Alpha centrality

In software, find first set (ffs) or find first one is a bit operation that, given an unsigned machine word, identifies the least significant index or position of the bits set to one in the word. A nearly equivalent operation is count trailing zeros (ctz) or number of trailing zeros (ntz), which counts the number of zero bits following the least significant one bit. The complementary operation that finds the index or position of the most significant set bit is log base 2, so called because it computes the binary logarithm $\lfloor \log _{2}x\rfloor$. This is closely related to count leading zeros (clz) or number of leading zeros (nlz), which counts the number of zero bits preceding the most significant one bit. These four operations also have negated versions:

• find first zero (ffz), which identifies the index of the least significant zero bit;
• count trailing ones, which counts the number of one bits following the least significant zero bit.
• count leading ones, which counts the number of one bits preceding the most significant zero bit;
• The operation that finds the index of the most significant zero bit, which does not have a common name.

There are two common variants of find first set, the POSIX definition which starts indexing of bits at 1, herein labelled ffs, and the variant which starts indexing of bits at zero, which is equivalent to ctz and so will be called by that name.

Examples

Given the following 32-bit word:

00000000000000001000000000001000

The count trailing zeros operation would return 3, while the count leading zeros operation returns 16. The count leading zeros operation depends on the word size: if this 32-bit word were truncated to a 16-bit word, count leading zeros would return zero. The find first set operation would return 4, indicating the 4th position from the right. The log base 2 is 15.

Similarly, given the following 32-bit word, the bitwise negation of the above word:

11111111111111110111111111110111

The count trailing ones operation would return 3, the count leading ones operation would return 16, and the find first zero operation ffz would return 4.

If the word is zero (no bits set), count leading zeros and count trailing zeros both return the number of bits in the word, while ffs returns zero. Both log base 2 and zero-based implementations of find first set generally return an undefined result for the zero word.

Hardware support

Many architectures include instructions to rapidly perform find first set and/or related operations, listed below. The most common operation is count leading zeros (clz), likely because all other operations can be implemented efficiently in terms of it (see Properties and relations).

Platform	Mnemonic	Name	Word sizes	Description	Result on zero input
Intel 386 and later	bsf	Bit Scan Forward	16, 32, 64	ctz	Undefined, sets zero flag
Intel 386 and later	bsr	Bit Scan Reverse	16, 32, 64	log base 2	Undefined, sets zero flag
x86 supporting ABM	lzcnt	Count Leading Zeros	16, 32, 64	clz	input size, sets carry flag
x86 supporting BMI1	tzcnt	Count Trailing Zeros	16, 32, 64	ctz	input size, sets carry flag
Itanium	clz	Count Leading Zeros	64	clz	64
ARM 5 or later	clz	Count Leading Zeros	32	clz	32
POWER/PowerPC/Power Architecture	cntlz/cntlzw/cntlzd	Count Leading Zeros	32, 64	clz	input size
Oracle SPARC Architecture 2011 and later	lzcnt (synonym: lzd)	Leading Zero Count	64	clz	64
MIPS	clz	Count Leading Zeros in Word	32, 64	clz	input size
MIPS	clo	Count Leading Ones in Word	32, 64	clo	input size
DEC Alpha	ctlz	Count Leading Zeros	64	clz	64
DEC Alpha	cttz	Count Trailing Zeros	64	ctz	64
Motorola 68020 and later	bfffo	Find First One in Bit Field	arbitrary	log base 2	field offset + field width

Notes: On some Alpha platforms CTLZ and CTTZ are emulated in software.

Tool and library support

A number of compiler and library vendors supply compiler intrinsics or library functions to perform find first set and/or related operations, which are frequently implemented in terms of the hardware instructions above:

