Maximum Strength of a Group

Problem Description

You are given a 0-indexed integer array nums representing the score of students in an exam. The teacher would like to form one non-empty group of students with maximal strength, where the strength of a group of students of indices i0, i1, i2, ... , ik is defined as nums[i0] * nums[i1] * nums[i2] * ... * nums[ik]. Return the maximum strength of a group the teacher can create.

Key Insights

• The product of numbers can yield a maximum strength when considering both positive and negative values.
• Multiplying an even number of negative numbers results in a positive product, while multiplying an odd number results in a negative product.
• The maximal group can be formed by considering various combinations of positive and negative numbers.
• If only negative numbers exist, pairing them to form a positive product is optimal.

Space and Time Complexity

Time Complexity: O(2^n) - We explore all possible combinations of students to find the maximum strength. Space Complexity: O(1) - We utilize a constant amount of space for variables, independent of input size.

Solution

To solve the problem, we can utilize a backtracking approach or enumeration of all possible combinations of elements in the nums array. The algorithm involves:

1. Iterating through all possible subsets of the input array.
2. Calculating the product of each subset.
3. Keeping track of the maximum product found.
4. Handling edge cases where the product might be negative or where elements are zero.

We will use an array to store the scores and iterate through possible combinations using bit manipulation to generate subsets, calculating their products as we go.

Code Solutions