Doc: improve kerning of math expressions

Author: Morwenn

docs/Measures-of-disorder.md

@@ -50,33 +50,33 @@ The *monotonicity* property implies the *prefix monotonicity* one. A measure of
 
 Let $X$ be a sequence of elements, and let $S_X$ be set of all permutations of that sequence:
 
-$$below_M(X) = \{ \pi \vert \pi \in S_X \text{ and } M(\pi) \le M(X) \}$$
+$$\mathit{below}_M(X) = \{ \pi \vert \pi \in S_X \text{ and } M(\pi) \le M(X) \}$$
 
 Let $T_S(X)$ be the number of steps needed for an algorithm $S$ to sort $X$. A sorting algorithm is said to be $M$-optimal if and only if, for some constant $c$, we have for all $X$:
 
-$$T_S(X) \le c \cdot max\{\lvert X \rvert, \log{} |below_M(X)|\}$$
+$$T_S(X) \le c \cdot max\{\lvert X \rvert, \log{} |\mathit{below}_M(X)|\}$$
 
-In other words, a sorting algorithm is considered $M$-optimal if it takes a number of steps that is within a constant factor of the lower bound of $M$ to sort a sequence. For example a $Rem$-optimal algorithm should be able to sort any sequence in $O(\lvert X \rvert \log{} Rem(X))$ steps.
+In other words, a sorting algorithm is considered $M$-optimal if it takes a number of steps that is within a constant factor of the lower bound of $M$ to sort a sequence. For example a $\mathit{Rem}$-optimal algorithm should be able to sort any sequence in $O(\lvert X \rvert \log{} \mathit{Rem}(X))$ steps.
 
 ### Partial ordering of measures of disorder
 
 Early on, authors have been wanting to prove that some measures of disorder were "better" than other, though it quickly appeared that just comparing the raw numbers by the measures was not enough. Alistair Moffat proposed the following intuitive definition in *Ranking measures of sortedness and sorting nearly sorted lists*:
 
 > Let $M_1$ and $M_2$ be two measures of disorder:
 >
-> * $M_1$ is algorithmically finer than $M_2$ (denoted $M_1 \le_{alg} M_2$) if and only if any $M_1$-optimal sorting algorithm is also $M_2$-optimal.
-> * $M_1$ and $M_2$ are algorithmically equivalent (denoted $M_1 =_{alg} M_2$) if and only if $M_1 \le_{alg} M_2$ and $M_2 \le_{alg} M_1$.
+> * $M_1$ is algorithmically finer than $M_2$ (denoted $M_1 \le_\mathit{alg} M_2$) if and only if any $M_1$-optimal sorting algorithm is also $M_2$-optimal.
+> * $M_1$ and $M_2$ are algorithmically equivalent (denoted $M_1 =_\mathit{alg} M_2$) if and only if $M_1 \le_\mathit{alg} M_2$ and $M_2 \le_\mathit{alg} M_1$.
 
 While useful to understand what we want from a partial order on measures of disorder, the definition above does not help a lot when it comes to actually proving that a measure is algorithmically finer than another. To better compare two measures of disorder, Jingsen Chen introduces the following operator in *Computing and ranking measures of presortedness*:
 
 > Let $M_1$ and $M_2$ be two measures of disorder:
 >
-> * $M_1$ is superior to $M_2$ (denoted $M_1 \preceq M_2$) if and only if there exists a constant $c$ such as $\lvert below_{M_1}(X) \rvert \le c \cdot \lvert below_{M_2}(X) \rvert$ for any sequence $X$.
+> * $M_1$ is superior to $M_2$ (denoted $M_1 \preceq M_2$) if and only if there exists a constant $c$ such as $\lvert \mathit{below}_{M_1}(X) \rvert \le c \cdot \lvert \mathit{below}_{M_2}(X) \rvert$ for any sequence $X$.
 > * $M_1$ and $M_2$ are equivalent (denoted $M_1 \equiv M_2$) if and only if $M_1 \preceq M_2$ and $M_2 \preceq M_1$.
 
