Maximum Number of Groups With Increasing Length

Maximum Number of Groups With Increasing Length

What is this problem about?

The Maximum Number of Groups With Increasing Length coding problem gives you an array of usage limits for each category. You want to form the maximum number of groups numbered 1, 2, 3, ..., k such that group i contains exactly i distinct categories, each category appears at most as many times as its usage limit, and each group uses categories independently (categories can be reused across groups). Find the maximum k.

Why is this asked in interviews?

Amazon uses this problem because it combines sorting, greedy reasoning, and binary search to find the maximum feasible number of groups. Checking whether k groups are feasible requires verifying that the total capacity (sum of min(usage_limit, k) for all categories) is at least 1+2+...+k = k*(k+1)/2. Binary searching on k gives an efficient solution.

Algorithmic pattern used

Binary search + greedy feasibility: Sort usage limits in descending order. Binary search on the answer k. For a given k, compute the total slots available: for each category (sorted desc), take min(usage_limit[i], k - i) where i is 0-indexed (since group i+1 needs at most k-i instances from each category considering ordering). If total slots ≥ k*(k+1)/2, k is feasible. This check runs in O(n); binary search adds O(log(n)) factor.

Example explanation

Usage limits: [1, 2, 5], sorted desc: [5, 2, 1].

• k=1: need 1 slot. Slots: min(5,1)+min(2,1)+min(1,1)=1+1+1=3 ≥ 1. Feasible.
• k=2: need 3 slots. Slots: min(5,2)+min(2,2)+min(1,2)=2+2+1=5 ≥ 3. Feasible.
• k=3: need 6 slots. Slots: min(5,3)+min(2,3)+min(1,3)=3+2+1=6 ≥ 6. Feasible.
• k=4: need 10 slots. Slots: min(5,4)+min(2,4)+min(1,4)=4+2+1=7 < 10. Not feasible.
• Answer = 3.

Common mistakes candidates make

• Not sorting usage limits: The greedy capacity check assumes sorted order for correctness.
• Wrong feasibility formula: The capacity of category i in the sorted array contributes at most min(limit[i], k-i) slots for groups; using just min(limit, k) overcounts.
• Integer overflow: k*(k+1)/2 can be large; use 64-bit integers.

Interview preparation tip

For the Array Math Sorting Binary Search Greedy interview pattern, "binary search on the answer k with a greedy feasibility check" is a powerful template. When feasibility is monotone (if k groups are feasible, k-1 are too), binary search applies. Practice extracting the feasibility condition before coding the binary search.