Problem Description
There is a pizza with 3n slices of varying size, you and your friends will take slices of pizza as follows:
- You will pick any pizza slice.
- Your friend Alice will pick the next slice in the anti-clockwise direction of your pick.
- Your friend Bob will pick the next slice in the clockwise direction of your pick.
- Repeat until there are no more slices of pizzas.
Given an integer array slices that represent the sizes of the pizza slices in a clockwise direction, return the maximum possible sum of slice sizes that you can pick.
Key Insights
- The pizza has 3n slices, and you can select n slices while your friends are also picking from the remaining slices.
- The selection is constrained by the anti-clockwise and clockwise picking of your friends, which creates a need for strategic selection.
- Dynamic Programming can be used to find the optimal solution by considering the maximum sum you can achieve for various starting points and slice selections.
- The problem can be viewed as a variation of the "House Robber" problem where you cannot take adjacent slices.
Space and Time Complexity
Time Complexity: O(n^2)
Space Complexity: O(n)
Solution
To solve the problem, we can use a dynamic programming approach. We'll create a 2D DP array where dp[i][j] represents the maximum sum we can achieve by selecting j slices from the first i slices. We need to account for the fact that when we pick a slice, we cannot pick the slices immediately next to it.
The algorithm works by iterating through the slices and updating the DP table based on whether we pick the current slice or skip it. The final result will be the maximum value found in the last row of the DP table.