#2708

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Strength of a Group

Maximize the strength of a student group by carefully selecting students based on their scores, using dynamic programming and state transitions.

array dynamic programming backtracking greedy bit manipulation

Problem Statement

You are given a 0-indexed integer array nums representing the scores of students in an exam. The teacher wants to form a non-empty group of students with maximal strength, where the strength of a group of students is defined as the product of their scores at indices i0, i1, i2, ..., ik.

Return the maximum strength of any group the teacher can form. For example, with nums = [3, -1, -5, 2, 5, -9], the optimal group provides a strength of 1350.

Examples

Example 1

Input: nums = [3,-1,-5,2,5,-9]

Output: 1350

One way to form a group of maximal strength is to group the students at indices [0,2,3,4,5]. Their strength is 3 * (-5) * 2 * 5 * (-9) = 1350, which we can show is optimal.

Example 2

Input: nums = [-4,-5,-4]

Output: 20

Group the students at indices [0, 1] . Then, we’ll have a resulting strength of 20. We cannot achieve greater strength.

Constraints

1 <= nums.length <= 13
-9 <= nums[i] <= 9

Solution Approach

Subset Evaluation

Iterate through all possible subsets of students and calculate the product of their scores. Check each group’s strength to find the maximum possible value. This brute force approach, though simple, is costly when not optimized.

Dynamic Programming Approach

Using dynamic programming (DP), track the best possible score for each possible subset of students. Efficiently transition between states by combining the product results from previously calculated subsets.

Backtracking and Optimization

Optimize the search by eliminating subsets that cannot potentially result in a maximal group strength. Use backtracking to prune the search tree, speeding up the DP transitions and avoiding redundant computations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this problem depends on the chosen approach. A brute force method might involve O(2^n) subset evaluations, while a more optimized dynamic programming or backtracking solution could reduce the complexity to O(n^2) or better, depending on how states are transitioned and subsets are evaluated. Space complexity similarly depends on the method used to store and transition between intermediate states.

What Interviewers Usually Probe

The candidate demonstrates understanding of dynamic programming by implementing state transitions effectively.
The candidate optimizes subset evaluation to avoid redundant calculations, showing a good grasp of time complexity management.
The candidate exhibits proficiency in using backtracking to prune the search space and improve algorithm efficiency.

Common Pitfalls or Variants

Common pitfalls

Failing to prune redundant calculations, which leads to inefficient solutions and long execution times.
Not properly managing state transitions, causing incorrect results or excessive computations.
Forgetting to account for the full range of possible subsets, leading to missing optimal solutions.

Follow-up variants

Consider larger input sizes and how the algorithm could scale with more students (larger nums array).
Optimize the space complexity by reducing the amount of state tracking used during dynamic programming transitions.
Adapt the solution for problems with different constraints, such as varying the score range or array length.

FAQ

What is the most efficient way to solve the 'Maximum Strength of a Group' problem?

The most efficient way is to use dynamic programming with state transitions, optimizing subset evaluations to avoid redundant calculations.

How does backtracking help in solving the 'Maximum Strength of a Group' problem?

Backtracking helps by pruning unnecessary calculations, speeding up the solution and improving overall efficiency, especially for larger inputs.

What is the time complexity of the 'Maximum Strength of a Group' problem?

The time complexity depends on the approach chosen, but can range from O(2^n) for brute force to O(n^2) or better with dynamic programming optimizations.

How can dynamic programming help in the 'Maximum Strength of a Group' problem?

Dynamic programming helps by storing and transitioning between previously computed subset strengths, avoiding the need to re-evaluate subsets multiple times.

What are common mistakes when solving 'Maximum Strength of a Group'?

Common mistakes include failing to prune the search space, not managing state transitions efficiently, and missing optimal subsets due to incorrect subset evaluations.

terminal

Solution

Solution 1: Binary Enumeration

The problem is actually to find the maximum product of all subsets. Since the length of the array does not exceed $13$, we can consider using the method of binary enumeration.

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class Solution:
    def maxStrength(self, nums: List[int]) -> int:
        ans = -inf
        for i in range(1, 1 << len(nums)):
            t = 1
            for j, x in enumerate(nums):
                if i >> j & 1:
                    t *= x
            ans = max(ans, t)
        return ans

Solution 2: Sorting + Greedy

First, we can sort the array. Based on the characteristics of the array, we can draw the following conclusions:

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class Solution:
    def maxStrength(self, nums: List[int]) -> int:
        ans = -inf
        for i in range(1, 1 << len(nums)):
            t = 1
            for j, x in enumerate(nums):
                if i >> j & 1:
                    t *= x
            ans = max(ans, t)
        return ans

Continue Practicing

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