912. Sort an Array - Detailed Explanation

Problem Statement

You are given an integer array. Your task is to sort this array in ascending order. The sorted array should have all the numbers arranged from the smallest to the largest.

Example 1

Input:

nums = [5, 2, 3, 1]

Output:

[1, 2, 3, 5]

Explanation:
When you sort the array [5, 2, 3, 1] in ascending order, you get [1, 2, 3, 5].

Example 2

Input:

nums = [5, 1, 1, 2, 0, 0]

Output:

[0, 0, 1, 1, 2, 5]

Explanation:
Sorting the array [5, 1, 1, 2, 0, 0] results in [0, 0, 1, 1, 2, 5].

Constraints

• The number of elements in the array can be large (up to 10⁵ or more).
• The array can contain negative integers.
• The expected time complexity is O(n log n) for efficient solutions.

Hints

1. Built-in Methods:
   Most programming languages offer built-in sorting functions which are optimized for average-case performance (often O(n log n)).
2. Implementing Sorting Algorithms:
   Consider implementing efficient algorithms like Merge Sort or Quick Sort if you need to code the sorting logic manually.
3. Edge Cases:
   Think about arrays that are already sorted, reverse sorted, or contain duplicates and negative numbers.

Approach 1: Brute Force (Inefficient)

Explanation

A brute force approach might involve comparing every pair of elements and swapping them if they are out of order. This is similar to Bubble Sort or Selection Sort.

• Drawbacks:
  • Time Complexity: O(n²) – inefficient for large arrays.
  • Not Recommended: For arrays with many elements, this approach is impractical.

Approach 2: Optimal Sorting Algorithms

There are several optimal sorting algorithms with an average time complexity of O(n log n):

Option 1: Merge Sort

• Concept:
  Merge Sort is a divide-and-conquer algorithm that splits the array into two halves, recursively sorts each half, and then merges the two sorted halves.
• Complexity:
  • Time: O(n log n)
  • Space: O(n) (due to auxiliary arrays during merging)

Option 2: Quick Sort

• Concept:
  Quick Sort selects a pivot element and partitions the array so that elements less than the pivot come before it and elements greater come after it. It then recursively sorts the partitions.
• Complexity:
  • Average Time: O(n log n)
  • Worst-case Time: O(n²) (can be improved with random pivot selection)
  • Space: In-place version uses O(log n) stack space

Option 3: Heap Sort

• Concept:
  Heap Sort converts the array into a heap (usually a max-heap) and repeatedly extracts the maximum element, building the sorted array in reverse.
• Complexity:
  • Time: O(n log n)
  • Space: O(1) (in-place)

Most languages provide highly optimized built-in sorting algorithms (like Timsort in Python or Dual-Pivot Quick Sort in Java) that are efficient and reliable.

Step-by-Step Walkthrough (Merge Sort Example)

1. Divide:
   Recursively split the array into halves until each subarray has only one element.
2. Conquer:
   A single element is considered sorted.
3. Merge:
   Combine two sorted subarrays into one sorted array by comparing the smallest elements of each subarray and choosing the smaller one to add to the merged array.
4. Result:
   The fully merged and sorted array is returned.

Visual Example:
For input [5, 2, 3, 1]:

• Split into: [5, 2] and [3, 1]
• Sort [5, 2] to get [2, 5] and [3, 1] to get [1, 3]
• Merge [2, 5] and [1, 3] to form [1, 2, 3, 5]

Complexity Analysis

• Merge Sort:
  • Time: O(n log n)
  • Space: O(n)
• Quick Sort:
  • Average Time: O(n log n)
  • Worst-case Time: O(n²) (mitigated by randomization)
  • Space: O(log n) in average in-place implementations
• Heap Sort:
  • Time: O(n log n)
  • Space: O(1)

Python Code

Below is an implementation using Python's built-in sort (which uses Timsort) and an example of a custom Merge Sort implementation.