diff --git a/documentation/changes_list.txt b/documentation/changes_list.txt index 30fda5b329..d235897d50 100644 --- a/documentation/changes_list.txt +++ b/documentation/changes_list.txt @@ -1,4 +1,24 @@ +v.0.1.5: (2012-04-30 12:46:04) +------------------------------ +* Major changes in the scripts to generate the doc: + - the scripts have been completely rewritten and the directories + completely rearranged. +* Minor enhancements to the manual: + - html version: several graphic enhancements. + - preface: comments from Hakan (Thanks again and again! ;-) ) + - chap2: + + table with cryptarithmetic puzzle improved in both version. + + figures "at a glance" done anew. + + comments from Hakan. +* FAQ: + - several graphic enhancements. +* Tutorials: + - few corrections in the C++ code. +* Lab sessions: + - chap2: questions posted. + + v.0.1.4: (2012-04-23 22:55:42) ------------------------------ * Major changes in the scripts to generate the doc: diff --git a/documentation/documentation_hub.html b/documentation/documentation_hub.html index 95f66e42fe..425e64f714 100644 --- a/documentation/documentation_hub.html +++ b/documentation/documentation_hub.html @@ -72,17 +72,24 @@ Thank you very much.

What's new?

Here is a little summary:

-
v.0.1.4: (2012-04-23 22:55:42)
+
v.0.1.5: (2012-04-30 12:46:04)
 ------------------------------
 * Major changes in the scripts to generate the doc:
   - the scripts have been completely rewritten and the directories
     completely rearranged.
 * Minor enhancements to the manual:
-  chap3:
-  - a lots of small improvements from Hakan (Thanks again!).
-  - some images resized in html version.
+  - html version: several graphic enhancements.
+  - preface: comments from Hakan (Thanks again and again! ;-) )
+  - chap2:
+    + table with cryptarithmetic puzzle improved in both version.
+    + figures "at a glance" done anew.
+    + comments from Hakan.
+* FAQ:
+  - several graphic enhancements.
 * Tutorials:
   - few corrections in the C++ code.
+* Lab sessions:
+  - chap2: questions posted.
 
 

 
@@ -97,9 +104,9 @@ Thank you very much.
 
 
 
Progress at a glance:

@@ -341,8 +348,8 @@ Thank you very much.
 Documentation hub
 
 

-  

-    95%
+  

+    100%
   

 

 
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index 7f4924acec..cd5a5d3ddd 100644
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diff --git a/documentation/user_manual/_images/math/fc53204fb8443e399c9cdd4a4d1008c574c1d87a.png b/documentation/user_manual/_images/math/fc53204fb8443e399c9cdd4a4d1008c574c1d87a.png
index 9f47d5b116..3f474829b2 100644
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diff --git a/documentation/user_manual/index.html b/documentation/user_manual/index.html
index 3d97ed314b..a808221b35 100644
--- a/documentation/user_manual/index.html
+++ b/documentation/user_manual/index.html
@@ -169,7 +169,7 @@ and how to use the full potential of our library.

 
What you will not learn in this document

 
This document is by no means a tutorial on Operations Research nor on Contraint Programming.
 It is also NOT a reference manual (refer to the documentation hub to find the reference manual).
-There are way too many methods, parameters, functions, . . . to explain them all in details. Once
+There are way too many methods, parameters, functions, etc. to explain them all in details. Once
 you understand the concepts and methods explained in this manual, you shouldn’t have any
 trouble scanning the reference manual and find the right method, parameter, function, . . . or code
 them yourselves!

@@ -177,7 +177,8 @@ them yourselves!

 questions about the non-CP part of the library, don’t hesitate to ask them on the mailing list. See
 the section How to reach us? below.

 
This document will not describe how to use the library (and the syntactic sugar introduced when
-possible) with Python, Java nor C#. This could possibly change in the future.

+possible) with Python, Java nor C#. This could possibly change in the future. The tutorial examples
+(see below) exist also in Python, Java and C# though.

 

 

 
How to read this document?

@@ -190,7 +191,7 @@ probably the most efficient and rewarding one!

 
That said, the manual is kept short so that you can read it in its entirety. The first part (Basics)
 is an introduction on how to use the CP solver to solve small problems. For real problems, you
 need to customize your search and this is explained in the second part (Customization). If you are
-interested in the routing part of the library, the third part if for you (Routing). Finally, some utilities
+interested in the routing part of the library, the third part is for you (Routing). Finally, some utilities
 and tricks are explained in the appendices.

