What is the time complexity of Triples with Bitwise AND Equal To Zero?

The optimal solution has a time complexity of O(n^2 + 2^16*log(2^16) + n).

What is the space complexity of Triples with Bitwise AND Equal To Zero?

The space complexity is O(2^16).

What data structures are used to solve Triples with Bitwise AND Equal To Zero?

This problem uses Array, Hash Table, Bit Manipulation. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Triples with Bitwise AND Equal To Zero?

Insight 1: Direct triple iteration would be O(n^3), which is too slow given n can be 1000. Insight 2: Pre-calculate and store the result of all possible pairwise AND operations. This significantly reduces the redundant calculations within the triple loop.

Which companies ask LeetCode 982. Triples with Bitwise AND Equal To Zero in interviews?

This problem is frequently asked at Flipkart.

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Triples with Bitwise AND Equal To Zero - LeetCode 982 Solution

Q: How to solve Triples with Bitwise AND Equal To Zero?

The solution efficiently calculates the number of AND triples by using a frequency counting approach coupled with dynamic programming related to bitwise operations. It first computes all pairwise ANDs of numbers from the array and stores their counts. Then, it leverages dynamic programming to count, for each number, how many masks are its supersets such that the AND-ing result is zero. Finally, the solution iterates through the input array `nums` and uses precomputed frequency counts to quickly determine the number of triples that result in zero.

Triples with Bitwise AND Equal To Zero - Complete Solution Guide

Triples with Bitwise AND Equal To Zero is LeetCode problem 982, 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 integer array nums, return the number of AND triples . An AND triple is a triple of indices (i, j, k) such that: 0 <= i < nums.length 0 <= j < nums.length 0 <= k < nums.length nums[i] & nums[j] & nums[k] == 0 , where & represents the bitwise-AND operator. Example 1: Input: nums = [2,1,3] Output: 12 Explanation: We could choose the following i, j, k triples: (i=0, j=0, k=1) : 2 & 2 & 1 (i=0, j=1, k=0) : 2 & 1 & 2 (i=0, j=1, k=1) : 2 & 1 & 1 (i=0, j=1, k=2) : 2 & 1 & 3 (i=0, j=2, k=1) : 2

Detailed Explanation

The problem asks us to find the number of "AND triples" in a given integer array `nums`. An AND triple is a set of three indices (i, j, k) such that the bitwise AND of the elements at those indices (nums[i] & nums[j] & nums[k]) is equal to 0. The indices i, j, and k can be the same. The problem constraints specify the size of the array (`1 <= nums.length <= 1000`) and the range of values in the array (`0 <= nums[i] < 2^16`).

Solution Approach

The solution efficiently calculates the number of AND triples by using a frequency counting approach coupled with dynamic programming related to bitwise operations. It first computes all pairwise ANDs of numbers from the array and stores their counts. Then, it leverages dynamic programming to count, for each number, how many masks are its supersets such that the AND-ing result is zero. Finally, the solution iterates through the input array `nums` and uses precomputed frequency counts to quickly determine the number of triples that result in zero.

Step-by-Step Algorithm

Step 1: Initialize a frequency array `counts` of size 2^16 to store the counts of all possible pairwise AND results.
Step 2: Iterate through all pairs of numbers in `nums` and calculate their bitwise AND. Increment the count of this AND result in the `counts` array.
Step 3: Create a `dp` array which is a copy of `counts`. The `dp` array will be used for dynamic programming to count subsets that result in the bitwise AND equal to zero.
Step 4: Use dynamic programming to build the `dp` table. Iterate through each bit position (0 to 15). For each mask, if the i-th bit is set, accumulate values. Accumulate the counts of subsets by iterating through each bit and updating dp[mask] using dp[mask ^ (1 << i)]. This allows the code to rapidly count masks resulting in zero when ANDed together.
Step 5: Calculate the total number of AND triples. Iterate through the input array `nums`. For each number `num`, find its complement with respect to 2^16 - 1 (all bits set to 1). The `dp` array at that complement contains a pre-calculated sum: how many pairs when ANDed result in a mask that when ANDed with the current number result in zero. Add this count to the total number of triplets.
Step 6: Return the total count of AND triples.

Key Insights

Insight 1: Direct triple iteration would be O(n^3), which is too slow given n can be 1000.
Insight 2: Pre-calculate and store the result of all possible pairwise AND operations. This significantly reduces the redundant calculations within the triple loop.
Insight 3: The bitmask complement technique allows efficient counting of how many numbers, when ANDed, result in zero. Utilizing dynamic programming speeds this up.

Complexity Analysis

Time Complexity: O(n^2 + 2^16*log(2^16) + n)

Space Complexity: O(2^16)

Topics

This problem involves: Array, Hash Table, Bit Manipulation.