Tool/library	Name	Type	Input type(s)	Notes	Result for zero input
POSIX.1 compliant libc 4.3BSD libc OS X 10.3 libc	ffs	Library function	int	Includes glibc. POSIX does not supply the complementary log base 2 / clz.	0
FreeBSD 5.3 libc OS X 10.4 libc	ffsl fls, flsl	Library function	int, long	fls ("find last set") computes (log base 2) + 1.	0
FreeBSD 7.1 libc	ffsll, flsll	Library function	long long		0
GCC 3.2.2	__builtin_ffs	Built-in functions	unsigned int, unsigned long, unsigned long long	Possibly available before 3.2.2.	0
	__builtin_ctz				undefined
	__builtin_clz				undefined
Visual Studio 2005	_BitScanForward _BitScanReverse	Compiler intrinsics	unsigned long, unsigned __int64	Separate return value to indicate zero input	0
Visual Studio 2008	__lzcnt	Compiler intrinsic	unsigned short, unsigned int, unsigned __int64	Relies on x64-only lzcnt instruction	Input size in bits
Intel C++ Compiler	_bit_scan_forward _bit_scan_reverse	Compiler intrinsics	int		undefined
NVIDIA CUDA	__clz	Functions	32-bit, 64-bit	Compiles to fewer instructions on the GeForce 400 Series	32
	__ffs				0
LLVM	llvm.ctlz.* llvm.cttz.*	Intrinsic	8, 16, 32, 64, 256	LLVM assembly language	Input size if arg 2 is 0, else undefined

Properties and relations

The count trailing zeros and find first set operations are related by ctz(x) = ffs(x) − 1 (except for the zero input). Given w bits per word, the log base 2 is easily computed from the clz and vice versa by lg(x) = w − 1 − clz(x).

As demonstrated in the example above, the find first zero, count leading ones, and count trailing ones operations can be implemented by negating the input and using find first set, count leading zeros, and count trailing zeros. The reverse is also true.

On platforms with an efficient log base 2 operation such as M68000, ctz can be computed by:

ctz(x) = lg(x & (−x))

where "&" denotes bitwise AND and "−x" denotes the negative of x treating x as a signed integer in twos complement arithmetic. The expression x & (−x) clears all but the least-significant 1 bit, so that the most- and least-significant 1 bit are the same.

On platforms with an efficient count leading zeros operation such as ARM and PowerPC, ffs can be computed by:

ffs(x) = w − clz(x & (−x)).

Conversely, clz can be computed using ctz by first rounding up to the nearest power of two using shifts and bitwise ORs, as in this 32-bit example (note that this example depends on ctz returning 32 for the zero input):

function clz(x):
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return 32 − ctz(x + 1)

On platforms with an efficient Hamming weight (population count) operation such as SPARC's POPC or Blackfin's ONES, ctz can be computed using the identity:

ctz(x) = pop((x & (−x)) − 1),

ffs can be computed using:

ffs(x) = pop(x ^ (~(−x)))

where "^" denotes bitwise xor, and clz can be computed by:

function clz(x):
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return 32 − pop(x)

The inverse problem (given i, produce an x such that ctz(x)=i) can be computed with a left-shift (1 << i).

Find first set and related operations can be extended to arbitrarily large bit arrays in a straightforward manner by starting at one end and proceeding until a word that is not all-zero (for ffs/ctz/clz) or not all-one (for ffz/clo/cto) is encountered. A tree data structure that recursively uses bitmaps to track which words are nonzero can accelerate this.

Algorithms

Where find first set or a related function is not available in hardware, it must be implemented in software. The simplest implementation of ffs uses a loop:

function ffs (x)
    if x = 0 return 0
    t ← 1
    r ← 1
    while (x & t) = 0
        t ← t << 1
        r ← r + 1
    return r

where "<<" denotes left-shift. Similar loops can be used to implement all the related operations. On modern architectures this loop is inefficient due to a large number of conditional branches. A lookup table can eliminate most of these:

table[0..2n-1] = ffs(i) for i in 0..2n-1
function ffs_table (x)
    if x = 0 return 0
    r ← 0
    loop
        if (x & (2n-1)) ≠ 0
            return r + table[x & (2n-1)]
        x ← x >> n
        r ← r + n

The parameter n is fixed (typically 8) and represents a time-space tradeoff. The loop may also be fully unrolled.

