What is the time complexity of The Number of Good Subsets?

The optimal solution has a time complexity of O(n + 30 * 2^10 + 2^10).

What is the space complexity of The Number of Good Subsets?

The space complexity is O(n + 2^10).

What data structures are used to solve The Number of Good Subsets?

This problem uses Array, Hash Table, Math, Dynamic Programming, Bit Manipulation, Counting, Number Theory, Bitmask. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving The Number of Good Subsets?

Insight 1: The numbers in the input array are constrained to be between 1 and 30 inclusive. This limits the number of possible prime factors to a small, fixed set of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. This allows for a bitmask approach using these primes. Insight 2: Dynamic programming with bitmasking can be used to efficiently count the number of good subsets. The bitmask represents the primes included in the product of the current subset. `dp[mask]` stores the number of subsets whose product has the prime factors represented by `mask`.

Which companies ask LeetCode 1994. The Number of Good Subsets in interviews?

This problem is frequently asked at Lowe's, Media.net.

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The Number of Good Subsets - LeetCode 1994 Solution

The Number of Good Subsets - Complete Solution Guide

The Number of Good Subsets is LeetCode problem 1994, 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

You are given an integer array nums . We call a subset of nums good if its product can be represented as a product of one or more distinct prime numbers. For example, if nums = [1, 2, 3, 4] : [2, 3] , [1, 2, 3] , and [1, 3] are good subsets with products 6 = 2*3 , 6 = 2*3 , and 3 = 3 respectively. [1, 4] and [4] are not good subsets with products 4 = 2*2 and 4 = 2*2 respectively. Return the number of different good subsets in nums modulo 10 9 + 7 . A subset of nums is any array that can be obtai

Detailed Explanation

The problem asks us to find the number of "good" subsets from a given array of integers. A subset is considered "good" if the product of its elements can be expressed as a product of distinct prime numbers. We need to return the count of such good subsets modulo 10^9 + 7. Note that different subsets formed from the same elements at different indices count as distinct.

Solution Approach

The solution uses dynamic programming with bitmasking. First, it counts the occurrences of each number from 1 to 30. Then, it iterates through the numbers from 2 to 30. For each number, it checks if it is a valid candidate for a 'good' subset (i.e., its prime factorization consists of distinct primes). If it is, a bitmask is created representing the prime factors of the number. The DP table, `dp`, is updated iteratively. `dp[mask]` stores the number of subsets that result in a product with exactly the primes denoted by mask. Finally, the algorithm sums up the values in the `dp` table (excluding `dp[0]`, the empty set), multiplies by the power of 2 representing the number of ways to include 1s, and returns the result modulo 10^9 + 7.

Step-by-Step Algorithm

Step 1: Initialize `primes` array with prime numbers less than or equal to 30: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29].
Step 2: Create a `counts` array to store the frequency of each number from 1 to 30 in the input array `nums`.
Step 3: Initialize a `dp` array of size 2^10 (where 10 is the number of primes) to store the counts of subsets with different prime factor combinations. Set `dp[0] = 1` because the empty set is always valid and its product has no prime factors.
Step 4: Iterate through the numbers from 2 to 30. For each number `num`, check if it can contribute to a 'good' subset. If `num` is divisible by 4, 9, or 25, skip it, as its product won't be distinct primes.
Step 5: Create a bitmask representing the prime factors of `num`. For each prime in the `primes` array, if `num` is divisible by the prime, set the corresponding bit in the mask.
Step 6: Update the `dp` array. Iterate backwards from `(1 << num_primes) - 1` to 0. If the current mask `j` does not have any of the prime factors of `num` (i.e., `(j & mask) == 0`), then update `dp[j | mask]` by adding `dp[j] * counts[num]` (modulo MOD). This represents adding the current number `num` to all subsets represented by `dp[j]` which does not conflict with the `mask` of `num`.
Step 7: After iterating through all numbers from 2 to 30, calculate the sum of all values in the `dp` array, excluding `dp[0]` (representing the empty set). This gives the number of good subsets excluding '1'.
Step 8: Calculate 2 raised to the power of the number of 1s in the input array (`counts[1]`) modulo MOD. This represents the number of ways to include 1s in any subset.
Step 9: Multiply the sum calculated in step 7 by the power of 2 calculated in step 8 (modulo MOD) to account for all possible combinations including 1s. Return the result as an integer.