 

 

diff --git a/documentation/user_manual/manual/first_steps.html b/documentation/user_manual/manual/first_steps.html
index 0ad533e95c..482b587102 100644
--- a/documentation/user_manual/manual/first_steps.html
+++ b/documentation/user_manual/manual/first_steps.html
@@ -71,8 +71,8 @@ about the other supported languages (Python, Java and C#).

 steps to write a basic program.

 
Prerequisites:

 
 
Files:

 
You can find the code in the directory documentation/tutorials/C++/chap2.

diff --git a/documentation/user_manual/manual/first_steps/anatomy.html b/documentation/user_manual/manual/first_steps/anatomy.html
index 6d793e920d..fea0a09fea 100644
--- a/documentation/user_manual/manual/first_steps/anatomy.html
+++ b/documentation/user_manual/manual/first_steps/anatomy.html
@@ -63,8 +63,8 @@ the cryptarithmetic puzzle in C++. In the next chapters, we will cover some of t
 in more details.

 

 
2.3.1. At a glance

-../../_images/anatomy1.png
-../../_images/anatomy2.png
+../../_images/anatomy1_2.png
+../../_images/anatomy3_4.png
 

 

 
2.3.2. Headers

@@ -75,7 +75,13 @@ in more details.

 

 
The header logging.h is needed for some logging facilities and some
 assert-like macros. The header constraint_solver.h is the main entry point
-to the CP solver and must be includedfootnote{Directly or indirectly when it is included in another header you include.} whenever you intend to use it.

+to the CP solver and must be included[1] whenever you intend to use it.

+
+
+
+
+
+
[1]Directly or indirectly when it is included in another header you include.

 

 

 
2.3.3. The namespace operations_research

@@ -95,7 +101,7 @@ the same convention in all our examples and code inside this namespace:

 
MakeBaseLine2,  MakeBaseLine3 and MakeBaseLine4 are helper functions to create the model.

 
We detail these functions later in Constraints but for the moment,
 let’s concentrate on CPIsFun() where all the magic happens.

-
It is called from the main[1]
+
It is called from the main[2]
 function:

 
int main(int argc, char **argv) {
   operations_research::CPIsFun();
@@ -106,7 +112,7 @@ function:

 
-
@@ -135,11 +141,11 @@ functionalities.
IntVar*conste=solver.MakeIntVar(0,kBase-1,"E");
-
For each letter, we create an integer variable IntVar whose domain is [0,\mathtt{kBase} - 1]
+
For each letter, we create an integer variable IntVar whose domain is [0,\mathtt{kBase} - 1]
 except for the variables c, i, f and t that cannot take the value 0.
 The MakeIntVar(i, j, name) method is a factory method that creates an
 integer variable whose domain is [i,j]=\{i, i+1, \dotsc
-, j-1, j\} and has a name name. It returns a pointer to an IntVar. The declaration IntVar* const c
+, j-1, j\}" style="vertical-align: -5px"/> and has a name name. It returns a pointer to an IntVar. The declaration IntVar* const c
 may seem a little be complicated at first. It is easier to understand if read from right to left: c is a constant pointer to an
 IntVar. We can modify the object pointed by c but this pointer, because it is constant, always refers to the same object.

@@ -154,12 +160,12 @@ them appropriately.

 
Warning

 
Never delete explicitly an object created by
 a factory method! First, the solver deletes all the objects for you.
-Second, deleting a pointer twice in C++ gives undefined behaviour[2]!

+Second, deleting a pointer twice in C++ gives undefined behaviour[3]!

 

 
 
[1]The main function does not lie inside the namespace
+
[2]The main function does not lie inside the namespace
 operations_research, hence the use of the operations_research identifier to call
 the function CPIsFun().

 
         
 
 
 
 

-
+

 
 
[2]It is possible to bypass the undefined behaviour but you don’t know what the solver needs to do, so keep your hands off of the object pointers! ;-)
[3]It is possible to bypass the undefined behaviour but you don’t know what the solver needs to do, so keep your hands off of the object pointers! ;-)

 
 

 
Beside integer variables, the solver provides factory methods to create
@@ -184,13 +190,14 @@ Aborted

 
In Asserting, we cover assert-like macros in more details.

 

 

-
2.3.7. Constraints

-
To create a integer linear constraint, we need to know how to multiply an integer variable
+
2.3.7. Constraints

+
To create an integer linear constraint, we need to know how to multiply an integer variable
 with an integer constant and how to add two integer variables. We have seen that
 the solver creates a variable and only provides a pointer to that variable.
 The solver also provides factory methods to multiply an integer coefficient by
 an IntVar given by a pointer:

 
IntVar* const var1 = solver.MakeIntVar(0, 1, "Var1");
+// var2 = var1 * 36
 IntVar* const var2 = solver.MakeProd(var1,36)->Var();
 

 

@@ -202,7 +209,8 @@ any integer expression.

 to an IntVar and then the integer constant.

