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# /usr/man/cat.3/Bit::Vector::String.3.Z

```

```

## NAME

```       Bit::Vector - Efficient bit vector, set of integers and "big int" math
library

```

## SYNOPSIS

```       OVERLOADED OPERATORS

MORE STRING IMPORT/EXPORT

See Bit::Vector::String(3).

CLASS METHODS

Version
\$version = Bit::Vector->Version();

Word_Bits
\$bits = Bit::Vector->Word_Bits();  #  bits in a machine word

Long_Bits
\$bits = Bit::Vector->Long_Bits();  #  bits in an unsigned long

new
\$vector = Bit::Vector->new(\$bits);  #  bit vector constructor

@veclist = Bit::Vector->new(\$bits,\$count);

new_Hex
\$vector = Bit::Vector->new_Hex(\$bits,\$string);

new_Bin
\$vector = Bit::Vector->new_Bin(\$bits,\$string);

new_Dec
\$vector = Bit::Vector->new_Dec(\$bits,\$string);

new_Enum
\$vector = Bit::Vector->new_Enum(\$bits,\$string);

Concat_List
\$vector = Bit::Vector->Concat_List(@vectors);

OBJECT METHODS

new
\$vec2 = \$vec1->new(\$bits);  #  alternative call of constructor

@veclist = \$vec->new(\$bits,\$count);

\$vec2 = \$vec1->Shadow();  #  new vector, same size but empty

Clone
\$vec2 = \$vec1->Clone();  #  new vector, exact duplicate

Concat
\$vector = \$vec1->Concat(\$vec2);

Concat_List
\$vector = \$vec1->Concat_List(\$vec2,\$vec3,...);

Size
\$bits = \$vector->Size();

Resize
\$vector->Resize(\$bits);
\$vector->Resize(\$vector->Size()+5);
\$vector->Resize(\$vector->Size()-5);

Copy
\$vec2->Copy(\$vec1);

Empty
\$vector->Empty();

Fill
\$vector->Fill();

Flip
\$vector->Flip();

Primes
\$vector->Primes();  #  Sieve of Erathostenes

Reverse
\$vec2->Reverse(\$vec1);

Interval_Empty
\$vector->Interval_Empty(\$min,\$max);

Interval_Fill
\$vector->Interval_Fill(\$min,\$max);

Interval_Flip
\$vector->Interval_Flip(\$min,\$max);

Interval_Reverse
\$vector->Interval_Reverse(\$min,\$max);

Interval_Scan_inc
if ((\$min,\$max) = \$vector->Interval_Scan_inc(\$start))

Interval_Scan_dec
if ((\$min,\$max) = \$vector->Interval_Scan_dec(\$start))

Interval_Copy
\$vec2->Interval_Copy(\$vec1,\$offset2,\$offset1,\$length);

Interval_Substitute
\$vec2->Interval_Substitute(\$vec1,\$off2,\$len2,\$off1,\$len1);

is_empty
if (\$vector->is_empty())

is_full
if (\$vector->is_full())

equal
if (\$vec1->equal(\$vec2))

Lexicompare (unsigned)
if (\$vec1->Lexicompare(\$vec2) == 0)
if (\$vec1->Lexicompare(\$vec2) != 0)
if (\$vec1->Lexicompare(\$vec2) <  0)
if (\$vec1->Lexicompare(\$vec2) <= 0)
if (\$vec1->Lexicompare(\$vec2) >  0)
if (\$vec1->Lexicompare(\$vec2) >= 0)

Compare (signed)
if (\$vec1->Compare(\$vec2) == 0)
if (\$vec1->Compare(\$vec2) != 0)
if (\$vec1->Compare(\$vec2) <  0)
if (\$vec1->Compare(\$vec2) <= 0)
if (\$vec1->Compare(\$vec2) >  0)
if (\$vec1->Compare(\$vec2) >= 0)

to_Hex
\$string = \$vector->to_Hex();

from_Hex
\$vector->from_Hex(\$string);

to_Bin
\$string = \$vector->to_Bin();

from_Bin
\$vector->from_Bin(\$string);

to_Dec
\$string = \$vector->to_Dec();

from_Dec
\$vector->from_Dec(\$string);

to_Enum
\$string = \$vector->to_Enum();  #  e.g. "2,3,5-7,11,13-19"

from_Enum
\$vector->from_Enum(\$string);

Bit_Off
\$vector->Bit_Off(\$index);

Bit_On
\$vector->Bit_On(\$index);

bit_flip
