Jump Game VIII

Jump Game VIII

1. What is this problem about?

The Jump Game VIII interview question defines jumps based on local maximums and minimums. From index i, you can jump to index j ($i < j$) if:

1. nums[i] <= nums[j] and all elements between are strictly smaller than nums[j].
2. nums[i] > nums[j] and all elements between are greater than or equal to nums[j]. Each index has a cost, and you want the minimum cost to reach the end.

2. Why is this asked in interviews?

Amazon asks this to evaluate proficiency with Monotonic Stacks and Shortest Path logic. It evaluations whether you can correctly identify valid edges in a sparse graph using array properties. It’s an advanced Dynamic Programming interview pattern that tests your ability to handle complex conditional transitions.

3. Algorithmic pattern used

This problem follows the Monotonic Stack and DP pattern.

1. Find Edges: Use a monotonic stack to find, for each $i$, the "Next Greater or Equal" element and the "Next Smaller" element. These are the only two possible jumps from any index $i$ according to the rules.
2. DP: Since the graph is a DAG (you only jump forward), you can use a simple DP array.
3. Transition: dp[j] = min(dp[j], dp[i] + costs[j]) for all valid jump pairs $(i, j)$.
4. Complexity: Because each node has at most 2 edges, the total time is $O(N)$.

4. Example explanation

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

• From index 0 (value 3):
  • Next $\geq 3$ is 4 at index 2.
  • Next $< 3$ is 2 at index 1.
• From index 1 (value 2):
  • Next $\geq 2$ is 4 at index 2.
  • Next $< 2$ is none. The edges are $(0 o 1), (0 o 2), (1 o 2)$. Calculate min cost using these edges.

5. Common mistakes candidates make

• Building a full graph: Trying to check all $j > i$ for each $i$, which is $O(N^2)$.
• Incorrect Stack Property: Failing to realize that only the nearest elements satisfying the property are valid jump targets.
• Cost logic: Forgetting to add the cost of the destination index costs[j].

6. Interview preparation tip

Monotonic stacks are the key to many "nearest element" jump games. Practice identifying which elements are "visible" from a given index based on heights. This is a core Stack interview pattern for advanced array problems.