# 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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