\$bit = \$vector->bit_flip(\$index);

bit_test
contains
\$bit = \$vector->bit_test(\$index);
\$bit = \$vector->contains(\$index);
if (\$vector->bit_test(\$index))
if (\$vector->contains(\$index))

Bit_Copy
\$vector->Bit_Copy(\$index,\$bit);

LSB (least significant bit)
\$vector->LSB(\$bit);

MSB (most significant bit)
\$vector->MSB(\$bit);

lsb (least significant bit)
\$bit = \$vector->lsb();

msb (most significant bit)
\$bit = \$vector->msb();

rotate_left
\$carry = \$vector->rotate_left();

rotate_right
\$carry = \$vector->rotate_right();

shift_left
\$carry = \$vector->shift_left(\$carry);

shift_right
\$carry = \$vector->shift_right(\$carry);

Move_Left
\$vector->Move_Left(\$bits);  #  shift left "\$bits" positions

Move_Right
\$vector->Move_Right(\$bits);  #  shift right "\$bits" positions

Insert
\$vector->Insert(\$offset,\$bits);

Delete
\$vector->Delete(\$offset,\$bits);

increment
\$carry = \$vector->increment();

decrement
\$carry = \$vector->decrement();

inc
\$overflow = \$vec2->inc(\$vec1);

dec
\$overflow = \$vec2->dec(\$vec1);

subtract
\$carry = \$vec3->subtract(\$vec1,\$vec2,\$carry);
(\$carry,\$overflow) = \$vec3->subtract(\$vec1,\$vec2,\$carry);

Neg
Negate
\$vec2->Neg(\$vec1);
\$vec2->Negate(\$vec1);

Abs
Absolute
\$vec2->Abs(\$vec1);
\$vec2->Absolute(\$vec1);

Sign
if (\$vector->Sign() == 0)
if (\$vector->Sign() != 0)
if (\$vector->Sign() <  0)
if (\$vector->Sign() <= 0)
if (\$vector->Sign() >  0)
if (\$vector->Sign() >= 0)

Multiply
\$vec3->Multiply(\$vec1,\$vec2);

Divide
\$quot->Divide(\$vec1,\$vec2,\$rest);

GCD (Greatest Common Divisor)
\$vecgcd->GCD(\$veca,\$vecb);
\$vecgcd->GCD(\$vecx,\$vecy,\$veca,\$vecb);

Power
\$vec3->Power(\$vec1,\$vec2);

Block_Store
\$vector->Block_Store(\$buffer);

Word_Size
\$size = \$vector->Word_Size();  #  number of words in "\$vector"

Word_Store
\$vector->Word_Store(\$offset,\$word);

Word_List_Store
\$vector->Word_List_Store(@words);

Word_Insert
\$vector->Word_Insert(\$offset,\$count);

Word_Delete
\$vector->Word_Delete(\$offset,\$count);

Chunk_Store
\$vector->Chunk_Store(\$chunksize,\$offset,\$chunk);

Chunk_List_Store
\$vector->Chunk_List_Store(\$chunksize,@chunks);

Index_List_Remove
\$vector->Index_List_Remove(@indices);

Index_List_Store
\$vector->Index_List_Store(@indices);

Or
Union
\$vec3->Or(\$vec1,\$vec2);
\$set3->Union(\$set1,\$set2);

And
Intersection
\$vec3->And(\$vec1,\$vec2);
\$set3->Intersection(\$set1,\$set2);

AndNot
Difference
\$vec3->AndNot(\$vec1,\$vec2);
\$set3->Difference(\$set1,\$set2);

Xor
ExclusiveOr
\$vec3->Xor(\$vec1,\$vec2);
\$set3->ExclusiveOr(\$set1,\$set2);

Not
Complement
\$vec2->Not(\$vec1);
\$set2->Complement(\$set1);

subset
if (\$set1->subset(\$set2))  #  true if \$set1 is subset of \$set2

Norm
\$norm = \$set->Norm();
\$norm = \$set->Norm2();
\$norm = \$set->Norm3();

Min
\$min = \$set->Min();

Max
\$max = \$set->Max();

Multiplication
\$matrix3->Multiplication(\$rows3,\$cols3,
\$matrix1,\$rows1,\$cols1,
\$matrix2,\$rows2,\$cols2);

Product
\$matrix3->Product(\$rows3,\$cols3,
\$matrix1,\$rows1,\$cols1,
\$matrix2,\$rows2,\$cols2);

Closure
\$matrix->Closure(\$rows,\$cols);

Transpose
\$matrix2->Transpose(\$rows2,\$cols2,\$matrix1,\$rows1,\$cols1);

