Formatted question description: https://leetcode.ca/all/1815.html

# 1815. Maximum Number of Groups Getting Fresh Donuts

Hard

## 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.

Example 1:

Input: batchSize = 3, groups = [1,2,3,4,5,6]

Output: 4

Explanation: You can arrange the groups as [6,2,4,5,1,3]. Then the 1st, 2nd, 4th, and 6th groups will be happy.

Example 2:

Input: batchSize = 4, groups = [1,3,2,5,2,2,1,6]

Output: 4

Constraints:

• 1 <= batchSize <= 9
• 1 <= groups.length <= 30
• 1 <= groups[i] <= 10^9

## Solution

The aim of this problem is to find a permutation of groups such that the number of prefix sums of groups that can be divisible by batchSize is the most. Use dynamic programming with compressed states to solve this.

class Solution {
public int maxHappyGroups(int batchSize, int[] groups) {
long state = 0;
int mod0 = 0;
for (int group : groups) {
group %= batchSize;
if (group == 0)
mod0++;
else
state += 1L << (group * 5);
}
Map<Long, Integer> map = new HashMap<Long, Integer>();
return mod0 + dp(0, state, batchSize, map);
}

public int dp(int curr, long state, int batchSize, Map<Long, Integer> map) {
if (!map.containsKey(state)) {
if (state == 0)
map.put(state, 0);
else {
int val = 0;
for (int i = 1; i < batchSize; i++) {
if ((state >>> (i * 5)) % 32 != 0) {
int nextVal = dp((curr + i) % batchSize, state - (1L << (i * 5)), batchSize, map);
val = Math.max(val, curr != 0 ? nextVal : nextVal + 1);
}
}
map.put(state, val);
}
}
return map.get(state);
}
}