Minimize Unhappy Students
You are organizing a trip with limited capacity and need to assign students while minimizing dissatisfaction.
Problem
You are given an array groups of length n, where each element represents a student's friend group ID. Students with the same group ID belong to the same friend group.
You are also given an integer t, representing the maximum capacity of the trip.
Definitions
- Friend Group Size: The number of students sharing the same group ID.
- Unhappy Student: A student is unhappy if they are left behind while at least one member of their group is selected (i.e., their group is split).
- If none of a group is selected, no one from that group is unhappy.
Objective
Select a subset of students (not exceeding capacity t) such that the total number of unhappy students is minimized.
Key Insight
- Ideally, you want to select entire groups to avoid splitting.
- However, when exact packing is not possible, you may need to partially include a group, which creates unhappy students.
Examples
Example 1:
Input: groups = [1,1,2,2,2,3,3], t = 4
Output: 0
Explanation:
Group sizes are [2,3,2]. Choosing groups of size 2 and 2 fills capacity exactly (2+2=4), so no group is split.
Example 2:
Input: groups = [1,1,2,2,2,3,3,4], t = 3
Output: 0
Explanation:
Group sizes are [2,3,2,1]. Selecting the group of size 3 fills capacity perfectly.
Notes
- You may select students from multiple groups.
- If you partially select a group of size
kand includexstudents (0 < x < k), thenk - xstudents are unhappy.
Approach Hint
This problem can be modeled as a variation of the knapsack problem, where:
- Each group is an item with weight = size.
- You want to maximize fully included students first.
- Then consider partial inclusion if needed, minimizing leftover (unhappy students).
Example 1
Input
groups = [1,1,2,2,2,3,3], t = 4Output
0Explanation
Pick groups of size 2 and 2 to fill capacity exactly.
Example 2
Input
groups = [1,1,2,2,2,3,3,4], t = 3Output
0Explanation
Pick the group of size 3.
Constraints
- 1 <= groups.length <= 10⁵
- 1 <= groups[i] <= 10⁵
- 1 <= t <= groups.length
Approach
Step 1: Count Group Sizes
Convert the groups array into frequencies (sizes of each friend group).
Step 2: 0/1 Knapsack (Maximize Full Groups)
We first try to select whole groups to maximize the number of students included without exceeding capacity t.
Let dp[c] = maximum number of students we can include using full groups with capacity c.
Step 3: Compute Best Full Inclusion
Find the maximum number of students we can include using only full groups.
Step 4: Consider Partial Group Inclusion
If we cannot fill exactly t, we can take part of a group.
Let:
full_sum= best sum from full groups- remaining capacity =
t - full_sum
We choose the smallest group that can fill remaining capacity partially.
Unhappy students = (group_size - taken_from_group)
Step 5: Try All Possibilities
We try all possible capacities from knapsack and compute minimal unhappy students.
Python Solution
from collections import Counter
def minimize_unhappy_students(groups, t):
# Count group sizes
count = Counter(groups)
sizes = list(count.values())
# Knapsack DP
dp = [False] * (t + 1)
dp[0] = True
for size in sizes:
for c in range(t, size - 1, -1):
if dp[c - size]:
dp[c] = True
# Find all achievable full sums
possible_sums = [i for i in range(t + 1) if dp[i]]
# Sort group sizes for partial consideration
sizes.sort()
min_unhappy = float('inf')
for full_sum in possible_sums:
if full_sum == t:
return 0 # perfect fit
remaining = t - full_sum
# Try to fill remaining with partial group
for size in sizes:
if size > remaining:
unhappy = size - remaining
min_unhappy = min(min_unhappy, unhappy)
break
# If we never used partial group, answer is 0
return min_unhappy if min_unhappy != float('inf') else 0
Complexity
- Counting: O(n)
- Knapsack: O(n * t)
- Total: O(n * t)
Key Insight
This is a hybrid of subset sum (knapsack) and greedy partial selection, where we prioritize full groups and only split when necessary.
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