 That definition seems to match the one proposed much earlier by Alistair Moffat and Ola Petersson in *A Framework for Adaptive Sorting*, though the authors use the symbol $\supseteq$ instead of $\preceq$.
 
-To prove that two measures of disorder were equivalent, authors have used the simpler method of showing that there exists some non-0 constants $c$ and $d$ such as $M_1(X) \le c \cdot M_2(X) \le d \cdot M_1(X)$. For example, the result $Max \equiv Dis$ below was originally obtained by proving that $Max(X) \le Dis(X) \le 2 Max(X)$ for any sequence $X$.
+To prove that two measures of disorder were equivalent, authors have used the simpler method of showing that there exists some non-0 constants $c$ and $d$ such as $M_1(X) \le c \cdot M_2(X) \le d \cdot M_1(X)$. For example, the result $\mathit{Max} \equiv \mathit{Dis}$ below was originally obtained by proving that $\mathit{Max}(X) \le \mathit{Dis}(X) \le 2 \mathit{Max}(X)$ for any sequence $X$.
 
 The graph below shows the partial ordering of several measures of disorder:
 - *Reg* is a measure of presortedness superior to all other ones in the graph.
@@ -144,7 +144,7 @@ Measures of disorder are pretty formalized, so the names of the functions in the
 Let's consider the following functions to compare two elements elements of a sequence:
 
 $$
-comp(x, y)=
+\mathit{comp}(x, y)=
 \begin{cases}
 1 & \text{ if } x \lt y\\
 -1 & \text{ if } x \gt y\\
@@ -188,7 +188,7 @@ Our implementation is slightly different from the original description in *Subli
 
 **Note:** *Block* does not seem to respect Mannila's criterion 3 in the presence of *equivalent elements*.
 
-**Note²:** `probe::block` does not respect Mannila's criterion 4: $Block(\langle 1, 0 \rangle) = 1$ and $Block(\langle 2, 3 \rangle) = 0$, but $Block(\langle 1, 0, 2, 3 \rangle) = 2$.
+**Note²:** `probe::block` does not respect Mannila's criterion 4: $\mathit{Block}(\langle 1, 0 \rangle) = 1$ and $\mathit{Block}(\langle 2, 3 \rangle) = 0$, but $\mathit{Block}(\langle 1, 0, 2, 3 \rangle) = 2$.
 
 ### *Dis*
 
@@ -222,13 +222,13 @@ Computes an approximation of the number of encroaching lists that can be extract
 
 Those lists are called encroaching because the bounds of a given list "encroach" those of all lists on its right.
 
-The number of encroaching lists does not satisfy the formal definition of a measure of presortedness because it returns $1$ for non-empty sorted sequences instead of $0$, which does not respect first Mannila's criterion. Using $Enc(X) - 1$ does not work either because it does not respect Mannila's fourth criterion. To circumvent these issues, `probe::enc` implements an equivalent measure of disorder $M_{Enc}$ proposed by V. Estivill-Castro in *Sorting and Measures of Disorder*, which satisfies all of Mannila's criteria for what makes a measure of presortedness:
+The number of encroaching lists does not satisfy the formal definition of a measure of presortedness because it returns $1$ for non-empty sorted sequences instead of $0$, which does not respect first Mannila's criterion. Using $\mathit{Enc}(X) - 1$ does not work either because it does not respect Mannila's fourth criterion. To circumvent these issues, `probe::enc` implements an equivalent measure of disorder $M_\mathit{Enc}$ proposed by V. Estivill-Castro in *Sorting and Measures of Disorder*, which satisfies all of Mannila's criteria for what makes a measure of presortedness:
 
 $$
-M_{Enc}(X)=
+M_\mathit{Enc}(X)=
 \begin{cases}
 0 & \text{if } X \text{ is sorted,}\\
-Enc(X_{tail}) & \text{otherwise, where } X_{tail} \text{ is } X \text{ without its leading ascending run.}
+\mathit{Enc}(X_\mathit{tail}) & \text{otherwise, where } X_\mathit{tail} \text{ is } X \text{ without its leading ascending run.}
 \end{cases}
 $$
 
