Problem Description
In a mystic dungeon, n magicians are standing in a line. Each magician has an attribute that gives you energy. Some magicians can give you negative energy, which means taking energy from you. You have been cursed in such a way that after absorbing energy from magician i, you will be instantly transported to magician (i + k). This process will be repeated until you reach the magician where (i + k) does not exist. In other words, you will choose a starting point and then teleport with k jumps until you reach the end of the magicians' sequence, absorbing all the energy during the journey. You are given an array energy and an integer k. Return the maximum possible energy you can gain. Note that when you reach a magician, you must take energy from them, whether it is negative or positive energy.
Key Insights
- You can start at any magician and jump k positions forward repeatedly.
- The goal is to maximize the energy gained from the magicians encountered during the jumps.
- The problem can be approached using dynamic programming to evaluate the maximum energy from each starting point efficiently.
Space and Time Complexity
Time Complexity: O(n)
Space Complexity: O(n)
Solution
To solve this problem, we can use a dynamic programming approach. We create a DP array where dp[i] represents the maximum energy that can be gained starting from the i-th magician. We can initialize the DP array with the energy values, and then for each magician i, we calculate the total energy by adding the energy of magician i to the maximum energy that can be obtained from the magician at position (i + k) if it exists. The final answer will be the maximum value in the DP array.