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Count the number of set bits in a 32-bit integer

February 18, 2025

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🏷 Tags: Algorithm Binary Bit-Manipulation Hammingweight Iec10967
Count the number of set bits in a 32-bit integer

Effectively counting fit bits (bits with a worth of 1) successful a 32-spot integer is a cardinal cognition successful machine discipline, with functions ranging from debased-flat spot manipulation to analyzable algorithms. Knowing however to execute this cognition optimally tin importantly contact show, peculiarly successful situations involving ample datasets oregon existent-clip processing. This article explores assorted strategies for counting fit bits, ranging from elemental iterative strategies to precocious bitwise operations. We’ll delve into the logic down all attack, analyse their show traits, and supply applicable examples to exemplify their utilization. Mastering this accomplishment tin empower you to compose much businesslike and elegant codification.

The Naive Attack: Iterative Checking

The easiest methodology includes iterating done all spot of the integer and checking if it’s fit. This tin beryllium achieved utilizing a loop and bitwise AND operations. Piece casual to realize, this methodology isn’t the about businesslike, peculiarly for integers with less fit bits.

For case, see the integer 5 (binary 00000101). The iterative attack would cheque each 32 bits, equal although lone 2 are fit. This inefficiency turns into much pronounced once dealing with sparse bitsets.

Codification Illustration (Python):

def count_set_bits_iterative(n): number = zero piece n > zero: if n & 1: number += 1 n >>= 1 instrument number

The Brian Kernighan’s Algorithm: An Businesslike Iterative Resolution

Brian Kernighan’s algorithm gives a much businesslike iterative attack. It leverages the reflection that subtracting 1 from an integer flips the rightmost fit spot and each the bits to its correct. By repeatedly making use of this cognition and utilizing a bitwise AND with the first figure, the fit bits are counted effectively, iterating lone arsenic galore occasions arsenic location are fit bits. This importantly improves show, particularly for numbers with less fit bits.

For illustration, if n = 5 (0101), the algorithm performs the pursuing steps:

  1. n & (n-1) = 0101 & 0100 = 0100 (number = 1)
  2. n & (n-1) = 0100 & 0011 = 0000 (number = 2)

The algorithm terminates arsenic n turns into zero last 2 iterations.

The Lookup Array Technique: Buying and selling Abstraction for Clip

The lookup array methodology pre-calculates the figure of fit bits for smaller chunks of bits (e.g., four bits oregon eight bits) and shops them successful a array. The 32-spot integer is past divided into these smaller chunks, and the pre-calculated values are retrieved and summed. This attack trades representation abstraction for velocity, and it tin beryllium importantly quicker than iterative strategies, particularly once dealing with ample datasets oregon existent-clip processing necessities wherever representation is not a great constraint. This technique excels successful embedded programs oregon specialised hardware wherever representation limitations mightiness be.

The MIT Hack: Harnessing Bitwise Magic

The MIT Hack makes use of a order of intelligent bitwise operations to effectively number the fit bits. It progressively isolates and counts teams of fit bits, lowering the figure of operations required. This method is famed for its class and velocity, making it a fashionable prime successful show-captious functions. Although analyzable astatine archetypal glimpse, knowing the underlying logic reveals the brilliance of its plan. This is a much precocious method that leverages circumstantial properties of bitwise operations, and it tin beryllium precise businesslike for processors optimized for specified operations.

def count_set_bits_mit(n): n = (n & 0x55555555) + ((n >> 1) & 0x55555555) n = (n & 0x33333333) + ((n >> 2) & 0x33333333) n = (n & 0x0f0f0f0f) + ((n >> four) & 0x0f0f0f0f) n = (n & 0x00ff00ff) + ((n >> eight) & 0x00ff00ff) n = (n & 0x0000ffff) + ((n >> sixteen) & 0x0000ffff) instrument n

Selecting the Correct Attack

The champion attack relies upon connected the circumstantial exertion and constraints. Iterative strategies are elemental to instrumentality however mightiness beryllium little businesslike for numbers with less fit bits. Brian Kernighan’s algorithm presents a bully equilibrium betwixt simplicity and ratio. The lookup array methodology excels successful velocity however requires other representation. The MIT Hack affords distinctive velocity however tin beryllium much analyzable to realize. See the commercial-offs betwixt codification complexity, show, and representation utilization once making your action.

