Problem Explanation

The problem is about finding the minimum number of steps to get exactly n number of A on a notepad, where the only two operations allowed are Copy All and Paste. This means, initially we have a notepad with a single character A and you have to achieve n number of As on the notepad by using Copy All and Paste operations in the least possible number of steps.

As an example, if the input n is 3, the steps would be:

1. Copy All operation to copy the single A on the notepad.
2. Paste operation to yield AA.
3. Second Paste operation to get AAA.

Hence, the minimum number of steps to get n = 3 is 3.

The range of n will be between 1 to 1000.

Now, let's look at the solution approach and how it uses dynamic programming to solve this problem.

Solution Approach

This problem is a classic example of a dynamic programming problem. Here we start by initializing an array dp of size n + 1 where dp[i] indicates the minimum steps to get i number of As.

The array dp is initialized in such a way that dp[i] is i for all i as in the worst case we can get i number of As by doing Copy All and Paste i-1 times.

Then, we iterate from 2 to n and for each i, perform a nested iteration from i / 2 to 2 in descending order. If i is divisible by j, we update dp[i] as dp[j] + i / j. Finally, dp[n] will have our result.

Here, the idea is if i is divisible by j then we can get i number of As by copying the j number of As and pasting it i / j times.

Python Solution

Here is the equivalent Python solution:

1
2python
3class Solution:
4    def minSteps(self, n: int) -> int:
5        if n <= 1:
6            return 0
7            
8        dp = [i for i in range(n + 1)]
9        
10        for i in range(2, n + 1):
11            for j in range(i // 2, 2, -1):
12                if i % j == 0:
13                    dp[i] = dp[j] + i // j
14                    break
15        return dp[n]

Java Solution

1
2java
3class Solution {
4    public int minSteps(int n) {
5        int[] dp = new int[n + 1];
6        
7        for (int i = 0; i <= n; i++) {
8            dp[i] = i;
9        }
10        
11        for (int i = 2; i <= n; i++) {
12            for (int j = i / 2; j > 2; --j) {
13                if (i % j == 0) {
14                    dp[i] = dp[j] + i / j;
15                    break;
16                }
17            }
18        }
19        return dp[n];
20    }
21}

JavaScript Solution

1
2javascript
3var minSteps = function(n) {
4    let dp = Array(n + 1).fill(0).map((_, i) => i);
5
6    for (let i = 2; i <= n; i++) {
7        for (let j = Math.floor(i / 2); j > 2; j--) {
8            if (i % j === 0) {
9                dp[i] = dp[j] + i / j;
10                break;
11            }
12        }
13    }
14    return dp[n];
15};

C++ Solution

1
2c++
3class Solution {
4public:
5    int minSteps(int n) {
6        vector<int> dp(n+1);
7        iota(dp.begin(), dp.end(), 0);
8        
9        for (int i = 2; i <= n; i++) {
10            for (int j = i / 2; j > 2; --j) {
11                if (i % j == 0) {
12                    dp[i] = dp[j] + i / j;
13                    break;
14                }
15            }
16        }
17        return dp[n];
18    }
19};

C# Solution

1
2csharp
3public class Solution {
4    public int MinSteps(int n) {
5        int[] dp = new int[n+1];
6        for(int i=0; i <= n; i++) {
7            dp[i] = i;
8        }
9        for(int i=2; i <= n; i++) {
10            for(int j=i/2; j > 2; --j) {
11                if(i % j == 0) {
12                    dp[i] = dp[j] + i / j;
13                    break;
14                }
15            }
16        }
17        return dp[n];
18    }
19}

Complexity Analysis

All our solutions have time and space complexity of O(n^2) and O(n), respectively.

The time complexity is O(n^2) because of the nested iteration. The outer iteration is from 2 to n and the inner iteration is from i / 2 to 2. Hence, the total time complexity gets O(n^2).

The space complexity is O(n) because of the storage array dp of size n + 1 that is being used.

Concluding

With initially only one character A, achieving n number of As by using the least possible number of Copy All and Paste operations might seem like a plain brute force approach. However, through the use of dynamic programming, the problem becomes more of locating the smallest possible divisor. The speed and efficiency of dynamic programming speeds up calculations, reducing the search space required and leveraging previously-computed results.

Therefore, the problem is a great practice to understand and implement dynamic programming. Furthermore, the problem can be solved using dynamic programming in Python, JavaScript, C++, and other languages, showing the wide application of dynamic programming in different programming languages.

The key takeaway from this problem is to familiarize yourself with dynamic programming and apply it in suitable scenarios where it can significantly decrease the time complexity. It's also an excellent illustration of how much time and space complexity can be saved by using more advanced data manipulation techniques, specifically through dynamic programming. This problem should encourage you to think about how you can use dynamic programming to optimize your own coding challenges or when trying to optimize time and space in real-world tasks such as these.