Maximum Number of Groups Entering a Competition

Problem Description

You are given a positive integer array grades which represents the grades of students in a university. You would like to enter all these students into a competition in ordered non-empty groups, such that the ordering meets the following conditions:

• The sum of the grades of students in the i-th group is less than the sum of the grades of students in the (i + 1)-th group, for all groups (except the last).
• The total number of students in the i-th group is less than the total number of students in the (i + 1)-th group, for all groups (except the last).

Return the maximum number of groups that can be formed.

Key Insights

• The groups need to have increasing sums of grades.
• The groups need to have an increasing number of students.
• The challenge lies in finding the largest number of groups that can satisfy both conditions simultaneously.
• A greedy approach can be employed to maximize the number of groups formed by incrementally building groups.

Space and Time Complexity

Time Complexity: O(n), where n is the number of students (length of grades array).
Space Complexity: O(1), as we are using a constant amount of extra space.

Solution

To solve this problem, we can use a greedy approach. We will keep track of the number of groups we can form and the number of students in each group. Starting with a group size of 1, we will attempt to form groups incrementally, checking whether we can create a new group with the increasing total number of students and their grades. The idea is to continue forming groups while maintaining the constraints until we cannot form any more valid groups.

1. Initialize a variable to keep track of the current group size.
2. Use a loop to form groups as long as the conditions are met.
3. Keep a running total of the number of students and their grades in each group.
4. Return the maximum number of groups formed.

Code Solutions