intégration des remarques des relecteurs

[16dcc.git] / hamilton.tex

diff --git a/hamilton.tex b/hamilton.tex
index 17d93803a0d523ddadc31cfe930449bc6a996f41..a48fc6254648226359a098a9f4d7e8101dbb0ce2 100644 (file)
--- a/hamilton.tex
+++ b/hamilton.tex
@@ -1,13 +1,13 @@
 Many approaches have been developed to solve the problem of building
 Many approaches have been developed to solve the problem of building

-a Gray code in a $\mathsf{N}$ cube~\cite{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04,Bykov2016}, according to properties 
+a Gray code in a $\mathsf{N}$-cube~\cite{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04,Bykov2016}, according to properties 

 the produced code has to verify.
 For instance,~\cite{DBLP:journals/combinatorics/BhatS96,ZanSup04} focus on
 balanced Gray codes. In the transition sequence of these codes, 
 the number of transitions of each element must differ
 at most by 2.
 the produced code has to verify.
 For instance,~\cite{DBLP:journals/combinatorics/BhatS96,ZanSup04} focus on
 balanced Gray codes. In the transition sequence of these codes, 
 the number of transitions of each element must differ
 at most by 2.

-This uniformity is a global property on the cycle, \textit{i.e.}
+This uniformity is a global property on the cycle, \textit{i.e.},

 a property that is established while traversing the whole cycle.
 a property that is established while traversing the whole cycle.

-On the opposite side, when the objective is to follow a subpart 
+On the other hand, when the objective is to follow a subpart 

 of the Gray code and to switch each element approximately the 
 same amount of times,
 local properties are wished.
 of the Gray code and to switch each element approximately the 
 same amount of times,
 local properties are wished.


@@ -15,16 +15,17 @@ For instance, the locally balanced property is studied in~\cite{Bykov2016}
 and an algorithm that establishes locally balanced Gray codes is given.
  
 The current context is to provide a function 
 and an algorithm that establishes locally balanced Gray codes is given.
  
 The current context is to provide a function 

-$f:\Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ by removing a Hamiltonian 
-cycle in the $\mathsf{N}$ cube. Such a function is going to be iterated
-$b$ times to produce a pseudo random number,
-\textit{i.e.} a vertex in the 
-$\mathsf{N}$ cube.
+$f:\Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ by removing an Hamiltonian 
+cycle in the $\mathsf{N}$-cube. Such a function is going to be iterated
+$b$ times to produce a pseudorandom number,
+\textit{i.e.}, a vertex in the 
+$\mathsf{N}$-cube.

 Obviously, the number of iterations $b$ has to be sufficiently large 
 to provide a uniform output distribution.
 Obviously, the number of iterations $b$ has to be sufficiently large 
 to provide a uniform output distribution.

-To reduce the number of iterations, the provided Gray code
-should ideally possess the both balanced and locally balanced properties.
-However, none of the two algorithms is compatible with the second one:
+To reduce the number of iterations, it can be claimed
+that the provided Gray code
+should ideally possess both balanced and locally balanced properties.
+However, both algorithms are incompatible with the second one:

 balanced Gray codes that are generated by state of the art works~\cite{ZanSup04,DBLP:journals/combinatorics/BhatS96} are not locally balanced. Conversely,
 locally balanced Gray codes yielded by Igor Bykov approach~\cite{Bykov2016}
 are not globally balanced.
 balanced Gray codes that are generated by state of the art works~\cite{ZanSup04,DBLP:journals/combinatorics/BhatS96} are not locally balanced. Conversely,
 locally balanced Gray codes yielded by Igor Bykov approach~\cite{Bykov2016}
 are not globally balanced.


@@ -39,29 +40,29 @@ namely~\cite{Robinson:1981:CS},~\cite{DBLP:journals/combinatorics/BhatS96},
 and~\cite{ZanSup04} have addressed the problem of providing an approach
 to produce balanced gray code.
 The authors of~\cite{Robinson:1981:CS} introduced an inductive approach
 and~\cite{ZanSup04} have addressed the problem of providing an approach
 to produce balanced gray code.
 The authors of~\cite{Robinson:1981:CS} introduced an inductive approach

-aiming at producing balanced Gray codes, provided the user gives 
+aiming at producing balanced Gray codes, assuming the user gives 

 a special subsequence of the transition sequence at each induction step.
 a special subsequence of the transition sequence at each induction step.

