Bit Manipulation

Advanced & DP

O(1) low-level bitwise operations, masks, and XOR parity tricks

226 problems·45 Easy·107 Medium·74 Hard
Pattern Study Guide & Cheat Sheet

Perform bitwise operations (&, |, ^, ~, <<, >>) to check, set, toggle, or count bits in O(1) time and space without allocating memory.

Core Invariant: Essential Bit Identities: 1) `x ^ x = 0` and `x ^ 0 = x` (cancels pairs); 2) `n & (n - 1)` clears the lowest set bit (Brian Kernighan); 3) `n & -n` isolates lowest set bit; 4) `(1 << k)` creates mask for k-th bit.

Recognize it (Keywords & Signals)
  • Single number (every element appears twice except one)
  • Number of 1 bits (Hamming weight)
  • Counting bits from 0 to N
  • Subsets represented as bitmasks (N <= 20)
  • Power of two verification
When NOT to use

Numbers exceed 64-bit integer limits and bitwise representations become complex strings, or problem requires decimal digits.

How to solve (Step-by-step)
  1. 1.Identify if problem involves duplicate cancellation (XOR), bit counting (`n & (n - 1)`), or subset mask (`1 << n`).
  2. 2.To clear lowest set bit: `n &= (n - 1)` in a loop until `n == 0`.
  3. 3.To check if k-th bit is set: `(n & (1 << k)) != 0`.
  4. 4.To toggle k-th bit: `n ^= (1 << k)`.
  5. 5.To check power of two: `n > 0 and (n & (n - 1)) == 0`.
Watch for (Interview Traps)
  • Operator precedence: `==` has higher precedence than `&` in Python! Always wrap `(n & (1 << i)) != 0` in parentheses
  • Negative numbers in Python: Python uses arbitrary-precision integers, so bit shifts on negative numbers require masking with `0xFFFFFFFF`
  • Off-by-one in bit shifts: 0-indexed vs 1-indexed bit positions
Brian Kernighan's Bit Counting & XOR Reduction
def single_number(nums: list[int]) -> int:
    # XOR Property: x ^ x = 0 and x ^ 0 = x
    # All duplicate pairs cancel out, leaving the unique single number
    unique = 0
    for x in nums:
        unique ^= x
    return unique