 
To add two IntVar given by their respective pointers, the solver provides
 again a factory method:

-
IntVar* const var3 = solver.MakeSum(var1,var2)->Var();
+
//var3 = var1 + var2
+IntVar* const var3 = solver.MakeSum(var1,var2)->Var();
 

 

 

diff --git a/documentation/user_manual/manual/first_steps/cryptarithmetic.html b/documentation/user_manual/manual/first_steps/cryptarithmetic.html
index 843e2426fa..d6a722db38 100644
--- a/documentation/user_manual/manual/first_steps/cryptarithmetic.html
+++ b/documentation/user_manual/manual/first_steps/cryptarithmetic.html
@@ -110,7 +110,7 @@ have different values in a feasible solution. Second, it is implicit that the fi
 digit of a number can not be 0. Letters C, I, F and T can thus
 not represent 0. Third, there are 10 letters, so we need at least 10
 different digits. The traditional decimal base is sufficient but let’s be more general
-and allow for a bigger base. We will use a constant kBase. This last constraint is not a CP constraint.
+and allow for a bigger base. We will use a constant kBase. The fact that we need at least 10 digits is not really a CP constraint.
 After all, the base is not a variable but a given integer that is chosen once
 and for all for the whole program[2].

 
@@ -127,17 +127,20 @@ puzzle has less than xFor each letter, we have a decision variable (we keep the same letters to name the variables).
 Given a base b, digits range from 0 to b-1.
 Remember that variables corresponding to C, I, F and T should be different
-from 0. Thus C, I, F and T have [1,\mathtt{b}-1] as domain and P, S, U, N, R and  E
-have [0,\mathtt{b}-1] as domain. Another possibility is to keep the same domain [0,\mathtt{b}-1] for all
-variables and force C, I, F and T to be different from 0 by adding inequalities. However, restraining the domain to [1,\mathtt{b}-1] is more efficient.

+from 0. Thus C, I, F and T have [1,\mathtt{b}-1] as domain and P, S, U, N, R and  E
+have [0,\mathtt{b}-1] as domain. Another possibility is to keep the same domain [0,\mathtt{b}-1] for all
+variables and force C, I, F and T to be different from 0 by adding inequalities. However, restraining the domain to [1,\mathtt{b}-1] is more efficient.
To model the sum constraint in any base b, we add the linear equation:

-
\begin{align*}
-     &                                  &   &                                  &   & {\color{blue}{\mathtt{C}}} \cdot \mathtt{b} & + & {\color{blue}{\mathtt{P}}}\\
-   + &                                  &   &                                  &   & {\color{blue}{\mathtt{I}}} \cdot \mathtt{b} & + & {\color{blue}{\mathtt{S}}}\\
-   + &                                  &   & {\color{blue}{\mathtt{F}}} \cdot \mathtt{b}^2 & + & {\color{blue}{\mathtt{U}}} \cdot \mathtt{b} & + & {\color{blue}{\mathtt{N}}}\\
-   = & {\color{blue}{\mathtt{T}}} \cdot \mathtt{b}^3 & + & {\color{blue}{\mathtt{R}}} \cdot \mathtt{b}^2 & + & {\color{blue}{\mathtt{U}}} \cdot \mathtt{b} & + & {\color{blue}{\mathtt{E}}}.
-\end{align*}

+
\begin{center}
+\begin{tabular}{cccccccc}
+     &                                         &   &                                        & + & ${\color{blue}{\mathtt{C}}} \cdot b$ & + & ${\color{blue}{\mathtt{P}}}$\\
+     &                                         &   &                                        & + & ${\color{blue}{\mathtt{I}}} \cdot b$ & + & ${\color{blue}{\mathtt{S}}}$\\
+     &                                         & + & ${\color{blue}{\mathtt{F}}} \cdot b^2$ & + & ${\color{blue}{\mathtt{U}}} \cdot b$ & + & ${\color{blue}{\mathtt{N}}}$\BStrut\\
+ \hline
+   = & ${\color{blue}{\mathtt{T}}} \cdot b^3$ & + & ${\color{blue}{\mathtt{R}}} \cdot b^2$ & + & ${\color{blue}{\mathtt{U}}} \cdot b$ & + & ${\color{blue}{\mathtt{E}}}\TStrut$
+\end{tabular}
+\end{center}

