Can a Frog Jump Across All Platforms? Jump Game Solutions with DFS, DP, Greedy

Problem Statement

The frog princess, a renowned "big beautiful frog," seeks a clever husband by setting up a jumping contest: a series of wooden planks of varying material, each allowing a limited maximum jump distance. The input is a non‑negative integer array where the frog starts at the first index and each element represents the maximum jump length from that position. The task is to decide whether the frog can reach the last plank.

Solution Approaches

Depth‑First Search (DFS)

The most straightforward method enumerates every possible jump using recursive DFS. While conceptually simple, its time complexity is high and often leads to timeout on large inputs. An optimized variant searches backward from the target, recursively finding boards that can reach the current one, but it still suffers from exponential blow‑up.

Dynamic Programming

Define an array dp where dp[i] records the farthest index reachable up to position i. If dp[i‑1] < i, the current position is unreachable. Otherwise update dp[i] = max(dp[i‑1], i + nums[i]). This yields a linear‑time solution with dramatically reduced memory usage compared to DFS.

Greedy Algorithm

The greedy approach maintains the farthest reachable index while scanning the array. If the current board can extend the reachable range, update the maximum; otherwise stay on the current board. When the scan reaches the last board, success is confirmed; if a position becomes unreachable, the frog cannot finish.

Reverse (Backward) Method

Instead of moving forward, start from the target board and repeatedly find the nearest preceding board that can jump to the current target. Set that board as the new target and continue backward. If the process reaches the first board, the frog can reach the end; otherwise it cannot. This method runs in O(n) time, avoiding the exponential branching of DFS.

Conclusion

Multiple algorithmic perspectives—brute‑force DFS, optimal dynamic programming, concise greedy, and efficient reverse scanning—can solve the classic "Jump Game" problem. Understanding their trade‑offs deepens algorithmic intuition and prepares you for variations where one method may outperform the others.