An algorithm for 32-bit ctz by Leiserson, Prokop, and Randall uses de Bruijn sequences to construct a minimal perfect hash function that eliminates all branches:

table[0..31] initialized by: for i from 0 to 31: table[ ( 0x077CB531 * ( 1 << i ) ) >> 27 ] ← i
function ctz_debruijn (x)
    return table[((x & (-x)) × 0x077CB531) >> 27]

The expression (x & (-x)) again isolates the least-significant 1 bit. There are then only 32 possible words, which the unsigned multiplication and shift hash to the correct position in the table. (Note: this algorithm does not handle the zero input.) A similar algorithm works for log base 2, but rather than isolate the most-significant bit, it rounds up to the nearest integer of the form 2ⁿ−1 using shifts and bitwise ORs:

table[0..31] = {0, 9, 1, 10, 13, 21, 2, 29, 11, 14, 16, 18, 22, 25, 3, 30,
                8, 12, 20, 28, 15, 17, 24, 7, 19, 27, 23, 6, 26, 5, 4, 31}
function lg_debruijn (x)
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return table[(x × 0x07C4ACDD) >> 27]

Both the count leading zeros and count trailing zeros operations admit binary search implementations which take a logarithmic number of operations and branches, as in these 32-bit versions:

function clz (x)
    if x = 0 return 32
    n ← 0
    if (x & 0xFFFF0000) = 0: n ← n + 16, x ← x << 16
    if (x & 0xFF000000) = 0: n ← n +  8, x ← x <<  8
    if (x & 0xF0000000) = 0: n ← n +  4, x ← x <<  4
    if (x & 0xC0000000) = 0: n ← n +  2, x ← x <<  2
    if (x & 0x80000000) = 0: n ← n +  1
    return n

function ctz (x)
    if x = 0 return 32
    n ← 0
    if (x & 0x0000FFFF) = 0: n ← n + 16, x ← x >> 16
    if (x & 0x000000FF) = 0: n ← n +  8, x ← x >>  8
    if (x & 0x0000000F) = 0: n ← n +  4, x ← x >>  4
    if (x & 0x00000003) = 0: n ← n +  2, x ← x >>  2
    if (x & 0x00000001) = 0: n ← n +  1
    return n

These can be assisted by a table as well, replacing the last four lines of each with a table lookup on the high/low byte.

Just as count leading zeros is useful for software floating point implementations, conversely, on platforms that provide hardware conversion of integers to floating point, the exponent field can be extracted and subtracted from a constant to compute the count of leading zeros. Corrections are needed to account for rounding errors.

Applications

The count leading zeros (clz) operation can be used to efficiently implement normalization, which encodes an integer as m × 2^e, where m has its most significant bit in a known position (such as the highest position). This can in turn be used to implement Newton-Raphson division, perform integer to floating point conversion in software, and other applications.

Count leading zeros (clz) can be used to compute the 32-bit predicate "x = y" (zero if true, one if false) via the identity clz(x − y) >> 5, where ">>" is unsigned right shift. It can be used to perform more sophisticated bit operations like finding the first string of n 1 bits. The expression 16 − clz(x − 1)/2 is an effective initial guess for computing the square root of a 32-bit integer using Newton's method. CLZ can efficiently implement null suppression, a fast data compression technique that encodes an integer as the number of leading zero bytes together with the nonzero bytes. It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers.

The log base 2 can be used to anticipate whether a multiplication will overflow, since $\lceil \log _{2}xy\rceil \leq \lceil \log _{2}x\rceil +\lceil \log _{2}y\rceil$.

Count leading zeros and count trailing zeros can be used together to implement Gosper's loop-detection algorithm, which can find the period of a function of finite range using limited resources.

A bottleneck in the binary GCD algorithm is a loop removing trailing zeros, which can be replaced by a count trailing zeros (ctz) followed by a shift. A similar loop appears in computations of the hailstone sequence.

A bit array can be used to implement a priority queue. In this context, find first set (ffs) is useful in implementing the "pop" or "pull highest priority element" operation efficiently. The Linux kernel real-time scheduler internally uses sched_find_first_bit() for this purpose.

The count trailing zeros operation gives a simple optimal solution to the Tower of Hanoi problem: the disks are numbered from zero, and at move k, disk number ctz(k) is moved the minimum possible distance to the right (circling back around to the left as needed). It can also generate a Gray code by taking an arbitrary word and flipping bit ctz(k) at step k.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Template:Cite web

External links

• Bit Twiddling Hacks, Sean Eron Anderson, Stanford University. Lists several efficient public domain C implementations for count trailing zeros and log base 2.
• Chess Programming Wiki: BitScan: A detailed explanation of a number of implementation methods for ffs (called LS1B) and log base 2 (called MS1B).