Key Insights

Insight 1: The numbers in the input array are constrained to be between 1 and 30 inclusive. This limits the number of possible prime factors to a small, fixed set of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. This allows for a bitmask approach using these primes.
Insight 2: Dynamic programming with bitmasking can be used to efficiently count the number of good subsets. The bitmask represents the primes included in the product of the current subset. `dp[mask]` stores the number of subsets whose product has the prime factors represented by `mask`.
Insight 3: The number '1' can be included in any good subset without affecting whether it remains a good subset. Therefore, we can count the number of 1s separately and multiply the final result by 2^(number of 1s) to account for all possible combinations including 1s.
Insight 4: Numbers that are not products of distinct primes (e.g., 4, 9, 25) cannot be part of a good subset, as they introduce repeated prime factors. These should be skipped.

Complexity Analysis

Time Complexity: O(n + 30 * 2^10 + 2^10)

Space Complexity: O(n + 2^10)

Topics

This problem involves: Array, Hash Table, Math, Dynamic Programming, Bit Manipulation, Counting, Number Theory, Bitmask.

Companies

Asked at: Lowe's, Media.net.

The Number of Good Subsets | LeetCode 1994

Hard

Array Hash Table Math Dynamic Programming Bit Manipulation Counting Number Theory Bitmask

Time: O(n + 30 * 2^10 + 2^10)Space: O(n + 2^10)

Solution Code

Available in:

class Solution {
    public int numberOfGoodSubsets(int[] nums) {
        int MOD = (int) 1e9 + 7;

        int[] primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29};
        int num_primes = primes.length;

        int[] counts = new int[31];
        for (int num : nums) {
            counts[num]++;
        }

        int[] dp = new int[1 << num_primes];
        dp[0] = 1;

        for (int num = 2; num <= 30; num++) {
            if (counts[num] == 0) {
                continue;
            }

            if (num % 4 == 0 || num % 9 == 0 || num % 25 == 0) {
                continue;
            }

            int mask = 0;
            for (int i = 0; i < num_primes; i++) {
                if (num % primes[i] == 0) {
                    mask |= (1 << i);
                }
            }

            for (int j = (1 << num_primes) - 1; j >= 0; j--) {
                if ((j & mask) == 0) {
                    dp[j | mask] = (int) (((long) dp[j | mask] + (long) dp[j] * counts[num]) % MOD);
                }
            }
        }

        long ans = 0;
        for (int i = 1; i < (1 << num_primes); i++) {
            ans = (ans + dp[i]) % MOD;
        }

        long power_of_2_for_1s = 1;
        for (int i = 0; i < counts[1]; i++) {
            power_of_2_for_1s = (power_of_2_for_1s * 2) % MOD;
        }

        ans = (ans * power_of_2_for_1s) % MOD;

        return (int) ans;
    }
}

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Problem Info

Description

Problem 1994: The Number of Good Subsets - Detailed explanation and solution approaches.

Topics

Array Hash Table Math Dynamic Programming Bit Manipulation Counting Number Theory Bitmask

Companies

Lowe'sMedia.net

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:1994

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

9 steps

1
Step 1: Initialize `primes` array with prime numbers less than or equal to 30: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29].
2
Step 2: Create a `counts` array to store the frequency of each number from 1 to 30 in the input array `nums`.
3
Step 3: Initialize a `dp` array of size 2^10 (where 10 is the number of primes) to store the counts of subsets with different prime factor combinations. Set `dp[0] = 1` because the empty set is always valid and its product has no prime factors.
4
Step 4: Iterate through the numbers from 2 to 30. For each number `num`, check if it can contribute to a 'good' subset. If `num` is divisible by 4, 9, or 25, skip it, as its product won't be distinct primes.
5
Step 5: Create a bitmask representing the prime factors of `num`. For each prime in the `primes` array, if `num` is divisible by the prime, set the corresponding bit in the mask.
6
Step 6: Update the `dp` array. Iterate backwards from `(1 << num_primes) - 1` to 0. If the current mask `j` does not have any of the prime factors of `num` (i.e., `(j & mask) == 0`), then update `dp[j | mask]` by adding `dp[j] * counts[num]` (modulo MOD). This represents adding the current number `num` to all subsets represented by `dp[j]` which does not conflict with the `mask` of `num`.
7
Step 7: After iterating through all numbers from 2 to 30, calculate the sum of all values in the `dp` array, excluding `dp[0]` (representing the empty set). This gives the number of good subsets excluding '1'.
8
Step 8: Calculate 2 raised to the power of the number of 1s in the input array (`counts[1]`) modulo MOD. This represents the number of ways to include 1s in any subset.
9
Step 9: Multiply the sum calculated in step 7 by the power of 2 calculated in step 8 (modulo MOD) to account for all possible combinations including 1s. Return the result as an integer.

