2 Keys Keyboard Problem

2 Keys Keyboard Problem

Given a positive integer N and a string S initially it is “A”, the task is to minimize the number of operations required to form a string consisting of N numbers of A’s by performing one of the following operations in each step:

• Copy all the characters present in the string S.
• Append all the characters to the string S which are copied last time.

Examples:

Input: N = 3
Output: 3
Explanation:
Below are the operations performed:
Operation 1: Copy the initial string S once i.e., “A”.
Operation 2: Appending the copied string i.e., “A”, to the string S modifies the string S to “AA”.
Operation 3: Appending the copied string i.e., “A”, to the string S modifies the string S to “AAA”.
Therefore, the minimum number of operations required to get 3 A’s is 3.

Input: N = 7
Output: 7

Approach: The given problem can be solved based on the following observations: 

1. If N = P₁*P₂*P_m where {P₁, P₂, …, P_m} are the prime numbers then one can perform the following moves:
   1. First, copy the string and then paste it (P₁ – 1) times.
   2. Similarly, again copying the string and pasting it for (P₂ – 1) times.
   3. Performing M times with the remaining prime number, one will get the string with N number of A’s.
2. Therefore, the total number of minimum moves needed is given by (P₁ + P₂ + … + P_m).

Follow the steps below to solve the problem:

Below is the implementation of the above approach:

C++

#include <bits/stdc++.h> using namespace std;    int minSteps(int N) {          int ans = 0;             for (int d = 2; d * d <= N; d++) {                              while (N % d == 0) {                                          ans += d;                             N /= d;         }     }             if (N != 1) {         ans += N;     }             return ans; }    int main() {     int N = 3;     cout << minSteps(N);        return 0; }

Time Complexity: O(√N)
Auxiliary Space: O(1)

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