How to Rotate a Sorted Array

Array rotation is a process where the elements of an array are shifted cyclically. In a sorted array, this operation rearranges the elements while maintaining the relative order of the two segments created by the rotation point.

Understanding Rotation Through an Example

Based on the provided reference, let's examine a specific case of rotating a sorted array.

Consider a sorted array: {1, 2, 3, 4, 5}.

If we rotate this array at index 3 (assuming a 0-based index, so the element is 4), the array is transformed. According to the reference, it will become: `{4, 5, 1, 2, 3}}$.

• What happened? The elements starting from index 3 (4 and 5) have been moved to the front of the array. The elements that were originally at the beginning (1, 2, and 3) have been shifted to the end, following the moved segment. This particular example illustrates a left rotation by 3 positions.

The Fundamental Rotation Step (As Described in the Reference)

The reference also provides a description of the mechanism involved in rotation:

In essence, we moved the element at the last index to the front, while shifting the remaining elements to the right.

This describes a specific type of rotation operation, specifically a single-step right rotation. In this basic operation:

• The element currently at the very end of the array is moved to become the new first element.
• All other elements are shifted one position to their right to make space.

How These Concepts Relate

A complete rotation, like the example showing {1, 2, 3, 4, 5} becoming {4, 5, 1, 2, 3}, is achieved by performing a series of shifts. While the example shows a multi-position left rotation, the reference's description of moving the last element to the front highlights a fundamental way elements are cyclically shifted within the array structure during a rotation process.

Visualizing the Example Rotation

Here's how the example rotation from {1, 2, 3, 4, 5} to `{4, 5, 1, 2, 3}}$ can be visualized:

• Original Array:

  [ 1, 2, 3, 4, 5 ]
    0  1  2  3  4  (Indices)

• Rotation at Index 3: The array is conceptually split after index 2 (3 | 4).

  [ 1, 2, 3 | 4, 5 ]

• Segments Swap Positions: The right segment moves to the front, and the left segment moves to the end.

  [ 4, 5 | 1, 2, 3 ]

• Resulting Rotated Array:

  [ 4, 5, 1, 2, 3 ]
    0  1  2  3  4  (New Indices)

This demonstrates how the elements are rearranged based on the rotation point, consistent with the example provided in the reference.