Leading Zero Count
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Leading Zero Count

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 bit 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 .[1] 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 is a rounded version of the binary logarithm.

There are two common variants of find first set, the POSIX definition which starts indexing of bits at 1,[2] 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 Operand widths Description Result of operating on 0
ARM (ARMv5T architecture and later)
except Cortex-M0/M0+/M1/M23
clz[3] Count Leading Zeros 32 clz 32
ARM (ARMv8-A architecture) clz Count Leading Zeros 32, 64 clz Operand width
AVR32 clz[4] Count Leading Zeros 32 clz 32
DEC Alpha ctlz[5] Count Leading Zeros 64 clz 64
cttz[5] Count Trailing Zeros 64 ctz 64
Intel 80386 and later bsf[6] Bit Scan Forward 16, 32, 64 ctz Undefined; sets zero flag
bsr[6] Bit Scan Reverse 16, 32, 64 Log base 2 Undefined; sets zero flag
x86 supporting ABM lzcnt[7] Count Leading Zeros 16, 32, 64 clz Operand width; sets carry flag
x86 supporting BMI1 tzcnt[8] Count Trailing Zeros 16, 32, 64 ctz Operand width; sets carry flag
Itanium clz[9] Count Leading Zeros 64 clz 64
MIPS clz[10][11] Count Leading Zeros in Word 32, 64 clz Operand width
clo[10][11] Count Leading Ones in Word 32, 64 clo Operand width
Motorola 68020 and later bfffo[12] Find First One in Bit Field Arbitrary Log base 2 Field offset + field width
PDP-10 jffo Jump if Find First One 36 ctz 0; no operation
POWER/PowerPC/Power ISA cntlz/cntlzw/cntlzd[13] Count Leading Zeros 32, 64 clz Operand width
Power ISA 3.0 and later cnttzw/cnttzd[14] Count Trailing Zeros 32, 64 ctz Operand width
RISC-V ("B" Extension) (draft) clz[15] Count Leading Zeros TBD clz Operand width
ctz[15] Count Trailing Zeros TBD ctz Operand width
SPARC Oracle Architecture 2011 and later lzcnt (synonym: lzd) [16] Leading Zero Count 64 clz 64
VAX ffs[17] Find First Set 0-32 ctz Operand width; sets zero flag
z/Architecture vclz[18] Vector Count Leading Zeroes 8, 16, 32, 64 clz Operand width
vctz[19] Vector Count Trailing Zeroes 8, 16, 32, 64 ctz Operand width

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[2][20]
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[21]
ffsl
fls
flsl
Library function int,
long
fls ("find last set") computes (log base 2) + 1. 0
FreeBSD 7.1 libc[22] ffsll
flsll
Library function long long 0
GCC 3.4.0[23][24]
Clang 5.x [25][26]
__builtin_ffs[l,ll,imax]
__builtin_clz[l,ll,imax]
__builtin_ctz[l,ll,imax]
Built-in functions unsigned int,
unsigned long,
unsigned long long,
uintmax_t
GCC documentation considers result undefined clz and ctz on 0. 0 (ffs)
Visual Studio 2005 _BitScanForward[27]
_BitScanReverse[28]
Compiler intrinsics unsigned long,
unsigned __int64
Separate return value to indicate zero input 0
Visual Studio 2008 __lzcnt[29] 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[30]
Compiler intrinsics int Undefined
NVIDIA CUDA[31] __clz Functions 32-bit, 64-bit Compiles to fewer instructions on the GeForce 400 Series 32
__ffs 0
LLVM llvm.ctlz.*
llvm.cttz.*[32]
Intrinsic 8, 16, 32, 64, 256 LLVM assembly language Input size if arg 2
is 0, else undefined
GHC 7.10 (base 4.8), in Data.Bits countLeadingZeros
countTrailingZeros
Library function FiniteBits b => b Haskell programming language Input size in bits

Properties and relations

If bits are labeled starting at 1 (which is the convention used in this article), then count trailing zeros and find first set operations are related by ctz(x) = ffs(x) - 1 (except when the input is zero). If bits are labeled starting at 0, then count trailing zeros and find first set are exactly equivalent operations.