@@ -244,17 +244,17 @@ $$
 #include <cpp-sort/probes/exc.h>
 ```
 
-Computes the minimum number of exchanges required to sort $X$, which corresponds to $\lvert X \rvert$ minus the number of cycles in the sequence. A cycle corresponds to a number of elements in a sequence that need to be rotated to be in their sorted position; for example, let $\langle 2, 4, 0, 6, 3, 1, 5 \rangle$ be a sequence, the cycles are $\langle 0, 2 \rangle$ and $\langle 1, 3, 4, 5, 6 \rangle$ so $Exc(X) = \lvert X \rvert - 2 = 5$.
+Computes the minimum number of exchanges required to sort $X$, which corresponds to $\lvert X \rvert$ minus the number of cycles in the sequence. A cycle corresponds to a number of elements in a sequence that need to be rotated to be in their sorted position; for example, let $\langle 2, 4, 0, 6, 3, 1, 5 \rangle$ be a sequence, the cycles are $\langle 0, 2 \rangle$ and $\langle 1, 3, 4, 5, 6 \rangle$ so $\mathit{Exc}(X) = \lvert X \rvert - 2 = 5$.
 
-**Warning:** `probe::exc` generally returns a result higher than the minimum number of exchanges required to sort $X$ when it contains *equivalent elements*. This is because extending $Exc$ to *equivalent elements* is a NP-hard problem (see *On the Cost of Interchange Rearrangement in Strings* by Amir et al). The function does handle such elements in some simple cases, but not in the general case.
+**Warning:** `probe::exc` generally returns a result higher than the minimum number of exchanges required to sort $X$ when it contains *equivalent elements*. This is because extending $\mathit{Exc}$ to *equivalent elements* is a NP-hard problem (see *On the Cost of Interchange Rearrangement in Strings* by Amir et al). The function does handle such elements in some simple cases, but not in the general case.
 
 | Complexity  | Memory      | Iterators     | Monotonic |
 | ----------- | ----------- | ------------- | --------- |
 | n log n     | n           | Forward       | Yes       |
 
 `max_for_size`: $\lvert X \rvert - 1$ when every element in $X$ is one element away from its sorted position.
 
-**Note:** *Exc* does not respect Mannila's criterion 3 (a subsequence contains no more disorder than the whole sequence): $Exc(\langle 3, 1, 2, 0 \rangle) = 1$, but $Exc(\langle 3, 1, 2 \rangle) = 2$.
+**Note:** *Exc* does not respect Mannila's criterion 3 (a subsequence contains no more disorder than the whole sequence): $\mathit{Exc}(\langle 3, 1, 2, 0 \rangle) = 1$, but $\mathit{Exc}(\langle 3, 1, 2 \rangle) = 2$.
 
 *Warning: this algorithm might be noticeably slower when the passed range is not random-access.*
 
@@ -272,9 +272,9 @@ Computes the number of elements in $X$ that are not in their sorted position, wh
 
 `max_for_size`: $\lvert X \rvert$ when every element in $X$ is one element away from its sorted position.
 
-**Note:** *Ham* does not respect Mannila's criterion 3 (a subsequence contains no more disorder than the whole sequence): $Ham(\langle 3, 1, 2, 0 \rangle) = 2$, but $Ham(\langle 3, 1, 2 \rangle) = 3$.
+**Note:** *Ham* does not respect Mannila's criterion 3 (a subsequence contains no more disorder than the whole sequence): $\mathit{Ham}(\langle 3, 1, 2, 0 \rangle) = 2$, but $\mathit{Ham}(\langle 3, 1, 2 \rangle) = 3$.
 
