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content/Computer Science/Algorithms & Data Structures Analysis/Exercises/3.2.1.md
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| > [!exercise]+ Exercise 1 - Sort a list whose numbers span from 1 to 9. | ||
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| > [!exercise]+ 1. Create an array of size $k$, and increment each position by 1 for every occurrence of that position in the array. | ||
| > | ||
| > $$ | ||
| > \text{Unsorted Array:} \quad \begin{array}{|c|c|c|c|c|c|c|} | ||
| > 1. Create an array of size $k$, and increment each position by 1 for every occurrence of that position in the array. | ||
| > $$\begin{array}{l} | ||
| > \text{Unsorted Array:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 1 & 4 & 1 & 2 & 7 & 5 & 2 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > \end{array}$$ | ||
| > | ||
| > $$ | ||
| > \text{Element Counts:} \quad \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 0 & 2 & 2 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > 2. Starting from the second position, we add the value of the previous position to the current position. | ||
| > $$ | ||
| > \text{Element Counts:} \quad \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > 2. Starting from the second position, we add the value of the previous position to the current position. | ||
| > $$\begin{array}{l} | ||
| > \text{Element Counts:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 0 & 2 & 4 & 4 & 5 & 6 & 6 & 7 & 7 & 7 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > 3. Create a sorted array of length max of previous array (7) | ||
| > $$ | ||
| > \text{Sorted Array:} \quad \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \end{array}$$ | ||
| > | ||
| > 3. Create a sorted array of length max of previous array (7) | ||
| > $$\begin{array}{l} | ||
| > \text{Sorted Array:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > & & & & & & \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > 4. Iterate through the unsorted array. For each element $i$ in the unsorted array, find the value at position $i$ in the element counts array. This value is the position that $i$ will be in the sorted array. Decrement the value in the counts array by 1. | ||
| > - For example, the first element in the unsorted array is 1. We go to $counts[1]$ and find that the value is $2$. We place $1$ at position $2$ in the sorted array, and decrement $counts[1]$ by 1. | ||
| > $$ | ||
| > \text{Element Counts:} \quad \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \end{array}$$ | ||
| > | ||
| > 4. Iterate through the unsorted array. For each element $i$ in the unsorted array, find the value at position $i$ in the element counts array. This value is the position that $i$ will be in the sorted array. Decrement the value in the counts array by 1. | ||
| > - For example, the first element in the unsorted array is 1. We go to $\text{counts}[1]$ and find that the value is $2$. We place $1$ at position $2$ in the sorted array, and decrement $\text{counts}[1]$ by 1. | ||
| > $$\begin{array}{l} | ||
| > \text{Element Counts:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 0 & 1 & 4 & 4 & 5 & 6 & 6 & 7 & 7 & 7 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > $$ | ||
| > \text{Sorted Array:} \quad \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \end{array}$$ | ||
| > $$\begin{array}{l} | ||
| > \text{Sorted Array:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > & 1 & & & & & \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > 5. Repeat this for every element in the unsorted array | ||
| > $$ | ||
| > \text{Element Counts:} \quad \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \end{array}$$ | ||
| > | ||
| > 5. Repeat this for every element in the unsorted array | ||
| > $$\begin{array}{l} | ||
| > \text{Element Counts:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 0 & 0 & 2 & 4 & 4 & 5 & 6 & 6 & 7 & 7 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > $$ | ||
| > \text{Sorted Array:} \quad \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \end{array}$$ | ||
| > $$\begin{array}{l} | ||
| > \text{Sorted Array:} \\ | ||
| > \begin{array}{|c|c|c|c|c|c|c|} | ||
| > \hline | ||
| > 1 & 1 & 2 & 2 & 4 & 5 & 7 \\ | ||
| > \hline | ||
| > \end{array} | ||
| > $$ | ||
| > \end{array}$$ |