Add a visual explanation of the measure of disorder Osc

Author: Morwenn

README.md

@@ -231,7 +231,7 @@ Libraries*](https://arxiv.org/abs/1505.01962).
 * The algorithm behind `utility::quicksort_adversary` is a fairly straightforward adaptation of the
 one provided by M. D. McIlroy in [*A Killer Adversary for Quicksort*](https://www.cs.dartmouth.edu/~doug/mdmspe.pdf).
 
-* The algorithm used by [`utility::check_strict_weak_ordering`][utility-check-strict-weak-ordering] is a reimplementation of the one desribed in the README file of danlark1's [quadratic_strict_weak_ordering project](https://github.com/danlark1/quadratic_strict_weak_ordering).
+* The algorithm used by [`utility::check_strict_weak_ordering`][utility-check-strict-weak-ordering] is a reimplementation of the one desribed in the README file of Danila Kutenin's [quadratic_strict_weak_ordering project](https://github.com/danlark1/quadratic_strict_weak_ordering).
 
 * The test suite reimplements random number algorithms originally found in the following places:
   - [xoshiro256\*\*](https://prng.di.unimi.it/)

docs/Measures-of-disorder.md

@@ -332,6 +332,12 @@ The measure of disorder is slightly different from its original description in [
 
 Computes the *Oscillation* measure described by C. Levcopoulos and O. Petersson in *Adaptive Heapsort*, using an algorithm devised by J. Nehring.
 
+Let $\lvert \lvert \mathit{Cross}(x_i) \rvert \rvert$ be the number of links between adjacent pairs that "cross" the value $x_i$. We define the oscillation measure as $\mathit{Osc}(X) = \sum_{i=0}^{\lvert X \rvert - 1} \lvert \lvert \mathit{Cross}(x_i) \rvert \rvert$.
+
+![Plot of Cross(x_5) over the sequence 6, 3, 9, 8, 4, 7, 1, 10](images/measures-of-disorder-osc-cross.png)
+
+In the illustration above, we can see that a horizontal line drawn from $x_5$ crosses three pairs of adjacent elements in the geometrical representation of the sequence $X = \langle 6, 3, 9, 8, 4, 7, 1, 10 \rangle$. Thus $\lvert \lvert \mathit{Cross}(x_5) \rvert \rvert = 3$. In this example, we have $\mathit{Osc(X) = 17}$.
+
 | Complexity  | Memory      | Iterators     | Monotonic |
 | ----------- | ----------- | ------------- | --------- |
 | n log n     | n           | Forward       | No        |

docs/images/measures-of-disorder-osc-cross.png

@@ -0,0 +1,50 @@
+\documentclass[margin=0.5cm]{standalone}
+\usepackage{tikz}
+\pagestyle{empty}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw[thin,->] (-0.25,0) -- (8,0) node[right] {$pos$};
+\draw[thin,->] (0,-0.25) -- (0,6) node[above] {$val$};
+
+% ticks
+\foreach \x [count=\xi starting from 0] in {1,2,3,4,5,6,7}{
+    \draw (\x,2pt) -- (\x,-2pt);
+    \ifodd\xi
+        \node[anchor=north] at (\x,0) {$\x$};
+    \fi
+}
+\foreach \x [count=\xi starting from 0] in {1,2,3,4,5,6,7,8,9,10,11}{
+    \draw (2pt,\x/2) -- (-2pt,\x/2);
+    \ifodd\xi
+        \node[anchor=east] at (0,\x/2) {$\x$};
+    \fi
+}
+
+% points
+\foreach \y [count=\x] in {
+	6, 3, 9, 8, 4, 7, 1, 10
+}{
+    \fill (\x-1,\y/2) circle (2pt);
+}
+
+\tikzset{every edge/.append style = {gray}};
+
+\path[every node/.style={font=\sffamily\small}]
+  (0,6/2) edge (1,3/2)
+  (1,3/2) edge (2,9/2)
+  (2,9/2) edge (3,8/2)
+  (3,8/2) edge (4,4/2)
+  (4,4/2) edge (5,7/2)
+  (5,7/2) edge (6,1/2)
+  (6,1/2) edge (7,10/2);
+
+
+\tikzset{every edge/.append style = {dashed,gray}};
+
+\path[every node/.style={font=\sffamily\small}]
+  (0,7/2) edge (8,7/2);
+
+\end{tikzpicture}
+\end{document}