-This work have been strengthened in~\cite{DBLP:journals/combinatorics/BhatS96}
-where the authors have explicitly shown how to construct such a subsequence.
+This work has been strengthened in~\cite{DBLP:journals/combinatorics/BhatS96}
+where the authors have explicitly shown how to build such a subsequence.

 Finally the authors of~\cite{ZanSup04} have presented 
 the \emph{Robinson-Cohn extension} 
 Finally the authors of~\cite{ZanSup04} have presented 
 the \emph{Robinson-Cohn extension} 

-algorithm. There rigorous presentation of this one 
-have mainly allowed them to prove two properties.
+algorithm. Their rigorous presentation of this algorithm
+has mainly allowed them to prove two properties.

 The former states that if 
 $\mathsf{N}$ is a 2-power, a balanced Gray code is always totally balanced.
 The latter states that for every $\mathsf{N}$ there 
 The former states that if 
 $\mathsf{N}$ is a 2-power, a balanced Gray code is always totally balanced.
 The latter states that for every $\mathsf{N}$ there 

-exists a Gray code such that all transition count numbers are 
+exists a Gray code such that all transition count numbers 

 are 2-powers whose exponents are either equal
 or differ from each other by 1.
 However, the authors do not prove that the approach allows to build 
 are 2-powers whose exponents are either equal
 or differ from each other by 1.
 However, the authors do not prove that the approach allows to build 

-(totally balanced) Gray code. 
+(totally balanced) Gray codes. 

 What follows shows that this fact is established and first recalls the approach.
 
 
 Let be given a $\mathsf{N}-2$-bit Gray code whose transition sequence is
 $S_{\mathsf{N}-2}$. What follows is the 
  \emph{Robinson-Cohn extension} method~\cite{ZanSup04}
 What follows shows that this fact is established and first recalls the approach.
 
 
 Let be given a $\mathsf{N}-2$-bit Gray code whose transition sequence is
 $S_{\mathsf{N}-2}$. What follows is the 
  \emph{Robinson-Cohn extension} method~\cite{ZanSup04}

-which produces a $n$-bits Gray code.
+which produces a $\mathsf{N}$-bits Gray code.

 
 \begin{enumerate}
 \item \label{item:nondet}Let $l$ be an even positive integer. Find 
 
 \begin{enumerate}
 \item \label{item:nondet}Let $l$ be an even positive integer. Find 


@@ -83,12 +84,12 @@ two elements have been exchanged.
 \end{enumerate} 
 
 It has been proven in~\cite{ZanSup04} that 
 \end{enumerate} 
 
 It has been proven in~\cite{ZanSup04} that 

-$S_{\mathsf{N}}$ is transition sequence of a cyclic $\mathsf{N}$-bits Gray code 
+$S_{\mathsf{N}}$ is the transition sequence of a cyclic $\mathsf{N}$-bits Gray code 

 if $S_{\mathsf{N}-2}$ is. 
 if $S_{\mathsf{N}-2}$ is. 

-However, the step~(\ref{item:nondet}) is not a constructive 
-step that precises how to select the subsequences which ensures that 
+However, step~(\ref{item:nondet}) is not a constructive 
+step that precises how to select the subsequences which ensure that 

 yielded Gray code is balanced.
 yielded Gray code is balanced.

-Next section shows how to choose the sequence $l$ to have the balance property.
+Following sections show how to choose the sequence $l$ to have the balance property.

 
 \subsection{Balanced Codes}
 Let us first recall how to formalize the balance property of a Gray code.
 
 \subsection{Balanced Codes}
 Let us first recall how to formalize the balance property of a Gray code.


