StudyLabs

Lecture 09 - Jan 22

2 min read

Insertion Sort

Insertion sort builds the sorted array one element at a time by repeatedly inserting the next element into its correct position.

Example

Input Array: [51, 26, 3, 65, 91, 8]

Initial: [51 | 26, 3, 65, 91, 8] (first element is trivially sorted)

Step 1 (i=2): Insert 26

  • Compare 26 with 51: 26 < 51, so shift 51 right
  • Result: [26, 51 | 3, 65, 91, 8]

Step 2 (i=3): Insert 3

  • Compare 3 with 51: shift 51 right
  • Compare 3 with 26: shift 26 right
  • Result: [3, 26, 51 | 65, 91, 8]

Step 3 (i=4): Insert 65

  • Compare 65 with 51: 65 > 51, already in position
  • Result: [3, 26, 51, 65 | 91, 8]

Step 4 (i=5): Insert 91

  • Compare 91 with 65: 91 > 65, already in position
  • Result: [3, 26, 51, 65, 91 | 8]

Step 5 (i=6): Insert 8

  • Compare 8 with 91, 65, 51, 26: shift all right
  • Compare 8 with 3: 8 > 3, insert after 3
  • Result: [3, 8, 26, 51, 65, 91] ✓ Sorted

Algorithm

InsertionSort(A, n):

// Array A has indices 1 to n
for i = 2 to n:                      // Start from 2nd element
    key = A[i]                       // Element to be inserted
    j = i - 1                        // Index of last element in sorted portion
    
    // Shift elements greater than key to the right
    while (j > 0 AND A[j] > key):
        A[j + 1] = A[j]              // Shift element right
        j = j - 1                    // Move to previous element
    
    A[j + 1] = key                   // Insert key at correct position

Time Complexity Analysis

Worst Case: Array in reverse order [91, 65, 51, 26, 8, 3]

  • For each element i, we compare with all i-1 previous elements
  • Total comparisons:

Best Case: Array already sorted [3, 8, 26, 51, 65, 91]

  • For each element, only one comparison (no shifts needed)
  • Total comparisons:

Average Case: Random order

Space Complexity: (in-place sorting)

Characteristics

  • Stable: Equal elements maintain their relative order
  • In-place: Requires only additional space
  • Adaptive: Efficient for nearly sorted data
  • Online: Can sort data as it’s received

Best for: Small arrays or nearly sorted data

Graph View