Sum of Two Integers Without Using '+' Operator
Problem
Given two integers a and b, return the sum of the two integers without using the '+' or '-' operators.
- −10⁹ ≤ a, b ≤ 10⁹
Example
a = 1, b = 23The straightforward addition using '+' is disallowed. Instead, the algorithm uses bitwise operations to simulate addition. XOR (^) computes the sum bits without carry, while AND (&) followed by a left shift (<< 1) computes the carry bits. Iteratively applying these operations accumulates the sum until no carry remains. For example, starting with a=1 (01) and b=2 (10): - XOR: 01 ^ 10 = 11 (sum without carry) - AND and shift: (01 & 10) << 1 = 00 (carry) Since carry is zero, the sum is 3 (11 in binary).
Approach
Straightforward Solution
Using the '+' operator directly adds the two numbers in O(1) time. However, the problem forbids '+' and '-', so this approach is invalid.
Core Observation
Addition can be decomposed into two parts: sum without carry and carry bits. XOR operation yields the sum bits ignoring carry, while AND operation identifies carry bits. Shifting the carry bits left aligns them for the next addition step.
Path to Optimal
The key recognition signals are 'sum without using +' and 'bitwise operations'. These indicate Bit Manipulation because addition can be simulated by combining XOR and AND operations iteratively. XOR computes the sum bits ignoring carry, and AND followed by left shift computes the carry bits. Repeating this process until carry is zero yields the final sum.
Optimal Approach
Use a loop that continues while the carry (b) is non-zero. In each iteration, compute carry as (a & b) << 1, update a as a ^ b (sum without carry), and update b as carry. When carry becomes zero, a contains the final sum. Masking with 0xffffffff simulates 32-bit integer overflow behavior, ensuring correct results for negative numbers in Python's unlimited integer representation.
Time
O(1)
The loop runs at most 32 times for 32-bit integers, as each iteration propagates carry bits one position to the left, guaranteeing termination in constant time.
Space
O(1)
Only a fixed number of integer variables and bitwise operations are used, requiring constant auxiliary space.
Pattern Spotlight
Bit Manipulation (Iterative Carry Propagation)
Addition without '+' can be achieved by iteratively combining sum bits (XOR) and carry bits (AND shifted left), propagating carry until none remains, effectively simulating binary addition at the bit level.
Solution
| 1 | class Solution:
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| 2 | def getSum(self, a: int, b: int) -> int:
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| 3 | mask = 0xffffffff
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| 4 |
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| 5 | while (b & mask) > 0:
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| 6 | carry = (a & b) << 1
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| 7 | a = a ^ b
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| 8 | b = carry
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| 9 |
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| 10 | return (a & mask) if b > 0 else a |
Step-by-Step Solution
Simulate 32-bit Integer Behavior with Masking
| 3 | mask = 0xffffffff
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Objective
To constrain the integers to 32-bit representation, simulating overflow and enabling correct handling of negative numbers in Python.
Key Insight
Python integers are unbounded, so bitwise operations can produce unexpected results for negative numbers. Applying a mask of 0xffffffff limits values to 32 bits, mimicking fixed-width integer behavior. This ensures that operations like carry propagation and final result extraction behave as expected for signed 32-bit integers.
Interview Quick-Check
Core Logic
Masking with 0xffffffff confines all intermediate results to 32 bits, enabling correct simulation of signed integer overflow.
Common Pitfalls & Bugs
Omitting the mask leads to infinite loops or incorrect results for negative inputs due to Python's unlimited integer size.
Iteratively Compute Sum and Carry Until No Carry Remains
| 5 | while (b & mask) > 0:
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| 6 | carry = (a & b) << 1
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| 7 | a = a ^ b
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| 8 | b = carry
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Objective
To repeatedly update the sum and carry bits until the carry becomes zero, yielding the final sum.
Key Insight
Each iteration computes the carry bits by ANDing a and b and shifting left, while the sum bits are computed by XORing a and b. Updating a and b with these values simulates one step of binary addition. The loop continues until carry is zero, meaning no further bit propagation is needed and the sum is complete.
Interview Quick-Check
Core Logic
The loop uses XOR for sum bits and shifted AND for carry bits, iterating until carry is zero, which simulates binary addition without '+' operator.
State & Boundaries
The loop condition `(b & mask) > 0` ensures the process continues only while there are carry bits to propagate.
Common Pitfalls & Bugs
Incorrect loop condition or missing masking can cause infinite loops or incorrect results.
Return Final Result with Correct Sign Handling
| 10 | return (a & mask) if b > 0 else a |
Objective
To return the final sum, correctly handling negative results by applying the mask conditionally.
Key Insight
After the loop, if carry bits remain positive, the result is masked to 32 bits to simulate signed integer behavior. Otherwise, the raw value of a is returned. This conditional return ensures that negative sums are represented correctly in Python's integer model, which differs from fixed-width integers in other languages.
Interview Quick-Check
Core Logic
The conditional return applies the mask to simulate 32-bit signed integer representation when carry bits are positive.
Common Pitfalls & Bugs
Failing to mask the result can cause incorrect sign representation for negative sums.
Line Analysis
This solution has 4 Critical lines interviewers watch for.
carry = (a & b) << 1Calculate carry bits by ANDing a and b, then shifting left.
AND identifies bits where both a and b have 1s, indicating a carry is needed; shifting left aligns these carry bits with the next higher bit position for addition in the next iteration.
while (b & mask) > 0:Continue loop while carry bits exist within 32-bit range.
The loop condition ensures the algorithm iterates only while there are carry bits to propagate, preventing infinite loops and guaranteeing termination.
a = a ^ bCompute sum bits without carry using XOR.
XOR adds bits without considering carry, effectively summing bits where only one of a or b has a 1, which is the core of bitwise addition.
return (a & mask) if b > 0 else aReturn the final sum, applying mask if carry bits remain positive.
Applying the mask here ensures the result fits within 32 bits and correctly represents negative numbers in Python's integer model, which differs from fixed-width integers in other languages.
mask = 0xffffffffDefine a mask to limit integers to 32 bits.
This mask simulates 32-bit integer overflow behavior in Python, which uses arbitrary-length integers, ensuring bitwise operations behave as expected for signed 32-bit integers.
b = carryUpdate b to the carry bits for the next iteration.
Assigning carry to b prepares the next iteration to add these carry bits, propagating them until no carry remains.
Test Your Understanding
Why does the algorithm use XOR for sum and AND followed by left shift for carry in each iteration?
XOR computes the sum bits without considering carry, while AND identifies positions where both bits are 1, indicating a carry is needed. Shifting carry left aligns it with the next higher bit position for addition in the next iteration.
Related Problems
Bit Manipulation pattern
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