💡

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 + 30 * 2^10 + 2^10)

💾

Space Complexity

O(n + 2^10)

📝

Why?

The time complexity is dominated by the dynamic programming loop. The outer loop iterates a maximum of 30 times (from 1 to 30), and the inner loop iterates 2^10 (1024) times. The space complexity is dominated by the dp array, which needs to store the counts for each of the 2^10 possible subsets of primes.

📖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

Recursion with Memoization

A recursive approach with memoization could also be used. We could explore all possible subsets recursively, checking if their product is a product of distinct primes. Memoization can be used to store the results of subproblems (subsets) to avoid redundant calculations.

Time: O(n * 2^n) without memoization, where n is the number of unique elements in the array. With memoization this could be improved to O(n * 2^10) assuming small valuesSpace: O(2^n) without memoization, where n is the number of unique elements in the array. O(2^10) with memoization

Trade-offs: Recursion might be easier to understand conceptually, but dynamic programming is generally more efficient due to the way it handles dependencies and prevents repeated calculations. However the constraint allows the DP solution which gives a much better performance

✨

Key Insights

1Insight 1: The numbers in the input array are constrained to be between 1 and 30 inclusive. This limits the number of possible prime factors to a small, fixed set of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. This allows for a bitmask approach using these primes.
2Insight 2: Dynamic programming with bitmasking can be used to efficiently count the number of good subsets. The bitmask represents the primes included in the product of the current subset. `dp[mask]` stores the number of subsets whose product has the prime factors represented by `mask`.
3Insight 3: The number '1' can be included in any good subset without affecting whether it remains a good subset. Therefore, we can count the number of 1s separately and multiply the final result by 2^(number of 1s) to account for all possible combinations including 1s.
4Insight 4: Numbers that are not products of distinct primes (e.g., 4, 9, 25) cannot be part of a good subset, as they introduce repeated prime factors. These should be skipped.

📚

Key Terms Explained

For Students

Array

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

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Input array contains only 1s. The algorithm correctly calculates 2^(number of 1s) - 1.
Edge case 2: Input array contains only numbers divisible by 4, 9 or 25. The algorithm returns 0, as there are no valid good subsets.
Edge case 3: Input array contains only 1 and numbers divisible by 4, 9 or 25. The algorithm returns 2^(number of 1s) - 1, because even though the other numbers cannot contribute, the 1's can still be added/removed to make a valid set.
Edge case 4: An empty input array. The code should gracefully handle an empty input by initializing `counts` and continuing the DP as usual, resulting in zero good subsets as expected.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to handle the number 1 separately. Failing to account for the number of ways to include or exclude the '1' values leads to an incorrect count of good subsets.

✗Common mistake 2: Not handling modulo operations correctly. The problem requires the answer to be returned modulo 10^9 + 7. Failing to apply the modulo operation at each step can lead to integer overflow and an incorrect result.

✗Common mistake 3: Not properly filtering out numbers that introduce repeated prime factors (e.g., 4, 9, 25). These numbers cannot be part of a good subset, and failing to exclude them will lead to an incorrect count.

✗Common mistake 4: Incorrect bitmasking: It's crucial to have the bitmask logic correct and the associated checks (j & mask) == 0 be correct for choosing subsets correctly.

✅

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: This type of problem can be related to network security where you need to find combinations of secure channels (represented by prime numbers) to establish a connection without compromising the security (distinct primes).

🏦Real-world scenario 2: In combinatorial optimization problems where we need to find subsets that satisfy certain constraints related to prime factorization, this approach could be useful. For example, in scheduling or resource allocation where resources are linked to prime numbers and we want to create conflict-free schedules.

💡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.

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: June 15, 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 13

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

@DevExpert92Jul 6

❓

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 are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

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 "The Number of Good Subsets"?

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 hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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