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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:
Copy All: You can copy all the characters present on the notepad (partial copy is not allowed).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:
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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