Jump Game II

Jump Game II

1. What is this problem about?

The Jump Game II interview question is an optimization challenge. You are given an array of non-negative integers nums, where each element represents your maximum jump distance from that position. You start at the first index and your goal is to reach the last index in the minimum number of jumps. You are guaranteed that you can always reach the end.

2. Why is this asked in interviews?

Companies like Amazon, Microsoft, and Google use the Jump Game II coding problem to test a candidate's ability to move beyond basic search and apply Greedy or Dynamic Programming optimizations. While DP is $O(N^2)$, the greedy solution is $O(N)$. It evaluations your ability to maintain multiple "boundaries" (current reach vs. next reach) in a single pass.

3. Algorithmic pattern used

This problem is best solved using the Greedy (Level-by-Level) pattern.

1. Initialize: jumps = 0, current_jump_end = 0, farthest = 0.
2. Iterate: Loop through the array from index 0 to $n-2$.
3. Explore: At each index, update farthest = max(farthest, i + nums[i]). This is the furthest you can go with one more jump.
4. Step Up: When you reach current_jump_end, it means you must make another jump to proceed.
   • Increment jumps.
   • Set current_jump_end = farthest.
5. Success: The loop ends before the last element, and the total jumps is your answer.

4. Example explanation

nums = [2, 3, 1, 1, 4]

1. $i=0$: farthest = 2. We reached current_jump_end (0). jumps = 1, current_jump_end = 2.
2. $i=1$: farthest = max(2, 1+3) = 4.
3. $i=2$: farthest = max(4, 2+1) = 4. We reached current_jump_end (2). jumps = 2, current_jump_end = 4. Result: 2 jumps.

5. Common mistakes candidates make

• $O(N^2)$ DP: Using a DP table where dp[i] is the min jumps to reach i. This is correct but too slow for $N=10^5$.
• Off-by-one in loop: Iterating all the way to $n-1$. You should stop at $n-2$ because if you are already at the last element, you don't need to jump again.
• Initialization errors: Forgetting to handle arrays of length 1 (where 0 jumps are needed).

6. Interview preparation tip

Think of the greedy approach as a Breadth-First Search on an array. Each "jump" is a level in the BFS. You are trying to find the minimum number of levels to reach the target index. This is a vital Array interview pattern for shortest-path problems.