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Initially on a notepad only one character 'A' is present. You can perform two operations on this notepad for each step:

1. Copy All: You can copy all the characters present on the notepad (partial copy is not allowed).
2. Paste: You can paste the characters which are copied last time.

Given a number n. You have to get exactly n 'A' on the notepad by performing the minimum number of steps permitted. Output the minimum number of steps to get n 'A'.

Example 1:

Input: 3Output: 3Explanation:Intitally, we have one character 'A'.In step 1, we use Copy All operation.In step 2, we use Paste operation to get 'AA'.In step 3, we use Paste operation to get 'AAA'.

Note:

1. The n will be in the range [1, 1000].

Approach #1: DP. [Java]

class Solution {    public int minSteps(int n) {        int[] dp = new int[n+1];                for (int i = 2; i <= n; ++i) {            dp[i] = i;            for (int j = i-1; j > 1; --j) {                if (i % j == 0) {                    dp[i] = dp[j] + (i/j);                    break;                }            }        }                return dp[n];    }}

Approach #2: Greedy. [C++]

public int minSteps(int n) {        int s = 0;        for (int d = 2; d <= n; d++) {            while (n % d == 0) {                s += d;                n /= d;            }        }        return s;    }

Analysis:

We look for a divisor d so that we can make d copies of (n / d) to get n. The process of making d copies takes d steps (1 step of copy All and d-1 steps of Paste)

We keep reducing the problem to a smaller one in a loop. The best cases occur when n is decreasing fast, and method is almost O(log(n)). For example, when n = 1024 then n will be divided by 2 for only 10 iterations, which is much faster than O(n) DP method.

The worst cases occur when n is some multiple of large prime, e.g. n = 997 but such cases are rare.

Reference:

https://leetcode.com/problems/2-keys-keyboard/discuss/105897/Loop-best-case-log(n)-no-DP-no-extra-space-no-recursion-with-explanation

https://leetcode.com/problems/2-keys-keyboard/discuss/105899/Java-DP-Solution

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