@@ -112,20 +113,18 @@ $|\textit{TC}_{\mathsf{N}}(i) - \textit{TC}_{\mathsf{N}}(j)| \le  2$.
 
    
 \begin{xpl}
 
    
 \begin{xpl}

-Let $L^*=000,100,101,001,011,111,110,010$ be the Gray code that corresponds to 
+Let $L^*=000,100,101,001,011,111,$ $110,010$ be the Gray code that corresponds to 

 the Hamiltonian cycle that has been removed in $f^*$.
 Its transition sequence is $S=3,1,3,2,3,1,3,2$ and its transition count function is 
 $\textit{TC}_3(1)= \textit{TC}_3(2)=2$ and  $\textit{TC}_3(3)=4$. Such a Gray code is balanced. 
 
 the Hamiltonian cycle that has been removed in $f^*$.
 Its transition sequence is $S=3,1,3,2,3,1,3,2$ and its transition count function is 
 $\textit{TC}_3(1)= \textit{TC}_3(2)=2$ and  $\textit{TC}_3(3)=4$. Such a Gray code is balanced. 
 

-Let now  
-$L^4=0000, 0010, 0110, 1110, 1111, 0111, 0011, 0001, 0101,$
-$0100, 1100, 1101, 1001, 1011, 1010, 1000$
-be a cyclic Gray code. Since $S=2,3,4,1,4,3,2,3,1,4,1,3,2,1,2,4$ $\textit{TC}_4$ is equal to 4 everywhere, this code
+Let  $L^4$ $=0000,0010,0110,1110,1111,0111,0011,0001,0101,0100,1100,1101,1001,1011,1010,1000$
+be a cyclic Gray code. Since $S=2,3,4,1,4,$ $3,2,3,1,4,1,3,2,1,2,4$, $\textit{TC}_4$ is equal to 4 everywhere, this code

 is thus totally balanced.
 
 On the contrary, for the standard $4$-bits Gray code  
 is thus totally balanced.
 
 On the contrary, for the standard $4$-bits Gray code  

-$L^{\textit{st}}=0000,0001,0011,0010,0110,0111,0101,0100,1100,$
-$1101,1111,1110,1010,1011,1001,1000$,
+$L^{\textit{st}}=0000,0001,0011, 
+0010,0110,0111,0101,0100,$ \newline $1100,1101,1111,1110,1010,1011,1001,1000$,

 we have $\textit{TC}_4(1)=8$ $\textit{TC}_4(2)=4$ $\textit{TC}_4(3)=\textit{TC}_4(4)=2$ and
 the code is neither balanced nor totally balanced.
 \end{xpl}
 we have $\textit{TC}_4(1)=8$ $\textit{TC}_4(2)=4$ $\textit{TC}_4(3)=\textit{TC}_4(4)=2$ and
 the code is neither balanced nor totally balanced.
 \end{xpl}


@@ -133,7 +132,7 @@ the code is neither balanced nor totally balanced.
 
 \begin{thrm}\label{prop:balanced}
 Let $\mathsf{N}$ in $\Nats^*$, and $a_{\mathsf{N}}$ be defined by
 
 \begin{thrm}\label{prop:balanced}
 Let $\mathsf{N}$ in $\Nats^*$, and $a_{\mathsf{N}}$ be defined by

-$a_{\mathsf{N}}= 2 \lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \rfloor$. 
+$a_{\mathsf{N}}= 2 \left\lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \right\rfloor$. 

 There exists then a sequence $l$ in 
 step~(\ref{item:nondet}) of the \emph{Robinson-Cohn extension} algorithm
 such that all the transition counts $\textit{TC}_{\mathsf{N}}(i)$ 
 There exists then a sequence $l$ in 
 step~(\ref{item:nondet}) of the \emph{Robinson-Cohn extension} algorithm
 such that all the transition counts $\textit{TC}_{\mathsf{N}}(i)$ 


@@ -146,13 +145,13 @@ for any $i$, $1 \le i \le \mathsf{N}$.
 