Companies

Asked at: Flipkart.

Triples with Bitwise AND Equal To Zero | LeetCode 982

Hard

Array Hash Table Bit Manipulation

Time: O(n^2 + 2^16*log(2^16) + n)Space: O(2^16)

Solution Code

Available in:

class Solution {
    public int countTriplets(int[] nums) {
        int limit = 1 << 16;
        int[] counts = new int[limit];

        for (int num1 : nums) {
            for (int num2 : nums) {
                counts[num1 & num2]++;
            }
        }

        int[] dp = counts.clone();

        for (int i = 0; i < 16; i++) {
            for (int mask = 0; mask < limit; mask++) {
                if ((mask >> i) % 2 == 1) {
                    dp[mask] += dp[mask ^ (1 << i)];
                }
            }
        }

        int triplets = 0;
        int all_bits_mask = limit - 1;
        for (int num : nums) {
            int complement = num ^ all_bits_mask;
            triplets += dp[complement];
        }

        return triplets;
    }
}

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Problem Info

Description

Problem 982: Triples with Bitwise AND Equal To Zero - Detailed explanation and solution approaches.

Topics

Array Hash Table Bit Manipulation

Companies

Flipkart

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:982

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 frequency array `counts` of size 2^16 to store the counts of all possible pairwise AND results.
2
Step 2: Iterate through all pairs of numbers in `nums` and calculate their bitwise AND. Increment the count of this AND result in the `counts` array.
3
Step 3: Create a `dp` array which is a copy of `counts`. The `dp` array will be used for dynamic programming to count subsets that result in the bitwise AND equal to zero.
4
Step 4: Use dynamic programming to build the `dp` table. Iterate through each bit position (0 to 15). For each mask, if the i-th bit is set, accumulate values. Accumulate the counts of subsets by iterating through each bit and updating dp[mask] using dp[mask ^ (1 << i)]. This allows the code to rapidly count masks resulting in zero when ANDed together.
5
Step 5: Calculate the total number of AND triples. Iterate through the input array `nums`. For each number `num`, find its complement with respect to 2^16 - 1 (all bits set to 1). The `dp` array at that complement contains a pre-calculated sum: how many pairs when ANDed result in a mask that when ANDed with the current number result in zero. Add this count to the total number of triplets.
6
Step 6: Return the total count of AND triples.

💡

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^2 + 2^16*log(2^16) + n)

💾

Space Complexity

O(2^16)

📝

Why?

The time complexity is determined by the most time-consuming parts of the algorithm. The nested loops in the dynamic programming step determine the overall performance characteristics and the memory used is determined by the number of bits representing a number in nums.

📖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 all possible triples (i, j, k) and directly calculate nums[i] & nums[j] & nums[k]. If the result is 0, increment the count.

Time: O(n^3)Space: O(1)

Trade-offs: This approach is very simple to implement, but it is inefficient for larger input arrays. It would likely result in Time Limit Exceeded (TLE) on LeetCode. Use the provided solution for better performance.

✨

Key Insights

1Insight 1: Direct triple iteration would be O(n^3), which is too slow given n can be 1000.
2Insight 2: Pre-calculate and store the result of all possible pairwise AND operations. This significantly reduces the redundant calculations within the triple loop.
3Insight 3: The bitmask complement technique allows efficient counting of how many numbers, when ANDed, result in zero. Utilizing dynamic programming speeds this up.

📚

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!

Edge Cases

Edge case 1: nums = [0, 0, 0]. The code correctly counts all 27 possible combinations because 0 & 0 & 0 = 0. The precalculation and dynamic programming correctly handle this.
Edge case 2: nums = [2, 1, 3]. The code correctly enumerates and counts the 12 valid combinations as specified in the example.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not recognizing the constraints and attempting a brute-force O(n^3) approach. This will lead to TLE.

✗Common mistake 2: Failing to pre-compute the ANDs to avoid redundant calculations. This can significantly slow down the performance.

✗Common mistake 3: Not understanding the dynamic programming approach and failing to realize its impact on counting relevant supersets.

✅

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: Data analysis and filtering. This technique could be used to efficiently identify combinations of data points that satisfy certain bitwise conditions, such as finding data sets where the bitwise AND of certain attributes is zero (indicating a specific relationship or absence of certain features).

🏦Real-world scenario 2: Network security. Similar bit manipulation can be useful in firewall rules to filter network packets based on specific bit patterns in their headers. Although this specific problem doesn't directly translate to network security, the underlying bitwise operations and frequency analysis are relevant.

💡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

It's okay to not finish hard problems in interviews. Show your problem-solving process.

⚠️ 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: August 22, 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

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 14

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 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 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 "Triples with Bitwise AND Equal To Zero"?

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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