Jump Game II - Video Solutions

Problem Overview: You are given an array where each element represents the maximum jump length from that position. Starting at index 0, the goal is to reach the last index using the minimum number of jumps. The challenge is choosing jumps that maximize reach while minimizing the total number of moves.

Approach 1: Dynamic Programming (O(n²) time, O(n) space)

The dynamic programming approach builds the answer bottom‑up. Create a dp array where dp[i] stores the minimum number of jumps required to reach index i. For every position i, iterate through all previous indices j and check if j + nums[j] >= i. If so, index i is reachable from j, so update dp[i] = min(dp[i], dp[j] + 1). This brute-force exploration of reachable states guarantees the optimal answer but requires nested iteration across the array. Time complexity is O(n²) with O(n) extra space for the DP array. This method is useful for understanding the transition logic before optimizing.

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

The optimal strategy treats the array like levels in a breadth-first traversal. While scanning the array, track two boundaries: the current jump range (currentEnd) and the farthest index reachable from any position in the current range (farthest). Iterate once through the array and update farthest = max(farthest, i + nums[i]). When the iteration reaches currentEnd, a jump must be taken, so increment the jump count and extend the range to farthest. This greedy choice works because expanding the reach as far as possible at each step minimizes the number of jump boundaries encountered. The algorithm runs in O(n) time and uses O(1) extra space.

The greedy technique relies on the same reachability idea used in the related array traversal problems but optimizes it with a single pass. It avoids computing results for every state like the dynamic programming approach.

Recommended for interviews: Interviewers typically expect the greedy solution. It demonstrates the ability to recognize range expansion and level-based traversal patterns, a common technique in greedy algorithms. Showing the DP approach first proves you understand the state transition and correctness, but optimizing it to the linear-time greedy method shows stronger problem-solving skill.