## 1. Introduction

In this tutorial, we'll **learn Selection Sort**, see its implementation in Java, and analyze its performance.

## 2. Algorithm Overview

Selection Sort **begins with the element in the 1**^{st} position of an unsorted array and scans through subsequent elements to **find the smallest element**. Once found, the smallest element is swapped with the element in the 1^{st} position.

The algorithm then moves on to the element in the 2^{nd} position and scans through subsequent elements to find the index of the 2^{nd} smallest element. Once found, the second smallest element is swapped with the element in the 2^{nd} position.

This process goes on until we reach the n-1^{th} element of the array, which puts the n-1^{th} smallest element in the n-1^{th} position. The last element automatically falls in place, in the n-1^{th} iteration, thereby sorting the array.

We find** the largest element instead of the smallest element to sort the array in descending order.**

Let's see an example of an unsorted array and sort it in ascending order to visually understand the algorithm.

### 2.1. An Example

Consider the following unsorted array:

*int[] arr = {5, 4, 1, 6, 2}*

**Iteration 1**

Considering the above working of the algorithm, we start with the element in 1^{st} position – 5 – and scan through all subsequent elements to find the smallest element – 1. We then swap the smallest element with the element in 1^{st} position.

The modified array nows looks like:

*{1, 4, 5, 6, 2}*

Total comparisons made: 4

**Iteration 2**

In the second iteration, we move on to the 2^{nd} element – 4 – and scan through subsequent elements to find the second smallest element – 2. We then swap the second smallest element with the element in 2^{nd} position.

The modified array now looks like:

*{1, 2, 5, 6, 4}*

Total comparisons made: 3

Continuing similarly, we have the following iterations:

**Iteration 3**

*{1, 2, 4, 6, 5}*

Total comparisons made: 2

**Iteration 4**

*{1, 2, 4, 5, 6}*

Total comparisons made: 1

**3.** **Implementation**

Let's implement Selection Sort using a couple of *for* loops:

public static void sortAscending(final int[] arr) {
for (int i = 0; i < arr.length - 1; i++) {
int minElementIndex = i;
for (int j = i + 1; j < arr.length; j++) {
if (arr[minElementIndex] > arr[j]) {
minElementIndex = j;
}
}
if (minElementIndex != i) {
int temp = arr[i];
arr[i] = arr[minElementIndex];
arr[minElementIndex] = temp;
}
}
}

Of course, to reverse it we could do something quite similar:

public static void sortDescending(final int[] arr) {
for (int i = 0; i < arr.length - 1; i++) {
int maxElementIndex = i;
for (int j = i + 1; j < arr.length; j++) {
if (arr[maxElementIndex] < arr[j]) {
maxElementIndex = j;
}
}
if (maxElementIndex != i) {
int temp = arr[i];
arr[i] = arr[maxElementIndex];
arr[maxElementIndex] = temp;
}
}
}

And with a bit more elbow grease, we could combine these using *Comparator*s.

## 4. Performance Overview

### 4.1. Time

In the example that we saw earlier, **selecting the smallest element required a total of ***(n-1)* comparisons followed by swapping it to the 1^{st} position. Similarly, **selecting the next smallest element required total ***(n-2)* comparisons followed by swapping in the 2^{nd} position, and so on.

Thus, starting from index 0, we perform *n-1, n-2, n-3, n-4 …. 1* comparisons. The last element automatically falls in place due to previous iterations and swaps.

Mathematically, the **sum of the first ***n-1* natural numbers will tell us how many comparisons we need in order to sort an array of size *n* using Selection Sort.

**The formula for the sum of ***n* natural numbers is *n(n+1)/2*.

In our case, we need the sum of first *n-1* natural numbers. Therefore, we replace *n* with *n-1* in the above formula to get:

*(n-1)(n-1+1)/2 = (n-1)n/2 = ***(n^2-n)/2**

As *n^2* grows prominently as *n* grows, we consider the higher power of *n* as the performance benchmark, making this algorithm have a **time complexity of ***O(n^2).*

### 4.2. Space

In terms of auxiliary space complexity, Selection Sort requires one extra variable to hold the value temporarily for swapping. Therefore, Selection Sort's **space complexity is ***O(1)*.

## 5. Conclusion

Selection Sort is a very simple sorting algorithm to understand and implement. Unfortunately, **its quadratic time complexity makes it an expensive sorting technique**. Also, since the algorithm has to scan through each element, **the best case, average case, and worst-case time complexity is the same**.

Other sorting techniques like Insertion Sort and Shell Sort also have quadratic worst-case time complexity, but they perform better in best and average cases.

Check out the complete code for Selection Sort over on GitHub.

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