 
 The proof is done by induction on $\mathsf{N}$. Let us immediately verify 
 
 
 The proof is done by induction on $\mathsf{N}$. Let us immediately verify 

-that it is established for both odd and even smallest values, \textit{i.e.} 
+that it is established for both odd and even smallest values, \textit{i.e.}, 

 $3$ and $4$.
 $3$ and $4$.

-For the initial case where $\mathsf{N}=3$, \textit{i.e.} $\mathsf{N-2}=1$ we successively have:  $S_1=1,1$, $l=2$,  $u_0 = \emptyset$, and $v=\emptyset$.
+For the initial case where $\mathsf{N}=3$, \textit{i.e.}, $\mathsf{N-2}=1$ we successively have:  $S_1=1,1$, $l=2$,  $u_0 = \emptyset$, and $v=\emptyset$.

 Thus again the algorithm successively produces 
 $U= 1,2,1$, $V = 3$, $W= 2, 1, 1,3$, and $W' = 1,2,1,3$.
 Finally, $S_3$ is $1,2,1,3,1,2,1,3$ which obviously verifies the theorem.    
 Thus again the algorithm successively produces 
 $U= 1,2,1$, $V = 3$, $W= 2, 1, 1,3$, and $W' = 1,2,1,3$.
 Finally, $S_3$ is $1,2,1,3,1,2,1,3$ which obviously verifies the theorem.    

- For the initial case where $\mathsf{N}=4$, \textit{i.e.} $\mathsf{N-2}=2$ 
+ For the initial case where $\mathsf{N}=4$, \textit{i.e.}, $\mathsf{N-2}=2$ 

 we successively have:  $S_1=1,2,1,2$, $l=4$, 
 $u_0,u_1,u_2 = \emptyset,\emptyset,\emptyset$, and $v=\emptyset$.
 Thus again the algorithm successively produces 
 we successively have:  $S_1=1,2,1,2$, $l=4$, 
 $u_0,u_1,u_2 = \emptyset,\emptyset,\emptyset$, and $v=\emptyset$.
 Thus again the algorithm successively produces 


@@ -168,32 +167,30 @@ odd and even initial values.
 For the inductive case, let us first define some variables.
 Let $c_{\mathsf{N}}$ (resp. $d_{\mathsf{N}}$) be the number of elements 
 whose transition count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$).
 For the inductive case, let us first define some variables.
 Let $c_{\mathsf{N}}$ (resp. $d_{\mathsf{N}}$) be the number of elements 
 whose transition count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$).

-These two variables are defined by the system 
+Both of these variables are defined by the system 

  
  

-$$
+\[

 \left\{
 \begin{array}{lcl}
 c_{\mathsf{N}} + d_{\mathsf{N}} & = & \mathsf{N} \\
 c_{\mathsf{N}}a_{\mathsf{N}} + d_{\mathsf{N}}(a_{\mathsf{N}}+2) & = & 2^{\mathsf{N}} 
 \end{array}
 \right.
 \left\{
 \begin{array}{lcl}
 c_{\mathsf{N}} + d_{\mathsf{N}} & = & \mathsf{N} \\
 c_{\mathsf{N}}a_{\mathsf{N}} + d_{\mathsf{N}}(a_{\mathsf{N}}+2) & = & 2^{\mathsf{N}} 
 \end{array}
 \right.

-\qquad 

 \Leftrightarrow 
 \Leftrightarrow 

-\qquad 

 \left\{
 \begin{array}{lcl}
 d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N}} -\mathsf{N}.a_{\mathsf{N}}}{2} \\
 c_{\mathsf{N}} &= &\mathsf{N} -  d_{\mathsf{N}}
 \end{array}
 \right.
 \left\{
 \begin{array}{lcl}
 d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N}} -\mathsf{N}.a_{\mathsf{N}}}{2} \\
 c_{\mathsf{N}} &= &\mathsf{N} -  d_{\mathsf{N}}
 \end{array}
 \right.

-$$
+\]

 
 Since $a_{\mathsf{N}}$ is even, $d_{\mathsf{N}}$ is an integer.
 
 Since $a_{\mathsf{N}}$ is even, $d_{\mathsf{N}}$ is an integer.

