What is the time complexity of Check If It Is a Good Array?

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

What data structures are used to solve Check If It Is a Good Array?

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

What are the key insights for solving Check If It Is a Good Array?

Insight 1: Bézout's identity states that for integers a and b, there exist integers x and y such that ax + by = gcd(a, b), where gcd(a, b) is the greatest common divisor of a and b. More generally, for integers a1, a2, ..., an, there exist integers x1, x2, ..., xn such that a1x1 + a2x2 + ... + anxn = gcd(a1, a2, ..., an). Insight 2: The array `nums` is 'good' if and only if the greatest common divisor (GCD) of all the numbers in the array is equal to 1. This is a direct consequence of Bézout's identity.

Which companies ask LeetCode 1250. Check If It Is a Good Array in interviews?

This problem is frequently asked at Dropbox, Jump Trading, Nokia.

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Check If It Is a Good Array - LeetCode 1250 Solution

Check If It Is a Good Array - Complete Solution Guide

Check If It Is a Good Array is LeetCode problem 1250, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given an array nums of positive integers. Your task is to select some subset of nums , multiply each element by an integer and add all these numbers. The array is said to be good if you can obtain a sum of 1 from the array by any possible subset and multiplicand. Return True if the array is good otherwise return False . Example 1: Input: nums = [12,5,7,23] Output: true Explanation: Pick numbers 5 and 7. 5*3 + 7*(-2) = 1 Example 2: Input: nums = [29,6,10] Output: true Explanation: Pick numbers 29

Detailed Explanation

The problem asks us to determine if a given array of positive integers `nums` is a 'good' array. An array is considered 'good' if it is possible to select a subset of the numbers in the array, multiply each selected number by an integer (which can be positive, negative, or zero), and sum the results to obtain a value of 1. In simpler terms, we need to check if there exists a linear combination of the numbers in the array that equals 1.

Solution Approach

The solution calculates the greatest common divisor (GCD) of all the numbers in the array. If the final GCD is 1, it means that there exists a linear combination of the numbers that results in 1, and therefore the array is 'good'. The GCD is calculated iteratively. For each number in the array, the GCD of the current GCD and the number is calculated. If the GCD ever becomes 1, the function immediately returns `true`. Otherwise, the function continues iterating through the array. Finally, it returns `true` if GCD is 1, `false` otherwise.

Step-by-Step Algorithm

Step 1: Initialize a variable `g` to 0. This variable will store the running GCD.
Step 2: Iterate through the `nums` array.
Step 3: In each iteration, calculate the GCD of `g` and the current number `num` using `math.gcd(g, num)` in Python, `gcd(g, num)` in Java/C, or `std::gcd(g, num)` in C++.
Step 4: Update `g` with the calculated GCD.
Step 5: If `g` becomes 1 at any point, return `True` immediately because any subsequent GCD calculation will remain 1.
Step 6: After iterating through all numbers, if `g` is 1, return `True`. Otherwise, return `False`.

Key Insights

Insight 1: Bézout's identity states that for integers a and b, there exist integers x and y such that ax + by = gcd(a, b), where gcd(a, b) is the greatest common divisor of a and b. More generally, for integers a1, a2, ..., an, there exist integers x1, x2, ..., xn such that a1x1 + a2x2 + ... + anxn = gcd(a1, a2, ..., an).
Insight 2: The array `nums` is 'good' if and only if the greatest common divisor (GCD) of all the numbers in the array is equal to 1. This is a direct consequence of Bézout's identity.
Insight 3: The problem can be solved by iteratively calculating the GCD of the numbers in the array. If at any point the GCD becomes 1, the array is good and we can stop.

Complexity Analysis

Time Complexity: O(NlogM)

Space Complexity: O(1)

Topics

This problem involves: Array, Math, Number Theory.

Companies

Asked at: Dropbox, Jump Trading, Nokia.

Check If It Is a Good Array | LeetCode 1250

Hard

Array Math Number Theory

Time: O(NlogM)Space: O(1)

Solution Code

Available in:

class Solution {
    public boolean isGoodArray(int[] nums) {
        int g = 0;
        for (int num : nums) {
            g = gcd(g, num);
            if (g == 1) {
                return true;
            }
        }
        return g == 1;
    }

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

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Problem Info

Description

Problem 1250: Check If It Is a Good Array - Detailed explanation and solution approaches.

Topics

Array Math Number Theory

Companies

DropboxJump TradingNokia

Stats

Difficulty:Hard

Languages:4

Companies:3

Problem #:1250

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

6 steps

1
Step 1: Initialize a variable `g` to 0. This variable will store the running GCD.
2
Step 2: Iterate through the `nums` array.
3
Step 3: In each iteration, calculate the GCD of `g` and the current number `num` using `math.gcd(g, num)` in Python, `gcd(g, num)` in Java/C, or `std::gcd(g, num)` in C++.
4
Step 4: Update `g` with the calculated GCD.
5
Step 5: If `g` becomes 1 at any point, return `True` immediately because any subsequent GCD calculation will remain 1.
6
Step 6: After iterating through all numbers, if `g` is 1, return `True`. Otherwise, return `False`.

💡

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(NlogM)

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is determined by the number of iterations in the loop (N) and the time complexity of the `gcd` function (logM). The space complexity is constant because only a few variables are used, regardless of the input array size.

📖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

Iterative GCD Calculation (Same as Solution)

The approach described above is already quite efficient. There isn't a significantly better alternative in terms of asymptotic complexity.

Time: O(NlogM)Space: O(1)

Trade-offs: This is the standard and most efficient way to solve this problem, especially given the constraints. There are no significant tradeoffs to consider against alternative approaches.

✨

Key Insights

1Insight 1: Bézout's identity states that for integers a and b, there exist integers x and y such that ax + by = gcd(a, b), where gcd(a, b) is the greatest common divisor of a and b. More generally, for integers a1, a2, ..., an, there exist integers x1, x2, ..., xn such that a1x1 + a2x2 + ... + anxn = gcd(a1, a2, ..., an).
2Insight 2: The array `nums` is 'good' if and only if the greatest common divisor (GCD) of all the numbers in the array is equal to 1. This is a direct consequence of Bézout's identity.
3Insight 3: The problem can be solved by iteratively calculating the GCD of the numbers in the array. If at any point the GCD becomes 1, the array is good and we can stop.

📚

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: Empty array. The code handles this implicitly. An empty array is not considered a good array because the final GCD will be initialized value which equals 0, so the result is false. It will return the initial value of `g`, which is 0. So the condition g == 1 will be false.
Edge case 2: Array containing only 1. In this case, the GCD will immediately become 1, and the function will return `True`.
Edge case 3: Array containing only 0s. The code will correctly return `false` because the final GCD is 0.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not understanding Bézout's identity and its implications for solving this problem.

✗Common mistake 2: Using a naive GCD calculation algorithm that has a higher time complexity than O(logM).

✗Common mistake 3: Failing to handle the case where the array is empty or contains only zeros.

✅

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: Cryptography. GCD is used in key generation and encryption algorithms like RSA.

🏦Real-world scenario 2: Music theory. Finding the simplest ratio between frequencies to create harmonious intervals involves GCD calculation.

💡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

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ Watch Out

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

👨‍💻 From Experience

I encountered a variation of this problem in a system design interview. The key was recognizing the underlying pattern.

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: November 18, 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

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

@DevExpert92Jul 14

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 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.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

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 "Check If It Is a Good Array"?

For this hard-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.

🤖 Need help with this problem?