def count_set_bits(n: int) -> int:
    # Brian Kernighan's Algorithm: n & (n - 1) clears the lowest set bit
    count = 0
    while n > 0:
        n &= (n - 1)
        count += 1
    return count
Cost
O(1) or O(number of set bits) · O(1) strictly constant memory (Hardware-level instructions with blazing fast execution.)
Canonical problems
#136 Single Number: XOR all numbers to cancel duplicates in O(N) time and O(1) space
#191 Number of 1 Bits: Clear lowest bit with n & (n - 1) in loop
#231 Power of Two: n > 0 and (n & (n - 1)) == 0
#338 Counting Bits: DP on bits: dp[i] = dp[i >> 1] + (i & 1)
Medium·24 companies·Max freq 90%·Acc 64.6%
NianticNeetCode 150NeetCode 150+21
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CoupangFiverrNeetCode 150+20
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HuluSwiggyNeetCode 150+7
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persistent systemsIBMMeta+4
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Medium·7 companies·Max freq 68%·Acc 73.1%
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SiemensZomatoBloomberg+4
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Hard·7 companies·Max freq 52%·Acc 82.9%
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Medium·6 companies·Max freq 100%·Acc 54.8%
PayPal HoneyGrowwPalo Alto Networks+3
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fourkitesBloombergMeta+3
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PaytmBloombergMicrosoft+3
Hard·6 companies·Max freq 100%·Acc 0.5%
thoughtspotWeRideUber+3
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MathWorksCoupangAirbnb+3
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AdobeIBMAmazon+3
Hard·6 companies·Max freq 75%·Acc 68.2%
DE ShawTuringGoogle+3
Medium·6 companies·Max freq 66%·Acc 88.3%
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UberMicrosoftGoogle+3
Easy·6 companies·Max freq 42%·Acc 83.9%
IBMGoogleBloomberg+3
Easy·6 companies·Max freq 42%·Acc 60.5%
GoogleAccentureBloomberg+3
Medium·6 companies·Max freq 39%·Acc 73.8%
VisaGoogleAmazon+3
Medium·6 companies·Max freq 38%·Acc 75.6%
MicrosoftGoldman SachsBloomberg+3
Easy·6 companies·Max freq 25%·Acc 54.5%
MetatcsBloomberg+3
Easy·5 companies·Max freq 100%·Acc 63.4%
ClouderaBloombergGoogle+2
Medium·5 companies·Max freq 100%·Acc 0.6%
MeeshoGoogleMicrosoft+2
Medium·5 companies·Max freq 100%·Acc 0.4%
BaiduUberGoogle+2
Easy·5 companies·Max freq 100%·Acc 85.3%
AonAmazonBloomberg+2
Easy·5 companies·Max freq 100%·Acc 69.9%
YahooGoogleAmazon+2
Medium·5 companies·Max freq 97%·Acc 0.8%
Hudson River TradingJump TradingAdobe+2
Medium·5 companies·Max freq 94%·Acc 30.2%
Squarepoint CapitalZomatoRippling+2
Hard·5 companies·Max freq 91%·Acc 78.3%
ExpediaOracleServiceNow+2
Medium·5 companies·Max freq 88%·Acc 44.7%
GeicoLinkedInMicrosoft+2
Medium·5 companies·Max freq 68%·Acc 61.2%
TrilogyGoldman SachsGoogle+2
Medium·5 companies·Max freq 63%·Acc 46.4%
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Medium·5 companies·Max freq 59%·Acc 0.9%
PaytmAmazonBloomberg+2
Medium·5 companies·Max freq 55%·Acc 61.6%
GrammarlyGoogleAmazon+2
Medium·5 companies·Max freq 54%·Acc 31.5%
LinkedInAmazonMeta+2
Hard·5 companies·Max freq 53%·Acc 46.0%
PinterestAmazonMicrosoft+2
Easy·5 companies·Max freq 52%·Acc 90.1%
AdobeAmazonMicrosoft+2
Easy·5 companies·Max freq 50%·Acc 68.7%
UberBloombergMeta+2
Medium·5 companies·Max freq 41%·Acc 67.4%
MicrosoftGoogleMeta+2
Medium·5 companies·Max freq 38%·Acc 53.6%
GoogleBloombergAmazon+2
Easy·5 companies·Max freq 38%·Acc 79.3%
MicrosoftBloombergMeta+2
Medium·5 companies·Max freq 38%·Acc 65.3%
IBMAmazonBloomberg+2
Easy·5 companies·Max freq 25%·Acc 81.5%
MetaAmazonBloomberg+2
Easy·5 companies·Max freq 25%·Acc 65.8%
GoogleBloombergAmazon+2
Medium·5 companies·Max freq 25%·Acc 61.9%
AmazonBloombergGoogle+2
Medium·5 companies·Max freq 16%·Acc 49.1%
GoogleMicrosoftAmazon+2
Medium·4 companies·Max freq 100%·Acc 53.8%
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Hard·4 companies·Max freq 100%·Acc 69.4%
