and (3.26) in the first category, respectively. So we only need to verify the inequality (3.32). We have
θE(i, j + 1, k4) ≥θE(i, j, k) ⇔ ei,j+1−rsi,j+1+ti,j+1−si,j+1−1 ≥ eij−rsij+tij−sij−1
⇔ei,j+1− (r + 1)(si,j+1−sij)>eij