Minimum Moves to Reach Target Score

Problem Description

You are playing a game with integers. You start with the integer 1 and you want to reach the integer target. In one move, you can either increment the current integer by one or double the current integer. You can use the increment operation any number of times, however, you can only use the double operation at most maxDoubles times. Given the two integers target and maxDoubles, return the minimum number of moves needed to reach target starting with 1.

Key Insights

• The increment operation can be performed without limits, making it flexible for reaching the target.
• The double operation can significantly increase the current integer, but it is limited by the maxDoubles parameter.
• The strategy involves maximizing the use of the double operation early on, then using increments to reach the target.

Space and Time Complexity

Time Complexity: O(log(target)) - In the worst case, we may need to perform a logarithmic number of operations relative to the target. Space Complexity: O(1) - We are using a constant amount of space regardless of the input size.

Solution

The solution involves a greedy approach where we first try to reach as close to the target as possible using the double operation while it is available. Once we can no longer double, we use increments to reach the target. The algorithm can be summarized as follows:

1. Start from the target and work backwards to the initial value of 1.
2. If the current target is even and we still have double operations left, halve the target (simulating a reverse double).
3. If the current target is odd, increment the target (simulating a reverse increment).
4. Count the number of operations until reaching 1.

Code Solutions