 
The global constraint AllDifferent springs to mind to model that variables must all have different values:
AllDifferent(C,P,I,S,F,U,N,T,R,E)
 

diff --git a/documentation/user_manual/manual/first_steps/monitors.html b/documentation/user_manual/manual/first_steps/monitors.html
index b372fb3d80..1dd2797be5 100644
--- a/documentation/user_manual/manual/first_steps/monitors.html
+++ b/documentation/user_manual/manual/first_steps/monitors.html
@@ -200,8 +200,8 @@ also additional information. You can also act on some of the variables for insta
 LOG(INFO) << "Number of solutions: " << number_solutions << std::endl;
 
 for (int index = 0; index < number_solutions; ++index) {
-  Assignment* const solution = all_solutions->solution(index);
-  LOG(INFO) << "Solution found:";
+  Assignment* const solution = all_solutions->solution(index);
+  LOG(INFO) << "Solution found:";
   LOG(INFO) << "v1=" << solution->Value(v1);
 }
 

diff --git a/documentation/user_manual/manual/objectives.html b/documentation/user_manual/manual/objectives.html
index 0aff19fa94..69ac8579ad 100644
--- a/documentation/user_manual/manual/objectives.html
+++ b/documentation/user_manual/manual/objectives.html
@@ -89,11 +89,11 @@ In fact, you can completely skip them if you wish. The basic ideas behind these
 
You can find the code in the directory documentation/tutorials/C++/chap3.
The files inside this directory are:
    
-
  • golomb1.cc: A first implementation. We show how to tell the solver to optimize an objective function. We use the \frac{n(n-1)}{2} differences as variables.
    • 
+
  • golomb1.cc: A first implementation. We show how to tell the solver to optimize an objective function. We use the \frac{n(n-1)}{2} differences as variables.
    • 
 
  • golomb2.cc: Same file as golomb1.cc but with some global statistics about the search added so we can see how well or bad
 a model behave.
    • 
 
  • golomb3.cc: A second implementation. This time, we only use the marks as variables and introduce the quaternary inequality constraints.
    • 
-
  • golomb4.cc: We improve the second implementation by reintroducing the \frac{n(n-1)}{2} differences variables.
    • 
+
  • golomb4.cc: We improve the second implementation by reintroducing the \frac{n(n-1)}{2} differences variables.
    • 
 
  • golomb5.cc: In this third implementation, we replace the inequality constraints by the more powerful globlal AllDifferent constraint.
    • 
 
  • golomb6.cc: The last implementation is a tightening of the model used in the third implementation. We add better upper and lower bounds and break a symmetry in the search tree.
    • 
 

diff --git a/documentation/user_manual/manual/objectives/content_model.html b/documentation/user_manual/manual/objectives/content_model.html
index 414f6bd221..22b6139bcc 100644
--- a/documentation/user_manual/manual/objectives/content_model.html
+++ b/documentation/user_manual/manual/objectives/content_model.html
@@ -64,7 +64,7 @@
 
Most of the mathematical classes in or-tools inherit from the BaseObject class. Its only
 public method is a virtual DebugString(). If you are curious or just in doubt about
 the object you just constructed, DebugString() is for you.

-
Let’s have a closer look at the constraints that model the inner structure of the Golomb ruler of order 5:

+
Let’s have a closer look at the constraints that model the inner structure of the Golomb ruler of order 5:
const int n = 5;
 ...
 for (int i = 2; i <= n - 1; ++i) {
@@ -136,7 +136,7 @@ the object you just constructed, 
                                                                 (2..48))
 

 

-
The cast to transform the sum Y_1 + Y_0 into an IntVar.

+
The cast to transform the sum Y_1 + Y_0 into an IntVar.

 
And then:

 
...: Y_4(1..24) == Var<(Y_1(1..24) + Y_0(1..24))>(2..48)
 ...: Y_5(1..24) == Var<(Y_2(1..24) + Y_1(1..24))>(2..48)
diff --git a/documentation/user_manual/manual/objectives/data_search.html b/documentation/user_manual/manual/objectives/data_search.html
index e592d320b3..cfbc50f021 100644
--- a/documentation/user_manual/manual/objectives/data_search.html
+++ b/documentation/user_manual/manual/objectives/data_search.html
@@ -80,7 +80,7 @@ library offers a basic but portable timer. This timer starts to measure the time
 

 

 
As its name implies, the time measured is the wall time, i.e. ... TO BE COMPLETED.

