What is the time complexity of Sum of Two Integers?

The optimal solution has a time complexity of O(1).

What data structures are used to solve Sum of Two Integers?

This problem uses Math, Bit Manipulation. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Sum of Two Integers?

Insight 1: Addition can be broken down into two bitwise operations: XOR (^) for calculating the sum without carry, and AND (&) followed by a left shift (<< 1) for calculating the carry. Insight 2: The process of calculating the sum and carry needs to be repeated until the carry becomes zero.

Which companies ask LeetCode 371. Sum of Two Integers in interviews?

This problem is frequently asked at Hulu.

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Sum of Two Integers - LeetCode 371 Solution

Q: How to solve Sum of Two Integers?

The provided code uses bitwise operations to simulate the addition of two integers. It iteratively calculates the sum without carry (using XOR) and the carry (using AND followed by a left shift). The process continues until there is no carry left. The code then handles the conversion of the 32-bit representation to a signed integer if necessary.

Q: Which companies ask LeetCode 371. Sum of Two Integers in interviews?

This problem is frequently asked at Hulu.

Sum of Two Integers - Complete Solution Guide

Sum of Two Integers is LeetCode problem 371, 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

Given two integers a and b , return the sum of the two integers without using the operators + and - . Example 1: Input: a = 1, b = 2 Output: 3 Example 2: Input: a = 2, b = 3 Output: 5 Constraints: -1000 <= a, b <= 1000

Detailed Explanation

The problem requires us to calculate the sum of two integers, `a` and `b`, without using the standard addition (+) or subtraction (-) operators. This means we need to utilize bitwise operations to simulate the process of addition.

Solution Approach

The provided code uses bitwise operations to simulate the addition of two integers. It iteratively calculates the sum without carry (using XOR) and the carry (using AND followed by a left shift). The process continues until there is no carry left. The code then handles the conversion of the 32-bit representation to a signed integer if necessary.

Step-by-Step Algorithm

Step 1: Initialize a mask `mask` to `0xFFFFFFFF` (32 bits of 1s). This is used to ensure that we're dealing with 32-bit integers, regardless of the underlying language's integer representation.
Step 2: Enter a `while` loop that continues as long as `b` (the carry) is not zero.
Step 3: Calculate the sum without carry using `a ^ b`. This performs a bitwise XOR operation, where each bit in the result is 1 if the corresponding bits in `a` and `b` are different, and 0 if they are the same. Then AND this with the mask.
Step 4: Calculate the carry using `(a & b) << 1`. This performs a bitwise AND operation, where each bit in the result is 1 if the corresponding bits in `a` and `b` are both 1, and 0 otherwise. The result is then left-shifted by 1, effectively multiplying it by 2. Then AND this with the mask.
Step 5: Update `a` and `b` with the calculated sum without carry and carry, respectively. This is crucial for the iterative process.
Step 6: After the loop finishes (when there's no carry), check if `a` is greater than `0x7FFFFFFF`. This checks if the most significant bit of the 32-bit representation is set (i.e., if it's a negative number in 2's complement).
Step 7: If `a` is greater than `0x7FFFFFFF`, convert it to a signed integer using `~(a ^ mask)`. This essentially performs the 2's complement operation to get the negative representation.
Step 8: If `a` is not greater than `0x7FFFFFFF`, it's a positive number, so return `a` directly.

Key Insights

Insight 1: Addition can be broken down into two bitwise operations: XOR (^) for calculating the sum without carry, and AND (&) followed by a left shift (<< 1) for calculating the carry.
Insight 2: The process of calculating the sum and carry needs to be repeated until the carry becomes zero.
Insight 3: We need to handle the representation of integers in Python (and other languages) which may have a different default bit width or sign representation than what's assumed in bitwise operations designed for fixed-size integers (like 32-bit). This requires masking and sign extension handling.

Complexity Analysis

Time Complexity: O(1)

Space Complexity: O(1)

Topics

This problem involves: Math, Bit Manipulation.

Companies

Asked at: Hulu.

Sum of Two Integers | LeetCode 371

Medium

Math Bit Manipulation

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

Solution Code

Available in:

class Solution {
    public int getSum(int a, int b) {
        int mask = 0xFFFFFFFF;

        while (b != 0) {
            int tempA = (a ^ b) & mask;
            int tempB = ((a & b) << 1) & mask;
            a = tempA;
            b = tempB;
        }

        if (a > 0x7FFFFFFF) {
            return ~(a ^ mask);
        } else {
            return a;
        }
    }
}

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Problem Info

Description

Problem 371: Sum of Two Integers - Detailed explanation and solution approaches.

