Problem Description
There is a donuts shop that bakes donuts in batches of batchSize. They have a rule where they must serve all of the donuts of a batch before serving any donuts of the next batch. You are given an integer batchSize and an integer array groups, where groups[i] denotes that there is a group of groups[i] customers that will visit the shop. Each customer will get exactly one donut.
When a group visits the shop, all customers of the group must be served before serving any of the following groups. A group will be happy if they all get fresh donuts. That is, the first customer of the group does not receive a donut that was left over from the previous group.
You can freely rearrange the ordering of the groups. Return the maximum possible number of happy groups after rearranging the groups.
Key Insights
- Groups must be served in full before moving to the next group.
- A group is happy if the total number of donuts served before them does not leave any leftovers that could be served to them.
- The arrangement of groups can significantly affect the number of happy groups.
- Dynamic programming or backtracking with memoization can be utilized to explore group arrangements efficiently.
Space and Time Complexity
Time Complexity: O(2^n * n) - where n is the number of groups, due to the bitmasking approach to explore all subsets of groups. Space Complexity: O(n) - for storing the state of the current group arrangement and memoization.
Solution
The problem can be solved using a bitmasking approach combined with dynamic programming. The idea is to use a bitmask to represent the state of which groups have been used. For each state, we calculate the total number of donuts served and check if the current group can be served fresh donuts. If they can be served fresh, we increment the count of happy groups. The process is repeated for all possible combinations of groups until we find the maximum number of happy groups.