-
For instance, on my computer, the program in golomb1.cc for n = 9 takes

+
For instance, on my computer, the program in golomb1.cc for n = 9 takes

 
Time: 4,773 seconds
 

 

diff --git a/documentation/user_manual/manual/objectives/first_implementation.html b/documentation/user_manual/manual/objectives/first_implementation.html
index 816a4abda8..049ef8bfa8 100644
--- a/documentation/user_manual/manual/objectives/first_implementation.html
+++ b/documentation/user_manual/manual/objectives/first_implementation.html
@@ -65,18 +65,18 @@
 

 
3.3.1. An upper bound

 
Several upper bounds exist on Golomb rulers.
-For instance, we could take n^3 - 2n^2+ 2n -1. Indeed, it can be
+For instance, we could take n^3 - 2n^2+ 2n -1. Indeed, it can be
 shown that the sequence

 

 
\Phi(a) = na^2 + a \qquad 0 \leqslant a < n.

-
forms a Golomb ruler. As its largest member is n^3 - 2n^2+ 2n -1 (when a = n - 1), we have
-an upper bound on the length of a Golomb ruler of order n:

+
forms a Golomb ruler. As its largest member is n^3 - 2n^2+ 2n -1 (when a = n - 1), we have
+an upper bound on the length of a Golomb ruler of order n:

 

 
G(n) \leqslant n^3 - 2n^2+ 2n -1.

 
Most bounds are really bad and this one isn’t an exception. The great mathematician Paul Erdös conjectured that

 

 
G(n) < n^2.

-
This conjecture hasn’t been proved yet but computational evidence has shown that the conjecture holds for n < 65000 (see [Dimitromanolakis2002]).

+
This conjecture hasn’t been proved yet but computational evidence has shown that the conjecture holds for n < 65000 (see [Dimitromanolakis2002]).

 

 
 
 
 
 
 

diff --git a/documentation/user_manual/manual/objectives/golomb_first_model.html b/documentation/user_manual/manual/objectives/golomb_first_model.html
index b8f784857f..c77e3199bf 100644
--- a/documentation/user_manual/manual/objectives/golomb_first_model.html
+++ b/documentation/user_manual/manual/objectives/golomb_first_model.html
@@ -73,21 +73,21 @@ illustrates a Golomb ruler of order 4 and all its - distinct - differences.

 A non optimal Golomb ruler of order 4.
 
A non optimal Golomb ruler of order 4.

 
-
The Golomb ruler is \{0, 2, 7, 11\} and its length is 11. Because
-we are interested in Golomb rulers with minimal length, we can fix the first mark to 0.

+
The Golomb ruler is \{0, 2, 7, 11\} and its length is 11. Because
+we are interested in Golomb rulers with minimal length, we can fix the first mark to 0.

 
Figure An optimal Golomb ruler of order 4.
 illustrates an optimal Golomb ruler of order 4 and all its - distinct - differences.

 

 An optimal Golomb ruler of order 4.
 
An optimal Golomb ruler of order 4.

 

-
Its length, 6, is optimal: it is not possible to construct a Golomb ruler with 4 marks with
-a length smaller than 6. We denote this optimal value by G(4) = 6. More generally, for a
-Golomb ruler of order n, we denote by G(n) its optimal value. The Golomb Ruler Problem (GRP) is to find, for a given
-order n, the smallest Golomb ruler with n marks.

+
Its length, 6, is optimal: it is not possible to construct a Golomb ruler with 4 marks with
+a length smaller than 6. We denote this optimal value by G(4) = 6. More generally, for a
+Golomb ruler of order n, we denote by G(n) its optimal value. The Golomb Ruler Problem (GRP) is to find, for a given
+order n, the smallest Golomb ruler with n marks.

 
You might be surprised to learn that
-the largest order for which the experts have found an optimal Golomb ruler so far is... 26. And it was a huge hunt involving hundreds  of people!
-The next table compares the number of days, the number of participants on the Internet and the number of visited nodes in the search tree to find and prove G(24), G(25) and G(26)[2][3][4].

+the largest order for which the experts have found an optimal Golomb ruler so far is... 26. And it was a huge hunt involving hundreds  of people!
+The next table compares the number of days, the number of participants on the Internet and the number of visited nodes in the search tree to find and prove G(24), G(25) and G(26)[2][3][4].

 

 
 
@@ -120,7 +120,7 @@ The next table compares the number of days, the number of participants on the In
 
 
 

 
 


-
The search for G(27) started on February 24, 2009 and at that time was expected to take... 7 years!
+
The search for G(27) started on February 24, 2009 and at that time was expected to take... 7 years!
 Still think it is an easy[1] problem? You too can participate: The OGR Project.

 
You can find all the known optimal Golomb rulers and more information on Wikipedia.