Topics

Math Bit Manipulation

Companies

Hulu

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:371

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

8 steps

1
Step 1: Initialize a mask `mask` to `0xFFFFFFFF` (32 bits of 1s). This is used to ensure that we're dealing with 32-bit integers, regardless of the underlying language's integer representation.
2
Step 2: Enter a `while` loop that continues as long as `b` (the carry) is not zero.
3
Step 3: Calculate the sum without carry using `a ^ b`. This performs a bitwise XOR operation, where each bit in the result is 1 if the corresponding bits in `a` and `b` are different, and 0 if they are the same. Then AND this with the mask.
4
Step 4: Calculate the carry using `(a & b) << 1`. This performs a bitwise AND operation, where each bit in the result is 1 if the corresponding bits in `a` and `b` are both 1, and 0 otherwise. The result is then left-shifted by 1, effectively multiplying it by 2. Then AND this with the mask.
5
Step 5: Update `a` and `b` with the calculated sum without carry and carry, respectively. This is crucial for the iterative process.
6
Step 6: After the loop finishes (when there's no carry), check if `a` is greater than `0x7FFFFFFF`. This checks if the most significant bit of the 32-bit representation is set (i.e., if it's a negative number in 2's complement).
7
Step 7: If `a` is greater than `0x7FFFFFFF`, convert it to a signed integer using `~(a ^ mask)`. This essentially performs the 2's complement operation to get the negative representation.
8
Step 8: If `a` is not greater than `0x7FFFFFFF`, it's a positive number, so return `a` directly.

💡

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(1)

⚡ Constant time - Super fast! Same speed regardless of input size.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The `while` loop iterates at most a constant number of times, as the number of bits in the integers is fixed (32 bits). Bitwise operations are also constant time. Therefore, the overall time complexity is O(1). The algorithm uses a constant amount of extra space, regardless of the input values. Therefore the space complexity is O(1).

📖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

The addition can also be implemented using recursion, where each recursive call handles one iteration of the bitwise addition process.

Time: O(1)Space: O(1) due to limited recursion depth

Trade-offs: Recursion offers an alternative style but doesn't provide any performance benefits compared to the iterative method and may lead to stack overflow issues in certain environments. It's generally less efficient due to function call overhead.

✨

Key Insights

1Insight 1: Addition can be broken down into two bitwise operations: XOR (^) for calculating the sum without carry, and AND (&) followed by a left shift (<< 1) for calculating the carry.
2Insight 2: The process of calculating the sum and carry needs to be repeated until the carry becomes zero.
3Insight 3: We need to handle the representation of integers in Python (and other languages) which may have a different default bit width or sign representation than what's assumed in bitwise operations designed for fixed-size integers (like 32-bit). This requires masking and sign extension handling.

📚

Key Terms Explained

For Students

Edge Cases

Edge case 1: Negative numbers. The code handles negative numbers correctly by using the 2's complement representation and checking for the most significant bit to determine if the result should be converted back to a signed integer.
Edge case 2: Addition resulting in overflow/underflow. Although the constraints limit the range of input numbers, the 32-bit mask ensures that the intermediate results are treated as 32-bit numbers, effectively simulating potential overflows during the addition process. The final check and conversion to signed integers handle cases where the 32-bit result represents a negative number even if the mathematical sum would be positive.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to handle negative numbers correctly. Without handling 2's complement, the code will produce incorrect results for negative inputs.

✗Common mistake 2: Not using a mask. The mask is crucial for ensuring the operations are performed on 32-bit integers, which is essential for correct handling of carries and overflow, especially when languages use integers with larger default sizes.

✅

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: Implementing low-level arithmetic operations in embedded systems or hardware where dedicated addition circuits are not available or are being tested.

🏦Real-world scenario 2: Cryptography. Bitwise operations are fundamental in many cryptographic algorithms, often requiring custom arithmetic implementations.

💡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

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ Watch Out

Always test your solution with edge cases: empty input, single element, and boundary values.

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 12, 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

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

@AlgoMasterJul 13

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

@InterviewProJul 6

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

@CodeNinjaJun 29

❓

Frequently Asked Questions

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Sum of Two Integers"?

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

🤖 Need help with this problem?