```

## IMPORTANT NOTES

```       o Method naming conventions

Method names completely in lower case indicate a boolean return
value.

(Except for the bit vector constructor method ""new()"", of course.)

o Boolean values

Boolean values in this module are always a numeric zero ("0") for
"false" and a numeric one ("1") for "true".

o Negative numbers

All numeric input parameters passed to any of the methods in this
module are regarded as being UNSIGNED (as opposed to the contents of
the bit vectors themselves, which are usually considered to be
SIGNED).

As a consequence, whenever you pass a negative number as an argument
to some method of this module, it will be treated as a (usually very
large) positive number due to its internal two's complement binary
representation, usually resulting in an "index out of range" error
message and program abortion.

o Bit order

Note that bit vectors are stored least order bit and least order word
first internally.

I.e., bit #0 of any given bit vector corresponds to bit #0 of word #0
in the array of machine words representing the bit vector.

(Where word #0 comes first in memory, i.e., it is stored at the least
memory address in the allocated block of memory holding the given bit
vector.)

Note however that machine words can be stored least order byte first
or last, depending on your system's implementation.

When you are exporting or importing a whole bit vector at once using
the methods ""Block_Read()"" and ""Block_Store()"" (the only time in
this module where this could make any difference), however, a conver-
sion to and from "least order byte first" order is automatically sup-
plied.

In other words, what ""Block_Read()"" provides and what
""Block_Store()"" expects is always in "least order byte first"
order, regardless of the order in which words are stored internally

This is to make sure that what you export on one machine using
""Block_Store()"" on a different machine.

Note further that whenever bit vectors are converted to and from
(binary or hexadecimal) strings, the RIGHTMOST bit is always the
LEAST SIGNIFICANT one, and the LEFTMOST bit is always the MOST SIG-
NIFICANT bit.

This is because in our western culture, numbers are always repre-
sented in this way (least significant to most significant digits go
from right to left).

Of course this requires an internal reversion of order, which the
corresponding conversion methods perform automatically (without any
at the bottom or the top end).

o "Word" related methods

Note that all methods whose names begin with ""Word_"" are MACHINE-
DEPENDENT!

They depend on the size (number of bits) of an "unsigned int" (C

Therefore, you should only use these methods if you are ABSOLUTELY
CERTAIN that portability of your code is not an issue!

Note that you can use arbitrarily large chunks (i.e., fragments of
bit vectors) of up to 32 bits IN A PORTABLE WAY using the methods
whose names begin with ""Chunk_"".

o Chunk sizes

Note that legal chunk sizes for all methods whose names begin with
""Chunk_"" range from "1" to ""Bit::Vector->Long_Bits();"" bits ("0"
is NOT allowed!).

In order to make your programs portable, however, you shouldn't use
chunk sizes larger than 32 bits, since this is the minimum size of an
"unsigned long" (C type) on all systems, as prescribed by ANSI C.

o Matching sizes

In general, for methods involving several bit vectors at the same
time, all bit vector arguments must have identical sizes (number of
bits), or a fatal "size mismatch" error will occur.

Exceptions from this rule are the methods ""Concat()"", ""Con-
cat_List()"", ""Copy()"", ""Interval_Copy()"" and ""Interval_Substi-
tute()"", where no conditions at all are imposed on the size of their
bit vector arguments.

In method ""Multiply()"", all three bit vector arguments must in
principle obey the rule of matching sizes, but the bit vector in
which the result of the multiplication is to be stored may be larger
than the two bit vector arguments containing the factors for the mul-
tiplication.

In method ""Power()"", the bit vector for the result must be the same
size or greater than the base of the exponentiation term. The expo-
nent can be any size.

o Index ranges

All indices for any given bits must lie between "0" and ""\$vec-
tor->Size()-1"", or a fatal "index out of range" error will occur.