 
@@ -168,35 +168,35 @@ of a Golomb ruler. In information theory, Golomb rulers are used for error detec
 

 
3.2.2.1. Describe

 
What is the goal of the Golomb Ruler Problem? To find a minimal Golomb ruler for a given
-order n. Our objective function is the length of the ruler or the largest
+order n. Our objective function is the length of the ruler or the largest
 integer in the Golomb ruler sequence.

 
What are the decision variables (unknowns)? We have at least two choices. We can either view the unknowns
 as the marks of the ruler (and retrieve all the differences from these variables) or choose the unknowns to be the differences (and retrieve the marks).
 Let’s try this second approach and use the efficient AllDifferent constraint.
-There are \frac{n (n-1)}{2} such differences.

+There are \frac{n (n-1)}{2} such differences.

 
What are the constraints? Using the differences as variables, we need to construct a Golomb
 ruler, i.e. the structure of the Golomb ruler has to be respected (see next section).

 

 
3.2.2.2. Model

 
For each positive difference, we have a decision variable. We collect them in
-an array Y\hspace{-0.1cm}.  Let’s order the differences so that we know which difference is represented by Y[i].

+an array Y\hspace{-0.1cm}.  Let’s order the differences so that we know which difference is represented by Y[i].

 
Figure An ordered sequence of differences for the Golomb ruler of order 4.
 illustrates an ordered sequence of differences for a Golomb ruler of order 4.

 

 An ordered sequence of differences for the Golomb ruler of order 4.
 
An ordered sequence of differences for the Golomb ruler of order 4.

 

-
We want to minimize the last difference in Y i.e. Y[\frac{n (n-1)}{2}-1] since the first index of an array is 0.
-When the order is n = 4, we want to optimize Y[\frac{4 (4-1)}{2}-1] = Y[5] which represents the 6^\textrm{th} difference. Instead of writing Y[i], we will also use the more convenient notation Y_i.

+
We want to minimize the last difference in Y i.e. Y[\frac{n (n-1)}{2}-1] since the first index of an array is 0.
+When the order is n = 4, we want to optimize Y[\frac{4 (4-1)}{2}-1] = Y[5] which represents the 6^\textrm{th} difference. Instead of writing Y[i], we will also use the more convenient notation Y_i.

 
Figure The inner structure of a Golomb ruler of order 5.
 illustrates the structure than must be respected for a Golomb ruler of order 5. To impose the inner structure of the Golomb Ruler,
-we force Y_4 = Y_0 + Y_1, Y_5 = Y_1 + Y_2 and so on as illustrated in Figure The inner structure of a Golomb ruler of order 5..

+we force Y_4 = Y_0 + Y_1, Y_5 = Y_1 + Y_2 and so on as illustrated in Figure The inner structure of a Golomb ruler of order 5..

 

 The inner structure of a Golomb ruler of order 5.
 
The inner structure of a Golomb ruler of order 5.

 

-
An easy way to construct these equality constraints is to use an index index going from 4 to 9[5], an
+
An easy way to construct these equality constraints is to use an index index going from 4 to 9[5], an
 index i to count the number of terms in a given equality and an index j to indicate the rank of the starting term in each
 equality:

 
int index = n - 2;
@@ -212,7 +212,7 @@ equality:

 

+(n-1) to the index of the last difference \left( \frac{n(n-1)}{2} - 1 \right).

 
 
 
[5]Or more generally from the index of the first difference that is the sum of two differences in our sequence
-(n-1) to the index of the last difference \left( \frac{n(n-1)}{2} - 1 \right).

 
 

 

diff --git a/documentation/user_manual/manual/objectives/second_implementation.html b/documentation/user_manual/manual/objectives/second_implementation.html
index 6e8798c49b..cdea5e1a14 100644
--- a/documentation/user_manual/manual/objectives/second_implementation.html
+++ b/documentation/user_manual/manual/objectives/second_implementation.html
@@ -56,8 +56,8 @@
             
   

 
3.6. A second model and its implementation

-
Our first model is really bad. One of the reasons is that we use too many variables: \frac{n (n-1)}{2} differences.
-What happens if we only consider the n marks as variables instead of the differences?

+
Our first model is really bad. One of the reasons is that we use too many variables: \frac{n (n-1)}{2} differences.
+What happens if we only consider the n marks as variables instead of the differences?

