highschool/Grade 10/Computer Science/ICS4U1/Sorting Methods/Insertion Sort.md

# Inssertion sort

## What is Insertion sort
- a sorting algorithm
- dividng an array into sroted and unsorted sections
- removes elements in idividually from un unosrted section of the list and inserts it into its correct position in a new sorted seqeuence

## Algorithm
- Set a key for the sections after the 1st element
- repeat the following
    - select first unsorted element as the key 
    - move sorted elements to the right to create space
    - shift unsorted elemnet into correct position
    - advance the key to the right by 1 element
- Tests first unsorted element (key) to last element of sorted section
- If key is larger, insertion sort leaves the element in place and hte next index's element becomes the key
- Else it finds the correct position within the sorted list
     - Duplicate the larger element in to one ahead of its current index each time the key is tested
     - By looping this, it makes space and inserts the key into that correct position

## Pseudo0code
```
for x = 1 : n
    key = list[x]
    y = x-1
    while y >= 0 && key < list[y]
        insert list[x] into the sorted sequence list[1 .. x-1]
        list[y+1] = list[y]
        y--
    list[y+1] = key
```

## Code
```java
for(int i=1; i<numbers.length; i++) {
    key = numbers[i];
    j = i-1;
    while(j >= 0 && numbers[j] > key) {
        numbers[j + 1] = numbers[j];
        j--;
    }
    numbers[j + 1] = key;
}
```
## Attributes
- Stable
    - Instead of swapping elements, all elements are shifted one position ahead
- Adaptable
    - if input is already sorted then time complexity will be $`O(n)`$
- Space complexity
    - In-place sorting, original array is updated rather than creating a new one, space complexity $`O(1)`$
- Very low overhead    

## Time Complexity
- Best Case
    - $`O(N)`$
- Worst Case
    - $`O(N^2)`$
- Average Case
    - $`O(N^2)`$


Number of passes and comparisons

- Number of passes
    - Insetion sort always require n-1 passes
- Mininimum number of comparisons = $`n-1`$
- maximum number of comparisons = $`\dfrac{n^2-n}{2}`$ or $`n(n-1)/2`$
- Average number of comparisons = $`\dfrac{n^2-n}{4}`$ or $`n(n-1)/4`$

## When To Use
- Good to use when the array is `nearly sorted`, and only a few elements are misplaced
- Method is efficient - reduces the number of comparisons if in a pratially sorted array
- Method will simply insert the unsorted elements into its correct place.

## When not to use
- In efficient for `inverse sorting`
- Descending order is considered the worst unsorted case
- Not as efficient if list is completely unsorted

## Relation to Selection Sort
- Outer loop over every index
- Inner loop
- Each pass (within the inner loop) increases the number of sorted items by one until there are no more unsorted items

## Differences To Selection Sort

Selection Sort
- Taking the current item and swapping it with the smallest item on the right side of the list
- More simple
- One possible case $`O(n^2)`$

Insertion Sort
- Taking the current item and inserting into the right position by adjusting the list
- more efficient (best case $`O(n)`$)