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 two's complement of x. 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,[33] as in this 32-bit example (which depends on ctz returning 32 for the zero input):

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

On platforms with an efficient Hamming weight (population count) operation such as SPARC's POPC[34][35] or Blackfin's ONES,[36], there is:

ctz(x) = popcount((x & (-x)) - 1),[37][38]
ffs(x) = popcount(x ^ (~(-x)))[39]

where "^" denotes bitwise exclusive-or and "~" denotes bitwise negation.

Applying the same rounding-up technique, clz can be computed by:

function clz(x):
    for each y in {1, 2, 4, 8, 16}: x 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

FFS

Find first set can also be implemented in software. A simple loop implementation:

function ffs (x)
    if x = 0 return 0
    t while (x & t) = 0
        t 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 loop
        if (x & (2n-1)) ? 0
            return r + table[x & (2n-1)]
        x 

The parameter n is fixed (typically 8) and represents a time-space tradeoff. The loop may also be fully unrolled. Alternatively, one can use the de Bruijn hash function for CTZ and add one to the result.

CTZ

Count Trailing Zeros (ctz) counts the number of zero bits succeeding the least significant one bit. For example, the ctz of 0x00000F00 is 8, and the ctz of 0x80000000 is 31.

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.[40][41] This algorithm assumes that the result of the multiplication is truncated to 32 bit.

table[0..31] initialized by: for i from 0 to 31: table[ ( 0x077CB531 * ( 1 << i ) ) >> 27 ] 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. (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:[42]

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 return table[(x * 0x07C4ACDD) >> 27]

A binary search implementation which takes a logarithmic number of operations and branches, as in these 32-bit versions:[43][44] This algorithm can be assisted by a table as well, replacing the bottom three "if" statements with a 256 entry lookup table using the final byte as an index.

function ctz (x)
    if x = 0 return 32
    n if (x & 0x0000FFFF) = 0: n if (x & 0x000000FF) = 0: n if (x & 0x0000000F) = 0: n if (x & 0x00000003) = 0: n if (x & 0x00000001) = 0: n return n

CLZ

Count Leading Zeros (clz) counts the number of zero bits preceding the most significant one bit. For example, a 32-bit clz of 0x00000F00 is 20, and the clz of 0x00000001 is 31.

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.[43][45]

The non-optimized approach examines one bit at a time until a non-zero bit is found, as shown in this C language example, and slowest with an input value of 1 because of the many loops it has to perform to find it.

#include <stdint.h> /* This header defines uint32_t, uint16_t, uint8_t for the following examples */
unsigned int clz32a( uint32_t x ) /* 32-bit clz */
{
  unsigned int n;
  if (x == 0)
     n = 32;
  else
     for (n = 0; ((x & 0x80000000) == 0); n++, x <<= 1) ;
  return n;
}
unsigned int clz16a( uint16_t x ) /* 16-bit clz */
{
  unsigned int n;
  if (x == 0)
     n = 16;
  else
     for (n = 0; ((x & 0x8000) == 0); n++, x <<= 1) ;
  return n;
}
unsigned int clz8a( uint8_t x ) /* 8-bit clz */
{
  unsigned int n;
  if (x == 0)
     n = 8;
  else
     for (n = 0; ((x & 0x80) == 0); n++, x <<= 1) ;
  return n;
}

An evolution of the previous looping approach examines four bits at a time then using a lookup table for the final four bits, which is shown here with a 16 (24) entry lookup table. A faster looping approach would examine eight bits at a time with a 256 (28) entry table.