 

 
3.6.1. Variables

 
You can find the code in the file tutorials/C++/chap3/golomb3.cc

@@ -70,7 +70,7 @@ In the same vain, we redefine the array 

 

 
We use an std::vector slightly bigger (by one more element) than absolutely necessary. Because the solver doesn’t allow NULL pointers, we have
-to assign a value to X[0]. The first mark X[1] is 0. We use again n^2 - 1 as an upper bound for the marks:

+to assign a value to X[0]. The first mark X[1] is 0. We use again n^2 - 1 as an upper bound for the marks:

 
// Upper bound on G(n), only valid for n <= 65 000
 CHECK_LE(n, 65000);
 const int64 max = n * n - 1;
@@ -93,15 +93,15 @@ to assign a value to X[0]<
 [1]Quaternary constraints is just a fancy way to say that the constraints each involves four variables.
 
 
-
We don’t need all combinations of (i,j,k,l) with i \neq k and j \neq l. For instance, combination (3,2,6,4)
-and combination (2,3,4,6) would both give the same constraint. One way to avoid such redundancy is to impose an order on unique positive differences[2].

+
We don’t need all combinations of (i,j,k,l) with i \neq k and j \neq l. For instance, combination (3,2,6,4)
+and combination (2,3,4,6) would both give the same constraint. One way to avoid such redundancy is to impose an order on unique positive differences[2].

 

 
 
 
[2]In section Breaking symmetries we’ll use another trick.

 
 

-
Take again n=4 and define the sequence of differences as in Figure Another ordered sequence of differences for the Golomb ruler of order 4..

+
Take again n=4 and define the sequence of differences as in Figure Another ordered sequence of differences for the Golomb ruler of order 4..

 

 Another ordered sequence of differences for the Golomb ruler of order 4.
 
Another ordered sequence of differences for the Golomb ruler of order 4.

@@ -119,7 +119,7 @@ and so on as suggested in Figure 
 
How to generate the quaternary constraints, part II.

 

-
We define a helper function that, given a difference (i,j) corresponding to an interval X[j] - X[i] computes the next difference in the sequence:

+
We define a helper function that, given a difference (i,j) corresponding to an interval X[j] - X[i] computes the next difference in the sequence:

 
bool next_interval(const int n, const int i, const int j, int* next_i,
                                                          int* next_j)  {
   CHECK_LT(i, n);
@@ -171,10 +171,10 @@ and so on as suggested in Figure [3]Remember again the remark at the beginning of this chapter about the tricky sums.
 
 
-
Note that we set the minimum value of the difference to 1, diff1->SetMin(1), to ensure that the differences are positive and \geqslant 1. Note also that the method
+
Note that we set the minimum value of the difference to 1, diff1->SetMin(1), to ensure that the differences are positive and \geqslant 1. Note also that the method
 MakeDifference() doesn’t allow us to give a name to the new variable, which is normal as this new variable is the difference of two
 existing variables. Its name is simply name1 - name2.

-
Let’s compare the first and second implementation. The next table compares some global indicators about the search for G(9).

+
Let’s compare the first and second implementation. The next table compares some global indicators about the search for G(9).

 @@ -236,7 +236,7 @@ existing variables. Its name is simply }
 
 
-
and compare this improved version with the two others, again to compute G(9):

+
and compare this improved version with the two others, again to compute G(9):

 

 
 

diff --git a/documentation/user_manual/manual/objectives/summary.html b/documentation/user_manual/manual/objectives/summary.html
index 14461c12cf..8a286df8d9 100644
--- a/documentation/user_manual/manual/objectives/summary.html
+++ b/documentation/user_manual/manual/objectives/summary.html
@@ -56,7 +56,7 @@
             
   

 
3.10. Summary

-
Not a good approach. For example with our tightened implementation 3 we need 35.679  seconds to solve G(11).

+
Not a good approach. For example with our tightened implementation 3 we need 35.679  seconds to solve G(11).

 
Further reading: see [SmithEtAl]

 

 
 


diff --git a/documentation/user_manual/manual/objectives/third_implementation.html b/documentation/user_manual/manual/objectives/third_implementation.html
index 6e4bc48a3f..b89cabb014 100644
--- a/documentation/user_manual/manual/objectives/third_implementation.html
+++ b/documentation/user_manual/manual/objectives/third_implementation.html
@@ -71,7 +71,7 @@ propagations of the “quaternary constraints”. But these constraints
 s.AddConstraint(s.MakeAllDifferent(Y));
 
 
-
and compare our three implementations, again to compute G(9):

+
and compare our three implementations, again to compute G(9):

 

 
diff --git a/documentation/user_manual/manual/objectives/tighten_model.html b/documentation/user_manual/manual/objectives/tighten_model.html
index fb85613431..46a6dbcdc1 100644
--- a/documentation/user_manual/manual/objectives/tighten_model.html
+++ b/documentation/user_manual/manual/objectives/tighten_model.html
@@ -117,8 +117,8 @@ representing the marks of a Golomb ruler, we implicitly took for granted that 

 
 

[3]Declaring variables in an std::vector doesn’t tell anything about their respective values!