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Medium·4 companies·Max freq 100%·Acc 50.2%
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Hard·4 companies·Max freq 100%·Acc 60.3%
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Medium·4 companies·Max freq 100%·Acc 66.6%
AonMicrosoftAmazon+1
Easy·4 companies·Max freq 100%·Acc 82.4%
MapboxMicrosoftGoogle+1
Medium·4 companies·Max freq 100%·Acc 62.9%
YahooAmazonGoogle+1
Hard·4 companies·Max freq 89%·Acc 51.0%
IXLTikTokMeta+1
Medium·4 companies·Max freq 88%·Acc 88.4%
Morgan StanleyNvidiaMicrosoft+1
Easy·4 companies·Max freq 63%·Acc 73.8%
OracleAdobeIBM+1
Hard·4 companies·Max freq 38%·Acc 66.1%
GoogleMetaAmazon+1
Easy·4 companies·Max freq 38%·Acc 64.5%
MicrosoftBloombergAmazon+1
Hard·4 companies·Max freq 27%·Acc 38.9%
GoogleGoldman SachsAmazon+1
Easy·4 companies·Max freq 25%·Acc 76.9%
MetaBloombergGoogle+1
Hard·4 companies·Max freq 25%·Acc 76.1%
AmazonMetaMicrosoft+1
Medium·4 companies·Max freq 18%·Acc 58.2%
BloombergAmazonGoogle+1
Hard·4 companies·Max freq 15%·Acc 64.6%
GoogleMetaAmazon+1
Medium·4 companies·Max freq 13%·Acc 55.5%
BloombergMetaAmazon+1
Hard·3 companies·Max freq 100%·Acc 46.4%
MindtickleRobloxDE Shaw
Easy·3 companies·Max freq 100%·Acc 88.5%
RobinhoodBloombergGoogle
Medium·3 companies·Max freq 100%·Acc 66.9%
TrilogyMetaGoogle
Hard·3 companies·Max freq 100%·Acc 52.6%
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AirbnbGoogleAmazon
Hard·3 companies·Max freq 50%·Acc 31.1%
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AppleMetaBloomberg
Medium·3 companies·Max freq 31%·Acc 0.6%
InfosysBloombergGoogle
Hard·3 companies·Max freq 25%·Acc 81.5%
AmazonBloombergGoogle
Medium·3 companies·Max freq 25%·Acc 66.7%
AmazonBloombergGoogle
Medium·3 companies·Max freq 25%·Acc 78.6%
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Hard·3 companies·Max freq 16%·Acc 0.6%
GoogleAmazonMeta
Easy·3 companies·Max freq 16%·Acc 80.3%
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Hard·3 companies·Max freq 13%·Acc 72.5%
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Easy·3 companies·Max freq 13%·Acc 53.7%
AmazonMicrosoftGoogle
Medium·2 companies·Max freq 100%·Acc 68.4%
BCGAmazon
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GrowwSprinklr
Easy·2 companies·Max freq 88%·Acc 87.7%
Walmart LabsGoogle
Medium·2 companies·Max freq 69%·Acc 49.5%
MicrostrategyGoogle
Medium·2 companies·Max freq 63%·Acc 35.1%
SwiggyAmazon
Hard·2 companies·Max freq 51%·Acc 51.6%
AppleGoogle
Hard·2 companies·Max freq 50%·Acc 51.6%
CitadelGoogle
Hard·2 companies·Max freq 50%·Acc 0.6%
tcsAirbnb
Medium·2 companies·Max freq 50%·Acc 84.7%
UberBloomberg
Medium·2 companies·Max freq 39%·Acc 70.0%
ZohoAmazon
Hard·2 companies·Max freq 38%·Acc 45.5%
DE ShawGoogle
Hard·2 companies·Max freq 38%·Acc 61.7%
InfosysMeta
Medium·2 companies·Max freq 38%·Acc 72.0%
MicrosoftGoogle
Easy·2 companies·Max freq 27%·Acc 85.3%
MicrosoftMeta
Easy·2 companies·Max freq 26%·Acc 75.1%
GoogleApple
Medium·2 companies·Max freq 25%·Acc 61.4%
GoogleAmazon
Medium·2 companies·Max freq 25%·Acc 64.7%
MetaAmazon
Easy·2 companies·Max freq 25%·Acc 78.9%
AmazonGoogle
Medium·2 companies·Max freq 25%·Acc 68.5%
GoogleAmazon
Hard·2 companies·Max freq 25%·Acc 40.5%
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Medium·2 companies·Max freq 25%·Acc 66.6%
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Easy·2 companies·Max freq 13%·Acc 79.2%
MetaGoogle
Medium·2 companies·Max freq 13%·Acc 84.8%
MetaGoogle
Easy·2 companies·Max freq 13%·Acc 0.7%
GoogleMeta
Medium·2 companies·Max freq 13%·Acc 79.8%
AmazonGoogle
Medium·2 companies·Max freq 13%·Acc 65.6%
AmazonGoogle
Hard·1 companies·Max freq 100%·Acc 47.2%
Directi
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Trilogy
Hard·1 companies·Max freq 100%·Acc 57.7%
Trilogy
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Kakao
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Quora
Easy·1 companies·Max freq 100%·Acc 76.3%
ZScaler
Hard·1 companies·Max freq 100%·Acc 33.2%
