Update

sorting.md

@@ -32,7 +32,7 @@ for j from 0 to N - 2 (inclusive)
       swap array[i] and array[i + 1]
 ```
 
-How does this work? Firstly notice there are two loops. The outer loop, with counter variable *j*, runs *N - 1* times -- in each iteration of this loop we will ensure one value gets to its correct place; specifically the values will be getting to their correct places from the top -- highest values will be sorted first (you can also implement the algorithm the other way around too, to sort the lowest values first, try it as an exercise). This makes sense, imagine that we have e.g. a sequence of length *N = 4* -- then the outer loop will run *N - 1 = 3* times (*j* will have values 0, 1 and 2); after fist iteration 1 value will be in its correct place, after 2 iterations 3 values will be in place and after 3 iterations 3 values will be in place which also means the last (forth) value has to be in place too, i.e. the array must be sorted. Now for the inner loop (with variable *i*): this one ensures actually getting the value in its place. Notice it goes from 0 to the top and always compares two neighbors in the array -- if the bottom neighbor is higher than the top neighbor, the loop swaps them, ensuring that the highest value will get to the top (it kind of "bubbles" up, hence the algorithm name). Also notice this loop doesn't always go to the very end of the array! It subtracts the value *j* from its top boundary because there the values that are already in place reside, so we don't need to sort them anymore; the inner loop can end earlier and earlier as the outer loop progresses. The algorithm would still work if we went through the whole array every time (try it), but its [time complexity](time_complexity.md) would suffer, i.e. by noticing the inner loop can get progressively shorter we greatly [optimize](optimization.md) the algorithm. Anyway, how the algorithm actually works is best seen on an example, so let's now try to use the algorithm to sort the following sequence:
+How does this work? Firstly notice there are two loops. The outer loop, with counter variable *j*, runs *N - 1* times -- in each iteration of this loop we will ensure one value gets to its correct place; specifically the values will be getting to their correct places from the top -- highest values will be sorted first (you can also implement the algorithm the other way around too, to sort the lowest values first, try it as an exercise). This makes sense, imagine that we have e.g. a sequence of length *N = 4* -- then the outer loop will run *N - 1 = 3* times (*j* will have values 0, 1 and 2); after fist iteration 1 value will be in its correct place, after 2 iterations 3 values will be in place and after 3 iterations 3 values will be in place which also means the last (forth) value has to be in place too, i.e. the array must be sorted. Now for the inner loop (with variable *i*): this one ensures actually getting the value in its place. Notice it goes from 0 to the top and always compares two adjacent values in the array -- if the bottom neighbor is higher than the top neighbor, the loop swaps them, ensuring that the highest value will get to the top (it kind of "bubbles" up, hence the algorithm name). Also notice this loop doesn't always go to the very end of the array! It subtracts the value *j* from its top boundary because there the values that are already in place reside, so we don't need to sort them anymore; the inner loop can end earlier and earlier as the outer loop progresses. The algorithm would still work if we went through the whole array every time (try it), but its [time complexity](time_complexity.md) would suffer, i.e. by noticing the inner loop can get progressively shorter we greatly [optimize](optimization.md) the algorithm. Anyway, how the algorithm actually works is best seen on an example, so let's now try to use the algorithm to sort the following sequence:
 
 ```
 3 7 8 3 2