"Jump Game II" is an extension of the Jump Game problem, focusing more on optimization. While the original Jump Game asks whether you can reach the last index of an array, Jump Game II asks for the minimum number of jumps required to reach the last index.

Understanding the Problem

Given an array of non-negative integers, where each element represents your maximum jump length at that position, your task is to find the minimum number of jumps you need to reach the last index.

Example

Consider the array [2, 3, 1, 1, 4]. The minimum number of jumps to reach the last index is 2 (jump 1 step from index 0 to 1, then 3 steps to the last index).

Fundamental Concepts

1. Greedy Algorithms: Like in Jump Game, greedy algorithms play a crucial role in Jump Game II. However, the implementation is more complex because it's not just about reaching the end but doing so with the fewest jumps.

2. Dynamic Programming: This approach can also be used, but it tends to be less efficient for this problem, as it might require considering all possible jump combinations, leading to higher time complexity.

Solving Jump Game II

Greedy Approach

1. Initialization: Keep track of the current end of the jump and the farthest point you can reach (initially both are at the first index). Also, maintain a counter for the number of jumps.

2. Iteration:

   • Traverse the array. With each step, update the farthest point you can reach.
   • If you reach the current end of the jump, it means you must make a jump to proceed. Increase the jump counter and update the end of the jump to the farthest point you can currently reach.

3. Result: The jump counter at the end of the iteration gives the minimum number of jumps.

Example Walkthrough

Using [2, 3, 1, 1, 4]:

• Start at index 0, can reach up to index 2. Make a jump.
• From index 1, you can reach the end (index 4). Make another jump.

Total jumps = 2.

Time and Space Complexity

• Time Complexity: The greedy approach usually runs in O(n) time, as it requires traversing the array once.
• Space Complexity: O(1) - no additional space is required apart from variables for tracking the current end, farthest reach, and number of jumps.

Conclusion

Jump Game II, while building on the same principles as Jump Game, adds complexity by requiring the solution to optimize the number of jumps. It's an excellent problem for understanding and applying greedy algorithms in a more nuanced way, requiring careful tracking of multiple variables (current jump end, farthest reach, and number of jumps) throughout the iteration of the array.