ZScaler
Medium·1 companies·Max freq 100%·Acc 31.1%
Mitsogo
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Mitsogo
Easy·1 companies·Max freq 100%·Acc 44.8%
Mitsogo
Hard·1 companies·Max freq 100%·Acc 46.5%
Vimeo
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PhonePe
Hard·1 companies·Max freq 100%·Acc 47.9%
Dropbox
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Hard·1 companies·Max freq 100%·Acc 53.4%
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Medium·1 companies·Max freq 100%·Acc 32.8%
IVP
Easy·1 companies·Max freq 100%·Acc 63.6%
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Medium·1 companies·Max freq 100%·Acc 41.8%
Akuna Capital
Medium·1 companies·Max freq 100%·Acc 64.1%
Zomato
Easy·1 companies·Max freq 100%·Acc 62.6%
Deutsche Bank
Medium·1 companies·Max freq 100%·Acc 52.2%
Teradata
Hard·1 companies·Max freq 90%·Acc 47.0%
Sprinklr
Medium·1 companies·Max freq 90%·Acc 42.8%
Sprinklr
Hard·1 companies·Max freq 90%·Acc 44.9%
Sprinklr
Hard·1 companies·Max freq 89%·Acc 30.8%
Aon
Medium·1 companies·Max freq 88%·Acc 44.1%
Morgan Stanley
Easy·1 companies·Max freq 88%·Acc 86.2%
Accenture
Easy·1 companies·Max freq 88%·Acc 94.7%
Adobe
Medium·1 companies·Max freq 88%·Acc 72.8%
Walmart Labs
Easy·1 companies·Max freq 76%·Acc 66.6%
Zoho
Hard·1 companies·Max freq 75%·Acc 36.9%
DE Shaw
Hard·1 companies·Max freq 75%·Acc 35.6%
DE Shaw
Hard·1 companies·Max freq 75%·Acc 72.9%
Airbnb
Hard·1 companies·Max freq 75%·Acc 45.7%
Infosys
Hard·1 companies·Max freq 65%·Acc 25.1%
IBM
Hard·1 companies·Max freq 63%·Acc 31.5%
Oracle
Medium·1 companies·Max freq 63%·Acc 72.1%
Salesforce
Hard·1 companies·Max freq 63%·Acc 31.4%
Salesforce
Hard·1 companies·Max freq 63%·Acc 41.3%
Salesforce
Medium·1 companies·Max freq 50%·Acc 58.2%
Apple
Medium·1 companies·Max freq 38%·Acc 37.5%
Amazon
Medium·1 companies·Max freq 38%·Acc 50.4%
Microsoft
Hard·1 companies·Max freq 38%·Acc 28.3%
Google
Medium·1 companies·Max freq 38%·Acc 53.9%
Google
Medium·1 companies·Max freq 31%·Acc 40.8%
Microsoft
Hard·1 companies·Max freq 27%·Acc 41.4%
Microsoft
Hard·1 companies·Max freq 25%·Acc 32.2%
Amazon
Medium·1 companies·Max freq 25%·Acc 39.2%
Amazon
Hard·1 companies·Max freq 25%·Acc 64.8%
Amazon
Easy·1 companies·Max freq 25%·Acc 73.0%
Amazon
Hard·1 companies·Max freq 25%·Acc 38.0%
Amazon
Medium·1 companies·Max freq 25%·Acc 62.8%
Amazon
Medium·1 companies·Max freq 25%·Acc 67.2%
Amazon
Medium·1 companies·Max freq 25%·Acc 71.1%
Amazon
Easy·1 companies·Max freq 25%·Acc 82.9%
Amazon
Medium·1 companies·Max freq 25%·Acc 61.8%
Amazon
Hard·1 companies·Max freq 25%·Acc 58.9%
Google
Hard·1 companies·Max freq 25%·Acc 22.1%
Google
Hard·1 companies·Max freq 25%·Acc 50.4%
Google
Hard·1 companies·Max freq 25%·Acc 41.0%
Google
Medium·1 companies·Max freq 25%·Acc 64.4%
Google
Hard·1 companies·Max freq 25%·Acc 41.5%
Google
Hard·1 companies·Max freq 25%·Acc 50.9%
Google
Medium·1 companies·Max freq 25%·Acc 60.5%
Google
Medium·1 companies·Max freq 25%·Acc 55.9%
Google
Medium·1 companies·Max freq 25%·Acc 44.2%
Google
Medium·1 companies·Max freq 25%·Acc 76.2%
Google
Medium·1 companies·Max freq 25%·Acc 67.4%
Google
Medium·1 companies·Max freq 25%·Acc 33.2%
Google
Hard·1 companies·Max freq 25%·Acc 27.9%
Google
Medium·1 companies·Max freq 18%·Acc 27.4%
Google
Hard·1 companies·Max freq 15%·Acc 26.0%
Google
Hard·1 companies·Max freq 13%·Acc 31.2%
Meta
Medium·1 companies·Max freq 13%·Acc 35.3%
Meta
Medium·1 companies·Max freq 13%·Acc 59.3%
Amazon
Medium·1 companies·Max freq 13%·Acc 26.9%
Bloomberg
Hard·1 companies·Max freq 13%·Acc 46.6%
Bloomberg
Hard·1 companies·Max freq 13%·Acc 26.2%
Google
Hard·1 companies·Max freq 13%·Acc 35.3%
Google
Hard·1 companies·Max freq 13%·Acc 17.9%
Google
Hard·1 companies·Max freq 13%·Acc 34.2%
Google
Medium·1 companies·Max freq 13%·Acc 41.6%
Google
Medium·1 companies·Max freq 13%·Acc 39.9%
Google
Hard·1 companies·Max freq 13%·Acc 22.5%
Google
Easy·1 companies·Max freq 13%·Acc 87.5%
Google
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