Jump Game V

Problem Description

Given an array of integers arr and an integer d. In one step you can jump from index i to index:

• i + x where: i + x < arr.length and 0 < x <= d.
• i - x where: i - x >= 0 and 0 < x <= d.

You can only jump from index i to index j if arr[i] > arr[j] and arr[i] > arr[k] for all indices k between i and j. Your goal is to return the maximum number of indices you can visit starting from any index of the array.

Key Insights

• The problem requires careful tracking of the indices you can jump to based on the values in the array.
• A depth-first search (DFS) or dynamic programming (DP) approach can be used to explore all possible jumps from any starting index.
• The constraints on jumping (based on the value at the indices) create a directed acyclic graph where each index can be seen as a node.
• The maximum number of reachable indices can be calculated by exploring all possible paths starting from each index.

Space and Time Complexity

Time Complexity: O(n^2) - In the worst case, for each index, we may check up to d indices to the left and right, leading to quadratic behavior.

Space Complexity: O(n) - We may use an array to store the results for each index, which requires linear space.

Solution

To solve the problem, we can use a depth-first search (DFS) or dynamic programming approach. The idea is to explore each index and track how far we can jump based on the rules provided. We will maintain an array to store the maximum indices reachable from each starting index. Here’s a step-by-step breakdown:

1. Initialize an array dp of the same length as arr to store the maximum indices reachable from each index.
2. For each index, check both left and right directions up to a distance of d.
3. For each valid jump, update the dp array accordingly.
4. Finally, return the maximum value from the dp array as the answer.

Code Solutions