What is the time complexity of Find the Maximum Factor Score of Array?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Find the Maximum Factor Score of Array?

This problem uses Array, Math, Number Theory. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Find the Maximum Factor Score of Array?

Insight 1: Pre-compute prefix and suffix GCD and LCM arrays to efficiently calculate GCD and LCM after removing an element. Insight 2: The constraints on the input numbers (1 to 30) imply that the numbers are relatively small, therefore the intermediate results of GCD and LCM won't overflow easily, however, the final factor score(product of GCD and LCM) should use a long data type to store the results.

Which companies ask LeetCode 3334. Find the Maximum Factor Score of Array in interviews?

This problem is frequently asked at Info Edge.

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Find the Maximum Factor Score of Array - LeetCode 3334 Solution

Find the Maximum Factor Score of Array - Complete Solution Guide

Find the Maximum Factor Score of Array is LeetCode problem 3334, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an integer array nums . The factor score of an array is defined as the product of the LCM and GCD of all elements of that array. Return the maximum factor score of nums after removing at most one element from it. Note that both the LCM and GCD of a single number are the number itself, and the factor score of an empty array is 0. Example 1: Input: nums = [2,4,8,16] Output: 64 Explanation: On removing 2, the GCD of the rest of the elements is 4 while the LCM is 16, which gives a maxi

Detailed Explanation

The problem asks us to find the maximum 'factor score' of an integer array `nums`. The factor score is defined as the product of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) of all elements in the array. We are allowed to remove at most one element from the array to maximize this factor score. If the array is empty after removal, the factor score is 0. The constraints are that the array length is between 1 and 100, and each element is between 1 and 30.

Solution Approach

The solution uses a dynamic programming approach by pre-computing prefix and suffix GCD and LCM arrays. These arrays help us to quickly calculate the GCD and LCM of the remaining elements after removing a single element from the original array. We iterate through the array and for each element, we calculate the factor score after removing that element. We then take the maximum of all these factor scores, including the factor score of the original array (without removing any element).

Step-by-Step Algorithm

Step 1: Handle the edge case where the array has only one element. In this case, the maximum factor score is the square of that element.
Step 2: Create four arrays: `pre_gcd`, `suf_gcd`, `pre_lcm`, and `suf_lcm` to store the prefix GCD, suffix GCD, prefix LCM, and suffix LCM, respectively.
Step 3: Calculate the prefix GCD and prefix LCM arrays. `pre_gcd[i]` stores the GCD of `nums[0]` to `nums[i]`, and `pre_lcm[i]` stores the LCM of `nums[0]` to `nums[i]`.
Step 4: Calculate the suffix GCD and suffix LCM arrays. `suf_gcd[i]` stores the GCD of `nums[i]` to `nums[n-1]`, and `suf_lcm[i]` stores the LCM of `nums[i]` to `nums[n-1]`.
Step 5: Initialize `max_score` to the factor score of the original array, which is `pre_gcd[n-1] * pre_lcm[n-1]`.
Step 6: Iterate through the array `nums`. For each element `nums[i]`, calculate the factor score of the array after removing `nums[i]`.
Step 7: If `i` is 0, the GCD and LCM are `suf_gcd[1]` and `suf_lcm[1]`, respectively.
Step 8: If `i` is `n-1`, the GCD and LCM are `pre_gcd[n-2]` and `pre_lcm[n-2]`, respectively.
Step 9: Otherwise, the GCD is `gcd(pre_gcd[i-1], suf_gcd[i+1])` and the LCM is `lcm(pre_lcm[i-1], suf_lcm[i+1])`.
Step 10: Update `max_score` with the maximum factor score found so far.
Step 11: Return `max_score`.

Key Insights

Insight 1: Pre-compute prefix and suffix GCD and LCM arrays to efficiently calculate GCD and LCM after removing an element.
Insight 2: The constraints on the input numbers (1 to 30) imply that the numbers are relatively small, therefore the intermediate results of GCD and LCM won't overflow easily, however, the final factor score(product of GCD and LCM) should use a long data type to store the results.
Insight 3: Consider edge cases where the array has only one element, or when removing the first or last element of the array.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Array, Math, Number Theory.

Companies

Asked at: Info Edge.

Find the Maximum Factor Score of Array | LeetCode 3334

Medium

Array Math Number Theory

Time: O(n)Space: O(n)

Solution Code

Available in:

class Solution {
    public long maxScore(int[] nums) {
        int n = nums.length;

        if (n == 1) {
            return (long) nums[0] * nums[0];
        }

        long[] pre_gcd = new long[n];
        long[] suf_gcd = new long[n];
        long[] pre_lcm = new long[n];
        long[] suf_lcm = new long[n];

        pre_gcd[0] = nums[0];
        pre_lcm[0] = nums[0];
        for (int i = 1; i < n; i++) {
            pre_gcd[i] = gcd(pre_gcd[i - 1], nums[i]);
            pre_lcm[i] = lcm(pre_lcm[i - 1], nums[i]);
        }

        suf_gcd[n - 1] = nums[n - 1];
        suf_lcm[n - 1] = nums[n - 1];
        for (int i = n - 2; i >= 0; i--) {
            suf_gcd[i] = gcd(suf_gcd[i + 1], nums[i]);
            suf_lcm[i] = lcm(suf_lcm[i + 1], nums[i]);
        }

