Triples with Bitwise AND Equal To Zero

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class Solution:
    def countTriplets(self, nums: List[int]) -> int:
        freq = defaultdict(int)
        for x in nums: 
            for y in nums: 
                freq[x&y] += 1

        ans = 0
        for x in nums: 
            mask = x = x ^ 0xffff
            while x: 
                ans += freq[x]
                x = mask & (x-1)
            ans += freq[0]
        return ans 

Identifying Problem Isomorphism

An approximate isomorphism: “4Sum II”.

In “Triples with Bitwise AND Equal To Zero”, you are given an array of integers A, and the task is to find the count of triples of indices (i,j,k) such that A[i] & A[j] & A[k] == 0.

In “4Sum II”, you are given four integer arrays each of length n. The objective is to find the count of tuples (i, j, k, l) where A[i] + B[j] = -C[k] - D[l].

In both problems, the task is to find tuples or triples from the given arrays such that a certain condition involving these tuple elements is met. In the first problem, the condition is a bitwise AND operation resulting in zero, while in the second problem, it is a sum equation.

“4Sum II” is simpler, since it deals with sum operations, while “Triples with Bitwise AND Equal To Zero” involves bitwise operations which are a bit more complex to handle.

A common technique used to solve these problems is using hashmaps. In “4Sum II”, you can store the sum of every pair of elements from the first two arrays in a hashmap and then for every pair in the remaining two arrays, you check if their negated sum is present in the hashmap. Similarly, in “Triples with Bitwise AND Equal To Zero”, you can also use a hashmap to store the frequency of bitwise AND results of every pair in the array, and then iterate through the array once more to check if any element bitwise AND with the hashmap keys results in zero.

10 Prerequisite LeetCode Problems

“982. Triples with Bitwise AND Equal To Zero” requires understanding of bitwise operations and possibly dynamic programming or hash maps to optimize the calculation process. Here are 10 problems to prepare for this:

LeetCode 136. Single Number
- This problem introduces you to the concept of bitwise XOR operation.
LeetCode 191. Number of 1 Bits
- This problem is a good start to understand bit manipulation, specifically counting the number of ‘1’ bits.
LeetCode 201. Bitwise AND of Numbers Range
- This problem requires a deep understanding of bitwise AND operations.
LeetCode 260. Single Number III
- This problem expands on Single Number I and gives you more practice with bitwise operations.
LeetCode 371. Sum of Two Integers
- This problem involves implementing addition using bitwise operations, which can deepen your understanding of how these operations work.
LeetCode 338. Counting Bits
- This is a good problem to understand dynamic programming related to bitwise operations.
LeetCode 78. Subsets
- This problem can be solved with bit manipulation, and it helps to understand the concept of generating all subsets (powerset).
LeetCode 389. Find the Difference
- This problem is a simple exercise in using XOR to find an element that’s different in two similar arrays.
LeetCode 268. Missing Number
- This problem is about finding a missing number in an array and can be solved with bitwise operations.
LeetCode 421. Maximum XOR of Two Numbers in an Array

This is a more advanced problem that involves optimizing the search for the maximum XOR of two numbers in an array.

Through these problems, you’ll be able to understand the usage and characteristics of bitwise operations and how they can be used in combination with data structures and algorithms to solve more complex problems. This will be useful in solving problem “982. Triples with Bitwise AND Equal To Zero”.

The “Triples with Bitwise AND Equal To Zero” problem is a counting problem involving bitwise operations. It requires understanding of bitwise AND operation and how to efficiently count certain triples in an array.

Here are 10 problems to prepare:

Single Number (LeetCode #136): This problem helps you understand the basic concept of bitwise operations, especially XOR operation.
Number of 1 Bits (LeetCode #191): This problem helps you understand how to deal with the individual bits of a number.
Bitwise AND of Numbers Range (LeetCode #201): This problem involves bitwise AND operation and can help you understand how bitwise AND works with multiple numbers.
Counting Bits (LeetCode #338): This problem requires understanding of bitwise operations and dynamic programming.
Single Number II (LeetCode #137): This problem extends the Single Number problem to include bitwise operations in a more complex scenario.
Majority Element (LeetCode #169): Although not directly related to bitwise operations, this problem involves counting elements in an array, which is part of the problem statement.
Subsets (LeetCode #78): This problem can help you understand how to generate all combinations of elements, which might be useful in generating triples.
Power of Two (LeetCode #231): This problem can help you to understand properties of numbers in binary representation.
Binary Prefix Divisible By 5 (LeetCode #1018): This problem can deepen your understanding of binary number manipulation and counting.
Sum of Two Integers (LeetCode #371): This problem involves using bitwise operations to perform traditional arithmetic operations.

After practicing these problems, you should be better prepared to understand and solve the “Triples with Bitwise AND Equal To Zero” problem. Good luck!

Problem Classification

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 & 3 & 1 (i=1, j=0, k=0) : 1 & 2 & 2 (i=1, j=0, k=1) : 1 & 2 & 1 (i=1, j=0, k=2) : 1 & 2 & 3 (i=1, j=1, k=0) : 1 & 1 & 2 (i=1, j=2, k=0) : 1 & 3 & 2 (i=2, j=0, k=1) : 3 & 2 & 1 (i=2, j=1, k=0) : 3 & 1 & 2 Example 2:

Input: nums = [0,0,0] Output: 27

Constraints:

1 <= nums.length <= 1000 0 <= nums[i] < 216

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.