Jump Game VI

Jump Game VI

1. What is this problem about?

The Jump Game VI interview question is a score maximization challenge. You are given an array of integers nums and an integer k. You start at the first index and want to reach the last index. From index i, you can jump to any index in the range [i+1, min(i+k, n-1)]. Your goal is to find the maximum possible total score, where your score is the sum of all elements at the indices you visit.

2. Why is this asked in interviews?

Companies like Microsoft and Google ask the Jump Game VI coding problem to test a candidate's ability to optimize Dynamic Programming. While a standard DP approach is $O(N \cdot K)$, the correct solution uses a Monotonic Queue to achieve $O(N)$ time complexity. It evaluations your skills in handling sliding window extremas.

3. Algorithmic pattern used

This problem follows the DP with Monotonic Queue Optimization pattern.

1. DP State: dp[i] is the maximum score to reach index i.
2. Transition: dp[i] = nums[i] + max(dp[j]) for all $j$ in $[i-k, i-1]$.
3. Monotonic Queue: To find the maximum dp[j] in the window of size $k$ efficiently:
   • Use a deque to store indices of dp values in decreasing order.
   • For each new index i, remove indices from the front of the deque that are outside the window [i-k, i-1].
   • The current max is at the front of the deque.
   • Maintain the decreasing property by removing indices from the back whose dp values are $\leq dp[i]$.

4. Example explanation

nums = [1, -1, -2, 4, -7, 3], $k = 2$

• dp[0] = 1. Deque: [0].
• i=1: Max from deque is dp[0]=1. dp[1] = -1 + 1 = 0. Deque: [0, 1].
• i=2: Max is dp[0]=1. dp[2] = -2 + 1 = -1. Deque: [1, 2] (0 is out of window).
• i=3: Max is dp[1]=0. dp[3] = 4 + 0 = 4. Deque: [3] (popped 1 and 2 because 4 is larger). Result is the last value in the dp array.

5. Common mistakes candidates make

• $O(N \cdot K)$ approach: Using a nested loop to find the max in the window, which is too slow for large $K$.
• Heap instead of Deque: Using a Priority Queue ($O(N \log N)$) is acceptable but less optimal than the $O(N)$ deque approach.
• Window size errors: Forgetting to remove indices from the deque that have "fallen out" of the window.

6. Interview preparation tip

Master the Monotonic Queue pattern. It is the standard optimization for any DP where you need the maximum or minimum of a sliding window. This is a top-tier Sliding Window interview pattern for performance-focused roles.