Jump Game II - Solution & Explanation

Problem Overview: You are given an array where nums[i] represents the maximum jump length from index i. Starting at index 0, the goal is to reach the last index using the minimum number of jumps. Every jump can move anywhere within the allowed range from the current position.

Approach 1: Dynamic Programming (O(n^2) time, O(n) space)

Dynamic Programming models the problem as a minimum steps problem. Create a dp array where dp[i] stores the minimum number of jumps needed to reach index i. Initialize dp[0] = 0 and iterate through each index. For every position i, iterate through all previous indices j and check if j + nums[j] >= i. If reachable, update dp[i] = min(dp[i], dp[j] + 1). This explores all valid jump paths and guarantees the optimal answer. Time complexity becomes O(n^2) because each index may scan all previous indices, while space complexity is O(n) for the DP table. This approach clearly demonstrates the optimal substructure typical in dynamic programming problems.

Approach 2: Greedy Range Expansion (O(n) time, O(1) space)

The optimal solution uses a greedy strategy that treats the array like levels in a BFS traversal. Maintain three variables: jumps, currentEnd, and farthest. As you iterate through the array, continuously update farthest = max(farthest, i + nums[i]) to track the farthest index reachable from the current jump window. When the iteration reaches currentEnd, you must make another jump, so increment jumps and update currentEnd = farthest. This effectively expands the reachable range layer by layer, similar to exploring levels in a graph. The algorithm scans the array once, giving O(n) time complexity and O(1) space complexity. This technique is a classic example of a greedy decision where the locally optimal choice (extending the reach as far as possible) produces the globally optimal result. It frequently appears in problems tagged under greedy and array traversal patterns.

Recommended for interviews: The greedy approach is the expected answer in most interviews. It reduces the quadratic DP exploration to a single linear pass while maintaining correctness. Explaining the DP approach first shows you understand the state transition and optimal substructure. Then optimizing it to the greedy window expansion demonstrates strong algorithmic reasoning and pattern recognition.