Broken Calculator

On a broken calculator that has a number showing on its display, we can perform two operations:

  • Double: Multiply the number on the display by 2, or;
  • Decrement: Subtract 1 from the number on the display.

Initially, the calculator is displaying the number X.

Return the minimum number of operations needed to display the number Y.

Example 1:

Input: X = 2, Y = 3
Output: 2
Explanation: Use double operation and then decrement operation {2 -> 4 -> 3}.

Example 2:

Input: X = 5, Y = 8
Output: 2
Explanation: Use decrement and then double {5 -> 4 -> 8}.

Example 3:

Input: X = 3, Y = 10
Output: 3
Explanation:  Use double, decrement and double {3 -> 6 -> 5 -> 10}.

Example 4:

Input: X = 1024, Y = 1
Output: 1023
Explanation: Use decrement operations 1023 times.

Note:

  1. 1 <= X <= 10^9
  2. 1 <= Y <= 10^9

Intuition

Here we are going to start from Y and will reach X with minimum number of permissible operations. To do so, instead of multiplying by 2 or subtracting 1 from X, we could divide by 2 (when Y is even) or add 1 to Y. The motivation for this is that it turns out we always greedily divide by 2:

Algorithm

While Y is larger than X, add 1 if it is odd, else divide by 2. After, we need to do (X - Y) additions to reach X.

class Solution {
    public int brokenCalculator(int X, int Y) {
        int ans = 0;
        while (Y > X) {
            ans++;
            if (Y % 2 == 1)
                Y++;
            else
                Y /= 2;
        }
        return ans + (X - Y);
    }
}

Complexity Analysis

  • Time Complexity: O(logY).
  • Space Complexity: O(1).

Categories: Greedy

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