Jump Game IX

Jump Game IX

1. What is this problem about?

The Jump Game IX interview question introduces a set of rules based on relative magnitudes. You are at index 0 and want to reach the last index with minimum cost. From index i, you can jump to index j ($i < j$) if:

1. nums[i] <= nums[j] and for all $k$ in $(i, j)$, nums[i] > nums[k]. (You jump to the first larger or equal element).
2. nums[i] > nums[j] and for all $k$ in $(i, j)$, nums[i] <= nums[k]. (You jump to the first smaller element). Each jump to index $j$ has a cost costs[j].

2. Why is this asked in interviews?

Companies like Medianet ask this to evaluate a candidate's proficiency with Monotonic Stacks and Dynamic Programming. It tests if you can identify "nearest" elements that satisfy a condition while skipping elements that don't. It evaluates your ability to build a sparse graph from array properties.

3. Algorithmic pattern used

This problem follows the Monotonic Stack and DP pattern.

1. Find Jump Targets: Use two monotonic stacks to find, for each index $i$, the nearest index $j$ that satisfies rule 1 and the nearest index $j$ that satisfies rule 2.
2. DP State: dp[i] is the minimum cost to reach index $i$.
3. Transition:
   • dp[0] = 0.
   • For each $i$, if you can jump to $j$, then dp[j] = min(dp[j], dp[i] + costs[j]).
4. Complexity: Since each index has at most two outgoing edges (one for each rule), the total edges are $O(N)$, making the DP $O(N)$.

4. Example explanation

nums = [3, 2, 4, 2], costs = [0, 5, 10, 2]

• From 0 (3):
  • Rule 1: First $\geq 3$ is 4 at index 2.
  • Rule 2: First $< 3$ is 2 at index 1.
• dp[1] = dp[0] + 5 = 5.
• dp[2] = dp[0] + 10 = 10.
• From 1 (2):
  • Rule 1: First $\geq 2$ is 4 at index 2.
  • ... Final answer is the min cost at the last index.

5. Common mistakes candidates make

• Brute Force: Checking every $j > i$ for each $i$ ($O(N^2)$), which will time out.
• Incorrect Stack Logic: Failing to find the first element that satisfies the criteria.
• Initialization: Forgetting to initialize the DP array with infinity.

6. Interview preparation tip

Practice using Monotonic Stacks to find "Nearest Greater/Smaller Element." It is the most frequent optimization for linear array problems. Combine this with DP to solve complex pathfinding tasks in Dynamic Programming interview patterns.