Frog Jump II

Frog Jump II

What is this problem about?

The Frog Jump II interview question asks you to find the optimal path for a frog to jump across a river and back. You are given a sorted array of stone positions. The frog must travel from the first stone to the last stone and then back to the first stone. Crucially, the frog cannot step on the same stone twice (except for the first and last stones). The "cost" of the path is the maximum jump distance the frog makes. You need to minimize this maximum jump distance.

Why is this asked in interviews?

Google and Microsoft use this Greedy interview pattern to test a candidate's ability to simplify complex routing problems. It evaluates your mathematical intuition: can you prove that an alternating path is the best strategy? It's a "Medium" problem that looks like it requires DP or complex graph routing, but actually has a very simple $O(N)$ linear solution.

Algorithmic pattern used

This problem is solved using a Greedy Strategy.

1. To minimize the maximum jump, the frog should use roughly every other stone on the way forward, and the remaining stones on the way back.
2. If it skips two stones (e.g., jumps from $i$ to $i+3$), the gap on the way back for the skipped stones will be even larger.
3. Therefore, the optimal path is to evaluate the jump distance between every alternate stone: stones[i+2] - stones[i].
4. The maximum of all these alternating jumps is the answer.

Example explanation

Stones: [0, 2, 5, 6, 7]

1. Consider jumps skipping exactly one stone:
   • stones[2] - stones[0] = $5 - 0 = 5$
   • stones[3] - stones[1] = $6 - 2 = 4$
   • stones[4] - stones[2] = $7 - 5 = 2$
2. Max jump among these is 5. Path: Forward 0 -> 2 -> 5 -> 7. Backward 7 -> 6 -> 0. Max jump forward: 5. Max jump backward: 7. Wait, 7 -> 6 -> 0 has a jump of 6. Let's look at the alternating logic again. Actually, backward is 7 -> 6 -> 2 -> 0? No, 2 was used forward. If Forward: 0 -> 5 -> 7 (jumps 5, 2). Backward: 7 -> 6 -> 2 -> 0 (jumps 1, 4, 2). Max jump is 5. The logic stones[i+2] - stones[i] perfectly captures the bottleneck distances.

Common mistakes candidates make

• Binary Search on Answer: While you can binary search the answer and validate using a greedy check, it's overkill ($O(N \log M)$ vs $O(N)$).
• Complex Graph Routing: Trying to model it as a min-cost max-flow or shortest path problem.
• Ignoring the start/end: Miscalculating the first and last jumps if the array is very small (e.g., length 2 or 3).

Interview preparation tip

If a problem asks for a round trip without reusing nodes, the "alternating sequence" (even indices one way, odd indices the other way) is almost always the optimal greedy strategy to balance the load.