### Data 4

#### Domains of minimality

After four steps of the computation we get a file of the following form:

```[   5  78064 ]
[*** -2 -2 -2 ]
[    7675   -24   7579  7675      0 11299 ]
[   11247    13  11247 11299      0 12217 ]
[   12197     5  12197 12217      0 78064 ]
[   17451    13  17451 17503  12197 78064 ]
[   43802    33  43802 43934  17451 46903 ]
[   46839    16  46839 46903  17451 78064 ]
[   59073   -22  58985 59073  46839 78064 ]
[   70939    24  70939 71035  58985 78064 ]
[*** -2 -2 -1 ]
[    1666   -35   1526  1666      0  5835 ]
[    5759    19   5759  5835      0  9925 ]
[    9881    11   9881  9925      0 13983 ]
[   13963     5  13963 13983      0 78064 ]
[   18250     5  18250 18270      0 78064 ]
[   43724   -12  43676 43724  18250 46670 ]
[   46642     7  46642 46670  18250 63681 ]
[   53370   -36  53226 53370  46642 62637 ]
[   62637   -28  62525 62637  46642 63681 ]
[   63661     5  63661 63681      0 78064 ]
[*** -2 -2  0 ]
[    5726    30   5726  5846      0  6516 ]
[    6516   -26   6412  6516      0  7797 ]
.   .   .   .   .   .   .   .   .
```
The first line contains the length of the given word (5 letters in "HYNTM") and the length of the Book. Other lines are grouped into families, each family having its perturbation parameters placed in the family header. Within a family, each line describes a perturbed ELS as follows:
```         E L S     extention     domain
start  skip   min   max    min   max
[    7675   -24   7579  7675      0 11299 ]
```
If skip > 0 then extension.min = start and extension.max = start + (n-1)*skip; if skip < 0 then extension.min = start + (n-1)*skip and extension.max = start; here n is the length of the given word (5). Anyway, extension.min < extension.max. These are the first and last positions in the Book, occupied by the unperturbed ELS with the same start and skip. (As before, all position numbers are decreased by 1 for some technical reason.)

In order to define domain(min,max), introduce the notion of a competitor. Given an element of a family (a line describing a perturbed ELS), its competitors are, by definition, all elements of the family having smaller skips (in absolute values, ignoring signs). For example, the first line "[ 7675 -24 7579 7675 0 11299 ]" of the first family (-2,-2,-2) has 5 competitors: they are lines having skips 13, 5, 13, 16, and (-22), but not 33, neither 24.

Compare the interval [extension.min, extension.max] occupied by a considered element of a family with intervals occupied by its competitors. Intervals of competitors are always shorter. There are several possible configurations:

```                      the considered element
[--------------------]
|                    |
[----------]  <-- a left competitor
[--------]  <-- a left competitor
|                    |
[-----]  <-- an internal competitor
an internal competitor --> [---]
an internal competitor --> [---]
|                    |
a right competitor --> [-------]
a right competitor --> [-------]
|                    |
```
An internal competitor causes a collapse of the domain: domain.min = start, domain.max = start + 1. There is no good reason why collapse to this end (rather than the other end or the middle), but it exerts no influence on the subsequent steps of the calculation.

In the absence of internal competitors, the domain is restricted from the left by left competitors, and from the right by right competitors:

```                      the considered element
[--------------------]
|                                 |
[----------]
|                                 |
[-----------]
|                                 |
[------]
|                                 |
[----]
|                                 |
| <-------- the domain  --------> |
```
That is, domain.min is the maximum of extension.min (not extension.max!) over all left competitors (if position numbers are meant increasing from the left to the right, of course), while domain.max is the minimum of extension.max over all right competitors.

In the absence of left (and internal) competitors we let domain.min = 0. In the absence of right (and internal) competitors, domain.max = 78064, the length of the Book (though it should be 78063, since all position numbers are decreased by 1).

For example, the first element "[ 7675 -24 7579 7675 0 11299 ]" of the first family has 5 competitors: no left, no internal, 5 right competitors. Their extension.max are: 11299, 12217, 17503, 46903, 59073. Thus, domain.min = 0 and domain.max = 11299 for the first element.

 back to Step 3-4 Data 4 Step 4-5