-Let us first proove that both $c_{\mathsf{N}}$ and  $d_{\mathsf{N}}$ are positive
+Let us first prove that both $c_{\mathsf{N}}$ and  $d_{\mathsf{N}}$ are positive

 integers.
 Let $q_{\mathsf{N}}$ and $r_{\mathsf{N}}$, respectively, be  
 integers.
 Let $q_{\mathsf{N}}$ and $r_{\mathsf{N}}$, respectively, be  

-the quotient and the remainder in the Euclidean disvision
-of $2^{\mathsf{N}}$ by $2\mathsf{N}$, \textit{i.e.} 
+the quotient and the remainder in the Euclidean division
+of $2^{\mathsf{N}}$ by $2\mathsf{N}$, \textit{i.e.}, 

 $2^{\mathsf{N}} = q_{\mathsf{N}}.2\mathsf{N} + r_{\mathsf{N}}$, with $0 \le r_{\mathsf{N}} <2\mathsf{N}$.
 First of all, the integer $r$ is even since $r_{\mathsf{N}} = 2^{\mathsf{N}} - q_{\mathsf{N}}.2\mathsf{N}= 2(2^{\mathsf{N}-1} - q_{\mathsf{N}}.\mathsf{N})$. 
 Next,  $a_{\mathsf{N}}$ is $\frac{2^{\mathsf{N}}-r_{\mathsf{N}}}{\mathsf{N}}$. Consequently 
 $2^{\mathsf{N}} = q_{\mathsf{N}}.2\mathsf{N} + r_{\mathsf{N}}$, with $0 \le r_{\mathsf{N}} <2\mathsf{N}$.
 First of all, the integer $r$ is even since $r_{\mathsf{N}} = 2^{\mathsf{N}} - q_{\mathsf{N}}.2\mathsf{N}= 2(2^{\mathsf{N}-1} - q_{\mathsf{N}}.\mathsf{N})$. 
 Next,  $a_{\mathsf{N}}$ is $\frac{2^{\mathsf{N}}-r_{\mathsf{N}}}{\mathsf{N}}$. Consequently 


@@ -203,7 +200,7 @@ The proof for $c_{\mathsf{N}}$ is obvious.
 
 
 For any $i$, $1 \le i \le \mathsf{N}$, let $zi_{\mathsf{N}}$ (resp. $ti_{\mathsf{N}}$ and $bi_{\mathsf{N}}$) 
 
 
 For any $i$, $1 \le i \le \mathsf{N}$, let $zi_{\mathsf{N}}$ (resp. $ti_{\mathsf{N}}$ and $bi_{\mathsf{N}}$) 

-be the occurence number of element $i$ in the sequence $u_0, \dots, u_{l-2}$ 
+be the occurrence number of element $i$ in the sequence $u_0, \dots, u_{l-2}$ 

 (resp. in the sequences $s_{i_1}, \dots , s_{i_l}$ and $v$)
 in step (\ref{item:nondet}) of the algorithm.
 
 (resp. in the sequences $s_{i_1}, \dots , s_{i_l}$ and $v$)
 in step (\ref{item:nondet}) of the algorithm.
 


@@ -241,7 +238,7 @@ or to $a_{\mathsf{N}}+2$), the variable  $zi_{\mathsf{N}}$ is thus an integer.
 Let us now prove that the resulting system has always positive integer 
 solutions $z_i$, $t_i$, $0 \le z_i, t_i \le \textit{TC}_{\mathsf{N}-2}(i)$ 
 and s.t. their sum is equal to $\textit{TC}_{\mathsf{N}-2}(i)$. 
 Let us now prove that the resulting system has always positive integer 
 solutions $z_i$, $t_i$, $0 \le z_i, t_i \le \textit{TC}_{\mathsf{N}-2}(i)$ 
 and s.t. their sum is equal to $\textit{TC}_{\mathsf{N}-2}(i)$. 

-This latter consraint is obviously established if the system has a solution.
+This latter constraint is obviously established if the system has a solution.

 We thus have the following system.
 
 
 We thus have the following system.
 