 
 

-
Thanks to diff1->SetMin(1) and diff2->SetMin(1) and the two for loops, the ordered variables X[1], X[2], X[3], X[4]
-have only increasing values, i.e. if i \leqslant j then X[i] \leqslant X[j]. Solutions (1) and (2) are said to be symmetric and avoiding the second one while accepting the first one is called breaking symmetry.

+
Thanks to diff1->SetMin(1) and diff2->SetMin(1) and the two for loops, the ordered variables X[1], X[2], X[3], X[4]
+have only increasing values, i.e. if i \leqslant j then X[i] \leqslant X[j]. Solutions (1) and (2) are said to be symmetric and avoiding the second one while accepting the first one is called breaking symmetry.

 
There is a well-known symmetry in the Golomb Ruler Problem that we didn’t break. Whenever you have a Golomb ruler, there exist another Golomb
 ruler with the same length that is called the mirror ruler.

 
Figure Two mirror Golomb rulers of order 4.
@@ -127,7 +127,7 @@ illustrates two mirror Golomb rulers of order 4.

 Two mirror Golomb rulers of order 4.
 
Two mirror Golomb rulers of order 4.

 

-
Golomb ruler \{0,1,4,6\} has \{0,2,5,6\} as mirror Golomb ruler. Both have exactly the same length and can be considered symmetric solutions. To break this symmetry and allow the search for the first one but not the second one, just add X[2]-X[1] < X[n] - X[n-1]:

+
Golomb ruler \{0,1,4,6\} has \{0,2,5,6\} as mirror Golomb ruler. Both have exactly the same length and can be considered symmetric solutions. To break this symmetry and allow the search for the first one but not the second one, just add X[2]-X[1] < X[n] - X[n-1]:

 
s.AddConstraint(s.MakeLess(s.MakeDifference(X[2],X[1])->Var(),
                            s.MakeDifference(X[n],X[n-1])->Var()));
 

@@ -135,7 +135,7 @@ illustrates two mirror Golomb rulers of order 4.

 

 

 
3.8.2. Better bounds helps

-
In all implementations, we used n^2 - 1 as an upper bound on G(n). In the case of the Golomb Ruler Problem, finding good upper bounds is a false problem. Very efficient techniques exist to find optimal or near optimal upper bounds. If we use those bounds, we reduce dramatically the domains of the variables. We can actually use G(n) as an upper bound for n \leqslant 25 as these bounds can be obtained by projective and affine projections in the plane[4].

+
In all implementations, we used n^2 - 1 as an upper bound on G(n). In the case of the Golomb Ruler Problem, finding good upper bounds is a false problem. Very efficient techniques exist to find optimal or near optimal upper bounds. If we use those bounds, we reduce dramatically the domains of the variables. We can actually use G(n) as an upper bound for n \leqslant 25 as these bounds can be obtained by projective and affine projections in the plane[4].

 
@@ -159,7 +159,7 @@ See }
 
 
-
where kG[n] is G(n).

+
where kG[n] is G(n).

 
The AllDifferent constraint doesn’t take a 2-dimensional array as parameter but it is easy to create one by flattening the array:

 
Constraint * AllDifferent(Solver* s,
                      const std::vector<std::vector<IntVar *> > & vars) {
@@ -182,8 +182,8 @@ See 
 
Y[i][j] = X[n] - \left\{ Y[1][2] + Y[2][3] + ... + Y[i-1][i] + Y[j][j+1] + ... + Y[n-1][n] \right\}

-
The differences on the right hand side of this expression are a set of different integers and there are n-1-j+i of them.
-If we minimize the sum of these consecutive differences, we actually maximize the right hand side, i.e. we bound Y[i][j] from above:

+
The differences on the right hand side of this expression are a set of different integers and there are n-1-j+i of them.
+If we minimize the sum of these consecutive differences, we actually maximize the right hand side, i.e. we bound Y[i][j] from above:

 
Y[i][j] \leqslant X[n] - (n-1-j+i)(n-j+i)/2

 
We can add:

@@ -195,7 +195,7 @@ If we minimize the sum of these consecutive differences, we actually maximize th
 }
 
 
-
Let’s compare our tightened third implementation with the rest, again to compute G(9):

+
Let’s compare our tightened third implementation with the rest, again to compute G(9):