        long max_score = pre_gcd[n - 1] * pre_lcm[n - 1];

        for (int i = 0; i < n; i++) {
            long g = 0;
            long l = 1;
            if (i == 0) {
                g = suf_gcd[1];
                l = suf_lcm[1];
            } else if (i == n - 1) {
                g = pre_gcd[n - 2];
                l = pre_lcm[n - 2];
            } else {
                g = gcd(pre_gcd[i - 1], suf_gcd[i + 1]);
                l = lcm(pre_lcm[i - 1], suf_lcm[i + 1]);
            }
            
            long current_score = g * l;
            max_score = Math.max(max_score, current_score);
        }

        return max_score;
    }

    private long gcd(long a, long b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a % b);
    }

    private long lcm(long a, long b) {
        if (a == 0 || b == 0) {
            return 0;
        }
        return (a * b) / gcd(a, b);
    }
}

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Problem Info

Description

Problem 3334: Find the Maximum Factor Score of Array - Detailed explanation and solution approaches.

Topics

Array Math Number Theory

Companies

Info Edge

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:3334

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

11 steps

1
Step 1: Handle the edge case where the array has only one element. In this case, the maximum factor score is the square of that element.
2
Step 2: Create four arrays: `pre_gcd`, `suf_gcd`, `pre_lcm`, and `suf_lcm` to store the prefix GCD, suffix GCD, prefix LCM, and suffix LCM, respectively.
3
Step 3: Calculate the prefix GCD and prefix LCM arrays. `pre_gcd[i]` stores the GCD of `nums[0]` to `nums[i]`, and `pre_lcm[i]` stores the LCM of `nums[0]` to `nums[i]`.
4
Step 4: Calculate the suffix GCD and suffix LCM arrays. `suf_gcd[i]` stores the GCD of `nums[i]` to `nums[n-1]`, and `suf_lcm[i]` stores the LCM of `nums[i]` to `nums[n-1]`.
5
Step 5: Initialize `max_score` to the factor score of the original array, which is `pre_gcd[n-1] * pre_lcm[n-1]`.
6
Step 6: Iterate through the array `nums`. For each element `nums[i]`, calculate the factor score of the array after removing `nums[i]`.
7
Step 7: If `i` is 0, the GCD and LCM are `suf_gcd[1]` and `suf_lcm[1]`, respectively.
8
Step 8: If `i` is `n-1`, the GCD and LCM are `pre_gcd[n-2]` and `pre_lcm[n-2]`, respectively.
9
Step 9: Otherwise, the GCD is `gcd(pre_gcd[i-1], suf_gcd[i+1])` and the LCM is `lcm(pre_lcm[i-1], suf_lcm[i+1])`.
10
Step 10: Update `max_score` with the maximum factor score found so far.
11
Step 11: Return `max_score`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The prefix and suffix arrays each require O(n) space. The algorithm iterates through the input array once and performs a constant amount of operations per element.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Brute Force

Iterate through each element of the array, remove it, and then calculate the GCD and LCM of the remaining elements. Keep track of the maximum factor score found.

Time: O(n^2 * log(max(nums))), where n is the length of the input array. The outer loop is O(n), and calculating the GCD and LCM of the remaining n-1 elements takes O(n * log(max(nums))) time because GCD/LCM calculation has logarithmic complexity.Space: O(1). Constant extra space is used.

Trade-offs: This approach is simpler to implement but has a higher time complexity than the main solution. The pre-computation reduces the complexity significantly.

✨

Key Insights

1Insight 1: Pre-compute prefix and suffix GCD and LCM arrays to efficiently calculate GCD and LCM after removing an element.
2Insight 2: The constraints on the input numbers (1 to 30) imply that the numbers are relatively small, therefore the intermediate results of GCD and LCM won't overflow easily, however, the final factor score(product of GCD and LCM) should use a long data type to store the results.
3Insight 3: Consider edge cases where the array has only one element, or when removing the first or last element of the array.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: Array with only one element. The code handles this case explicitly by returning nums[0] * nums[0].
Edge case 2: Removing the first or last element of the array. The code handles these cases separately within the loop using conditional checks (i == 0 and i == n-1) to correctly compute the GCD and LCM of the remaining elements.
Edge case 3: Empty array (implicitly handled). If all elements were removed due to the at most one element restriction, the loop condition prevents access to invalid indices, avoiding errors. The algorithm initializes the result with the score of the original array and later updates with potentially higher scores obtained by removing elements. If no removals produce better results, the original array score, which handles an empty array if the array length is 1, is returned.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer overflow. The product of LCM and GCD can be a large number, so using `int` might lead to integer overflow. Use `long` to store the result.

✗Common mistake 2: Incorrectly calculating GCD and LCM. Ensure the GCD and LCM functions are implemented correctly. Also, ensure that division by zero is handled in LCM calculation.

✗Common mistake 3: Not handling edge cases correctly, such as the array with only one element, or when removing the first or last elements of the array.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Optimizing resource allocation. You might have a set of resources (represented by numbers) with varying capacities, and you want to maximize the combined effective capacity (factor score) after removing one potentially problematic resource.

🏦Real-world scenario 2: Signal processing. Finding common frequencies and their harmonics in a signal. GCD would help to find the common frequencies, LCM can help find frequencies with which they are harmonic to each other. Removing one interfering signal to improve detection.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 3, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Find the Maximum Factor Score of Array"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether math techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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