 


@@ -267,7 +264,7 @@ When $3 \le N \le 7$, values are defined as follows:
 \textit{TC}_{4} & = & [4,4,4,4] \\
 \textit{TC}_{6} & = & [10,10,10,10,12,12] \\
 \end{eqnarray*}
 \textit{TC}_{4} & = & [4,4,4,4] \\
 \textit{TC}_{6} & = & [10,10,10,10,12,12] \\
 \end{eqnarray*}

-It is not hard to verify that all these instanciations verify the aformentioned contraints. 
+It is not difficult to check that all these instanciations verify the aforementioned constraints. 

 
 When  $N  \ge 8$, $\textit{TC}_{\mathsf{N}}(i)$ is defined as follows:
 \begin{equation}
 
 When  $N  \ge 8$, $\textit{TC}_{\mathsf{N}}(i)$ is defined as follows:
 \begin{equation}


@@ -282,7 +279,7 @@ a_{\mathsf{N}}+2 \textrm{ if } c_{\mathsf{N}} +1 \le i \le c_{\mathsf{N}} + d_{\
 
 
 We thus  have
 
 
 We thus  have

-$$
+\[

 \begin{array}{rcl} 
 \textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i) 
 &\ge& 
 \begin{array}{rcl} 
 \textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i) 
 &\ge& 


@@ -294,13 +291,13 @@ a_{\mathsf{N}} - 2(a_{\mathsf{N}-2}+2) \\
 \frac{2^{\mathsf{N}}-2N}{\mathsf{N}}
 -2 \left( \frac{2^{\mathsf{N-2}}}{\mathsf{N-2}}+2\right)\\
 &\ge& 
 \frac{2^{\mathsf{N}}-2N}{\mathsf{N}}
 -2 \left( \frac{2^{\mathsf{N-2}}}{\mathsf{N-2}}+2\right)\\
 &\ge& 

-\dfrac{(\mathsf{N} -2).2^{\mathsf{N}}-2N.2^{\mathsf{N-2}}-6N(N-2)}{\mathsf{N.(N-2)}}\\
+\frac{(\mathsf{N} -2).2^{\mathsf{N}}-2N.2^{\mathsf{N-2}}-6N(N-2)}{\mathsf{N.(N-2)}}\\

 \end{array}
 \end{array}

-$$
+\]

 
 A simple variation study of the function $t:\R \rightarrow \R$ such that 
 $x \mapsto t(x) = (x -2).2^{x}-2x.2^{x-2}-6x(x-2)$ shows that 
 
 A simple variation study of the function $t:\R \rightarrow \R$ such that 
 $x \mapsto t(x) = (x -2).2^{x}-2x.2^{x-2}-6x(x-2)$ shows that 

-its derivative is strictly postive if $x \ge 6$ and $t(8)=224$.
+its derivative is strictly positive if $x \ge 6$ and $t(8)=224$.

 The integer $\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i)$ is thus positive 
 for any $\mathsf{N} \ge 8$ and the proof is established.
 
 The integer $\textit{TC}_{\mathsf{N}}(i) - 2.\textit{TC}_{\mathsf{N}-2}(i)$ is thus positive 
 for any $\mathsf{N} \ge 8$ and the proof is established.
 


@@ -322,9 +319,15 @@ for any $\mathsf{N} \ge 8$ and the proof is established.
 For each element $i$, we are then left to choose $zi_{\mathsf{N}}$ positions 
 among $\textit{TC}_{\mathsf{N}}(i)$, which leads to 
 ${\textit{TC}_{\mathsf{N}}(i) \choose zi_{\mathsf{N}} }$ possibilities.
 For each element $i$, we are then left to choose $zi_{\mathsf{N}}$ positions 
 among $\textit{TC}_{\mathsf{N}}(i)$, which leads to 
 ${\textit{TC}_{\mathsf{N}}(i) \choose zi_{\mathsf{N}} }$ possibilities.

-Notice that all such choices lead to a hamiltonian path.
+Notice that all such choices lead to an Hamiltonian path.

 
 
 
 
 
 
 
 
 
 

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