-**Note²:** *Ham* does not respect Mannila's criterion 5: $Ham(\langle 4, 1, 2, 3 \rangle) \not \le \lvert \langle 1, 2, 3 \rangle \rvert + Ham(\langle 1, 2, 3 \rangle)$.
+**Note²:** *Ham* does not respect Mannila's criterion 5: $\mathit{Ham}(\langle 4, 1, 2, 3 \rangle) \not \le \lvert \langle 1, 2, 3 \rangle \rvert + \mathit{Ham}(\langle 1, 2, 3 \rangle)$.
 
 ### *Inv*
 
@@ -322,7 +322,7 @@ The measure of disorder is slightly different from its original description in [
 
 `max_for_size`: $\frac{\lvert X \rvert + 1}{2} - 1$ when $X$ is a sequence of elements that are alternatively greater then lesser than their previous neighbour.
 
-**Note:** `probe::mono` does not respect Mannila's criterion 4: $Mono(\langle 1, 2, 3, 4, 5 \rangle) = 0$ and $Mono(\langle 10, 9, 8, 7, 6 \rangle) = 0$, but $Mono(\langle 1, 2, 3, 4, 5, 10, 9, 8, 7, 6 \rangle) = 1$.
+**Note:** `probe::mono` does not respect Mannila's criterion 4: $\mathit{Mono}(\langle 1, 2, 3, 4, 5 \rangle) = 0$ and $\mathit{Mono}(\langle 10, 9, 8, 7, 6 \rangle) = 0$, but $\mathit{Mono}(\langle 1, 2, 3, 4, 5, 10, 9, 8, 7, 6 \rangle) = 1$.
 
 ### *Osc*
 
@@ -340,9 +340,9 @@ Computes the *Oscillation* measure described by C. Levcopoulos and O. Petersson
 
 **Note:** *Osc* does not seem to respect Mannila's criterion 3 in the presence of *equivalent elements*.
 
-**Note²:** *Osc* does not respect Mannila's criterion 4: $Osc(\langle 0 \rangle) = 0$ and $Osc(\langle 3, 2, 1 \rangle) = 0$, but $Osc(\langle 0, 3, 2, 1 \rangle) = 2$.
+**Note²:** *Osc* does not respect Mannila's criterion 4: $\mathit{Osc}(\langle 0 \rangle) = 0$ and $\mathit{Osc}(\langle 3, 2, 1 \rangle) = 0$, but $\mathit{Osc}(\langle 0, 3, 2, 1 \rangle) = 2$.
 
-**Note³:** *Osc* does not respect Mannila's criterion 5: $Osc(\langle 3, 0, 4, 2, 5, 1 \rangle) \not \le \lvert \langle 0, 4, 2, 5, 1 \rangle \rvert + Osc(\langle 0, 4, 2, 5, 1 \rangle)$, simplified: $11 \not \le 5 + 5$.
+**Note³:** *Osc* does not respect Mannila's criterion 5: $\mathit{Osc}(\langle 3, 0, 4, 2, 5, 1 \rangle) \not \le \lvert \langle 0, 4, 2, 5, 1 \rangle \rvert + \mathit{Osc}(\langle 0, 4, 2, 5, 1 \rangle)$, simplified: $11 \not \le 5 + 5$.
 
 ### *Rem*
 
@@ -386,7 +386,7 @@ Spearman's footrule distance: sum of distances between the position of individua
 
 `max_for_size`: $\frac{\lvert X \rvert²}{2}$ when $X$ is sorted in reverse order.
 
-**Note:** *Spear* does not respect Mannila's criterion 5: $Spear(\langle 4, 1, 2, 3 \rangle) \not \le \lvert \langle 1, 2, 3 \rangle \rvert + Spear(\langle 1, 2, 3 \rangle)$.
+**Note:** *Spear* does not respect Mannila's criterion 5: $\mathit{Spear}(\langle 4, 1, 2, 3 \rangle) \not \le \lvert \langle 1, 2, 3 \rangle \rvert + \mathit{Spear}(\langle 1, 2, 3 \rangle)$.
 
 ### *SUS*
 
@@ -428,7 +428,7 @@ The following definition is also given to determine whether a sequence is *p*-so
 
 *Right invariant metrics and measures of presortedness* by V. Estivill-Castro, H. Mannila and D. Wood mentions that:
 
-> In fact, *Par*($X$) = *Dis*($X$), for all $X$.
+> In fact, $\mathit{Par}(X) = \mathit{Dis}(X)$, for all $X$.
 
 In their subsequent papers, those authors consistently use *Dis* instead of *Par*, often accompanied by a link to *A New Measure of Presortedness*.