#include <stdint.h>
static const uint8_t clz_table_4bit[16] = { 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 };
unsigned int clz32b( uint32_t x ) /* 32-bit clz */
{
  unsigned int n;
  if (x == 0)
  {
    n = 32;
  }
  else
  {
    for (n = 0; ((x & 0xF0000000) == 0); n += 4, x <<= 4) ;
    n += (unsigned int)clz_table_4bit[x >> (32-4)];
  }
  return n;
}

Faster than the looping method is a binary search implementation which takes a logarithmic number of operations and branches, as in these 32-bit versions:[43][44] The initial check for zero was removed because this method doesn't have a loop.

#include <stdint.h>
unsigned int clz32c( uint32_t x ) /* 32-bit clz */
{
  unsigned int n = 0;
  if ((x & 0xFFFF0000) == 0) {n  = 16; x <<= 16;}
  if ((x & 0xFF000000) == 0) {n +=  8; x <<=  8;}
  if ((x & 0xF0000000) == 0) {n +=  4; x <<=  4;}
  if ((x & 0xC0000000) == 0) {n +=  2; x <<=  2;}
  if ((x & 0x80000000) == 0) {n +=  1;}
  return n;
}

The binary search algorithm can be assisted by a table as well, replacing the bottom two "if" statements with a 16 (24) entry lookup table using the final 4-bits as an index, which is shown here. An alternate approach replaces the bottom three "if" statements with a 256 (28) entry table using the final 8 bits as an index.

#include <stdint.h>
static const uint8_t clz_table_4bit[16] = { 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 };
unsigned int clz32d( uint32_t x ) /* 32-bit clz */
{
  unsigned int n = 0;
  if ((x & 0xFFFF0000) == 0) {n  = 16; x <<= 16;}
  if ((x & 0xFF000000) == 0) {n +=  8; x <<=  8;}
  if ((x & 0xF0000000) == 0) {n +=  4; x <<=  4;}
  n += (unsigned int)clz_table_4bit[x >> (32-4)];
  return n;
}

As with many algorithms, the fastest approach to simulate clz would be a precomputed lookup table. In two previous examples, clz_table_4bit[16] was a lookup table for all 16 (24) values of a 4-bit clz operation. For common byte-width increments, an 8-bit clz requires 256 (28) values, a 16-bit clz requires 64K (216) values, a 32-bit clz requires 4G (232) values. Though a 32-bit clz table is possible on some computers, wasting 4GB of memory for a table is not practical. The next step down requires 64KB, which may be a reasonable amount for some computers.

#include <stdint.h>
static uint8_t clz_table_16bit[65536]; /* This table MUST be computed, using clz16a above, before calling the following functions */
unsigned int clz32e( uint32_t x ) /* 32-bit clz */
{
  if ((x & 0xFFFF0000) == 0)
    return (unsigned int)clz_table_16bit[x] + 16;
  else
    return (unsigned int)clz_table_16bit[x >> 16];
}
unsigned int clz16e( uint16_t x ) /* 16-bit clz */
{
  return (unsigned int)clz_table_16bit[x];
}
unsigned int clz8e( uint8_t x ) /* 8-bit clz */
{
  return (unsigned int)clz_table_16bit[x] - 8;
}

Applications

The count leading zeros (clz) operation can be used to efficiently implement normalization, which encodes an integer as m × 2e, 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.[43][46]

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.[47] It can be used to perform more sophisticated bit operations like finding the first string of n 1 bits.[48] The expression 1 << (16 - clz(x - 1)/2) is an effective initial guess for computing the square root of a 32-bit integer using Newton's method.[49] 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.[50] It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers.[43]

The log base 2 can be used to anticipate whether a multiplication will overflow, since .[51]

Count leading zeros and count trailing zeros can be used together to implement Gosper's loop-detection algorithm,[52] which can find the period of a function of finite range using limited resources.[44]

The binary GCD algorithm spends many cycles removing trailing zeros; this 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.[53]

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.[44]

See also

References

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Further reading

External links


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