Problem Description
You are given a 0-indexed array nums of n integers and an integer target. You are initially positioned at index 0. In one step, you can jump from index i to any index j such that:
0 <= i < j < n-target <= nums[j] - nums[i] <= target
Return the maximum number of jumps you can make to reach index n - 1. If there is no way to reach index n - 1, return -1.
Key Insights
- The problem is about finding a path in an array where each jump must satisfy a specific condition based on the
target. - A dynamic programming approach can be utilized to keep track of the maximum number of jumps possible to reach each index.
- The solution requires considering all valid jumps from the current index to future indices, which can be efficiently managed using a loop.
Space and Time Complexity
Time Complexity: O(n^2) - In the worst case, we may need to check each index against all subsequent indices. Space Complexity: O(n) - We maintain an array to store the maximum jumps to each index.
Solution
We can utilize dynamic programming to solve this problem. We will maintain an array dp where dp[i] represents the maximum jumps to reach index i. Initially, dp[0] is 0 since we start there, and all other indices are set to -1 to indicate they are unreachable.
For each index i, we will look for all subsequent indices j (where j > i) and check if the jump from i to j is valid based on the given condition. If it is valid, we will update dp[j] to be the maximum of its current value or dp[i] + 1. Finally, if dp[n-1] is still -1, it means we can't reach the last index, and we return -1. Otherwise, we return dp[n-1].