  • See the frequence of this cognition successful your codification. If it’s a center portion of your algorithm, optimizing it tin output important show positive aspects.
  • Profiling your codification tin aid find the existent show bottlenecks and usher your optimization efforts.

Additional exploration of bitwise operations tin beryllium recovered successful this Wikipedia article connected bitwise operations.

β€œSpot manipulation is similar magic. Erstwhile you realize the basal spells, you tin conjure astonishing feats of ratio.” - Nameless Elder Package Technologist

For applicable functions successful information compression, cheque retired this assets: Information Compression Methods.

Larn much astir spot manipulation.Infographic Placeholder: Ocular cooperation of bitwise operations and their contact connected fit spot counting.

  • HakerRank Challenges: Trial your spot manipulation expertise connected platforms similar HackerRank.
  • Research the fascinating planet of spot twiddling hacks for much precocious strategies.

Often Requested Questions

Q: What is the quickest manner to number fit bits?

A: The quickest technique relies upon connected the circumstantial circumstances. The MIT Hack and lookup array strategies are mostly thought of the quickest, however they travel with commercial-offs successful status of codification complexity and representation utilization.

By knowing these methods, you tin take the about due technique for your circumstantial wants and compose much businesslike codification. Dive deeper into the planet of spot manipulation and research the sources offered to heighten your knowing additional. This cognition volition undoubtedly be invaluable successful your programming travel, permitting you to trade much businesslike and optimized codification. Delve into precocious strategies and lend to the ongoing optimization of these cardinal operations. Research applicable libraries and frameworks that message optimized implementations of these methods for circumstantial programming languages.

Research associated matters specified arsenic Hamming Importance, bitwise algorithms, and information compression strategies to broaden your knowing of spot manipulation and its purposes. Stack Overflow discussions tin supply insightful assemblage views and additional studying alternatives.

Question & Answer :
eight bits representing the figure 7 expression similar this:

00000111

3 bits are fit.

What are the algorithms to find the figure of fit bits successful a 32-spot integer?

This is recognized arsenic the ‘Hamming Importance’, ‘popcount’ oregon ‘sideways summation’.

Any CPUs person a azygous constructed-successful education to bash it and others person parallel directions which enactment connected spot vectors. Directions similar x86’s popcnt (connected CPUs wherever it’s supported) volition about surely beryllium quickest for a azygous integer. Any another architectures whitethorn person a dilatory education carried out with a microcoded loop that assessments a spot per rhythm (quotation wanted - hardware popcount is usually accelerated if it exists astatine each.).

The ‘champion’ algorithm truly relies upon connected which CPU you are connected and what your utilization form is.

Your compiler whitethorn cognize however to bash thing that’s bully for the circumstantial CPU you’re compiling for, e.g. C++20 std::popcount(), oregon C++ std::bitset<32>::number(), arsenic a transportable manner to entree builtin / intrinsic features (seat different reply connected this motion). However your compiler’s prime of fallback for mark CPUs that don’t person hardware popcnt mightiness not beryllium optimum for your usage-lawsuit. Oregon your communication (e.g. C) mightiness not exposure immoderate transportable relation that may usage a CPU-circumstantial popcount once location is 1.

Transportable algorithms that don’t demand (oregon payment from) immoderate HW activity

A pre-populated array lookup methodology tin beryllium precise accelerated if your CPU has a ample cache and you are doing tons of these operations successful a choky loop. Nevertheless it tin endure due to the fact that of the disbursal of a ‘cache girl’, wherever the CPU has to fetch any of the array from chief representation. (Expression ahead all byte individually to support the array tiny.) If you privation popcount for a contiguous scope of numbers, lone the debased byte is altering for teams of 256 numbers, making this precise bully.

If you cognize that your bytes volition beryllium largely zero’s oregon largely 1’s past location are businesslike algorithms for these situations, e.g. clearing the lowest fit with a bithack successful a loop till it turns into zero.

I accept a precise bully broad intent algorithm is the pursuing, recognized arsenic ‘parallel’ oregon ‘adaptable-precision SWAR algorithm’. I person expressed this successful a C-similar pseudo communication, you whitethorn demand to set it to activity for a peculiar communication (e.g. utilizing uint32_t for C++ and >>> successful Java):

GCC10 and clang 10.zero tin acknowledge this form / idiom and compile it to a hardware popcnt oregon equal education once disposable, giving you the champion of some worlds. (Godbolt)

int numberOfSetBits(uint32_t i) { // Java: usage int, and usage >>> alternatively of >>. Oregon usage Integer.bitCount() // C oregon C++: usage uint32_t i = i - ((i >> 1) & 0x55555555); // adhd pairs of bits i = (i & 0x33333333) + ((i >> 2) & 0x33333333); // quads i = (i + (i >> four)) & 0x0F0F0F0F; // teams of eight i *= 0x01010101; // horizontal sum of bytes instrument i >> 24; // instrument conscionable that apical byte (last truncating to 32-spot equal once int is wider than uint32_t) }

For JavaScript: coerce to integer with |zero for show: alteration the archetypal formation to i = (i|zero) - ((i >> 1) & 0x55555555);

This has the champion worst-lawsuit behaviour of immoderate of the algorithms mentioned, truthful volition effectively woody with immoderate utilization form oregon values you propulsion astatine it. (Its show is not information-babelike connected average CPUs wherever each integer operations together with multiply are changeless-clip. It doesn’t acquire immoderate sooner with “elemental” inputs, however it’s inactive beautiful first rate.)

References:

However this SWAR bithack plant:

i = i - ((i >> 1) & 0x55555555);

The archetypal measure is an optimized interpretation of masking to isolate the unusual / equal bits, shifting to formation them ahead, and including. This efficaciously does sixteen abstracted additions successful 2-spot accumulators (SWAR = SIMD Inside A Registry). Similar (i & 0x55555555) + ((i>>1) & 0x55555555).

The adjacent measure takes the unusual/equal 8 of these 16x 2-spot accumulators and provides once more, producing 8x four-spot sums. The i - ... optimization isn’t imaginable this clip truthful it does conscionable disguise earlier / last shifting. Utilizing the aforesaid 0x33... changeless some instances alternatively of 0xccc... earlier shifting is a bully happening once compiling for ISAs that demand to concept 32-spot constants successful registers individually.

The last displacement-and-adhd measure of (i + (i >> four)) & 0x0F0F0F0F widens to 4x eight-spot accumulators. It masks last including alternatively of earlier, due to the fact that the most worth successful immoderate four-spot accumulator is four, if each four bits of the corresponding enter bits have been fit. four+four = eight which inactive matches successful four bits, truthful transportation betwixt nibble parts is intolerable successful i + (i >> four).

Truthful cold this is conscionable reasonably average SIMD utilizing SWAR methods with a fewer intelligent optimizations. Persevering with connected with the aforesaid form for 2 much steps tin widen to 2x sixteen-spot past 1x 32-spot counts. However location is a much businesslike manner connected machines with accelerated hardware multiply:

Erstwhile we person fewer adequate “components”, a multiply with a magic changeless tin sum each the parts into the apical component. Successful this lawsuit byte components. Multiply is performed by near-shifting and including, truthful a multiply of x * 0x01010101 outcomes successful x + (x<<eight) + (x<<sixteen) + (x<<24). Our eight-spot parts are broad adequate (and holding tiny adequate counts) that this doesn’t food transportation into that apical eight bits.

A sixty four-spot interpretation of this tin bash 8x eight-spot components successful a sixty four-spot integer with a 0x0101010101010101 multiplier, and extract the advanced byte with >>fifty six. Truthful it doesn’t return immoderate other steps, conscionable wider constants. This is what GCC makes use of for __builtin_popcountll connected x86 programs once the hardware popcnt education isn’t enabled. If you tin usage builtins oregon intrinsics for this, bash truthful to springiness the compiler a accidental to bash mark-circumstantial optimizations.

With afloat SIMD for wider vectors (e.g. counting a entire array)

This bitwise-SWAR algorithm might parallelize to beryllium executed successful aggregate vector components astatine erstwhile, alternatively of successful a azygous integer registry, for a speedup connected CPUs with SIMD however nary usable popcount education. (e.g. x86-sixty four codification that has to tally connected immoderate CPU, not conscionable Nehalem oregon future.)

Nevertheless, the champion manner to usage vector directions for popcount is normally by utilizing a adaptable-shuffle to bash a array-lookup for four bits astatine a clip of all byte successful parallel. (The four bits scale a sixteen introduction array held successful a vector registry).

Connected Intel CPUs, the hardware 64bit popcnt education tin outperform an SSSE3 PSHUFB spot-parallel implementation by astir a cause of 2, however lone if your compiler will get it conscionable correct. Other SSE tin travel retired importantly up. Newer compiler variations are alert of the popcnt mendacious dependency job connected Intel.

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