```

## DESCRIPTION

```       OVERLOADED OPERATORS

MORE STRING IMPORT/EXPORT

See Bit::Vector::String(3).

CLASS METHODS

o "\$version = Bit::Vector->Version();"

Returns the current version number of this module.

o "\$bits = Bit::Vector->Word_Bits();"

Returns the number of bits of an "unsigned int" (C type) on your
machine.

(An "unsigned int" is also called a "machine word", hence the name of
this method.)

o "\$bits = Bit::Vector->Long_Bits();"

Returns the number of bits of an "unsigned long" (C type) on your
machine.

o "\$vector = Bit::Vector->new(\$bits);"

This is the bit vector constructor method.

Call this method to create a new bit vector containing "\$bits" bits
(with indices ranging from "0" to ""\$bits-1"").

Note that - in contrast to previous versions - bit vectors of length
zero (i.e., with "\$bits = 0") are permitted now.

The method returns a reference to the newly created bit vector.

A new bit vector is always initialized so that all bits are cleared
(turned off).

An exception will be raised if the method is unable to allocate the
necessary memory.

Note that if you specify a negative number for "\$bits" it will be
interpreted as a large positive number due to its internal two's com-
plement binary representation.

In such a case, the bit vector constructor method will obediently
attempt to create a bit vector of that size, probably resulting in an
exception, as explained above.

o "@veclist = Bit::Vector->new(\$bits,\$count);"

You can also create more than one bit vector at a time if you specify
the optional second parameter "\$count".

The method returns a list of "\$count" bit vectors which all have the
same number of bits "\$bits" (and which are all initialized, i.e., all
bits are cleared).

If "\$count" is zero, an empty list is returned.

If "\$bits" is zero, a list of null-sized bit vectors is returned.

Note again that if you specify a negative number for "\$count" it will
be interpreted as a large positive number due to its internal two's
complement binary representation.

In such a case, the bit vector constructor method will obediently
attempt to create that many bit vectors, probably resulting in an
exception ("out of memory").

o "\$vector = Bit::Vector->new_Hex(\$bits,\$string);"

This method is an alternative constructor which allows you to create
a new bit vector object (with "\$bits" bits) and to initialize it all
in one go.

The method internally first calls the bit vector constructor method
""new()"" and then passes the given string to the method
""from_Hex()"".

However, this method is more efficient than performing these two
steps separately: First because in this method, the memory area occu-
pied by the new bit vector is not initialized to zeros (which is
pointless in this case), and second because it saves you from the

An exception will be raised if the necessary memory cannot be allo-
cated (see the description of the method ""new()"" immediately above
for possible causes) or if the given string cannot be converted suc-
cessfully (see the description of the method ""from_Hex()"" further
below for details).

In the latter case, the memory occupied by the new bit vector is
released first (i.e., "free"d) before the exception is actually
raised.

o "\$vector = Bit::Vector->new_Bin(\$bits,\$string);"

This method is an alternative constructor which allows you to create
a new bit vector object (with "\$bits" bits) and to initialize it all
in one go.

The method internally first calls the bit vector constructor method
""new()"" and then passes the given string to the method
""from_Bin()"".

However, this method is more efficient than performing these two
steps separately: First because in this method, the memory area occu-
pied by the new bit vector is not initialized to zeros (which is
pointless in this case), and second because it saves you from the

An exception will be raised if the necessary memory cannot be allo-
cated (see the description of the method ""new()"" above for possible
causes) or if the given string cannot be converted successfully (see
the description of the method ""from_Bin()"" further below for
details).

In the latter case, the memory occupied by the new bit vector is
released first (i.e., "free"d) before the exception is actually
raised.

o "\$vector = Bit::Vector->new_Dec(\$bits,\$string);"

This method is an alternative constructor which allows you to create
a new bit vector object (with "\$bits" bits) and to initialize it all
in one go.

The method internally first calls the bit vector constructor method
""new()"" and then passes the given string to the method
""from_Dec()"".

However, this method is more efficient than performing these two
steps separately: First because in this method, ""new()"" does not
initialize the memory area occupied by the new bit vector with zeros
(which is pointless in this case, because ""from_Dec()"" will do it
anyway), and second because it saves you from the associated overhead

An exception will be raised if the necessary memory cannot be allo-
cated (see the description of the method ""new()"" above for possible
causes) or if the given string cannot be converted successfully (see
the description of the method ""from_Dec()"" further below for
details).

In the latter case, the memory occupied by the new bit vector is
released first (i.e., "free"d) before the exception is actually
raised.

o "\$vector = Bit::Vector->new_Enum(\$bits,\$string);"

This method is an alternative constructor which allows you to create
a new bit vector object (with "\$bits" bits) and to initialize it all
in one go.

The method internally first calls the bit vector constructor method
""new()"" and then passes the given string to the method
""from_Enum()"".

However, this method is more efficient than performing these two
steps separately: First because in this method, ""new()"" does not
initialize the memory area occupied by the new bit vector with zeros
(which is pointless in this case, because ""from_Enum()"" will do it
anyway), and second because it saves you from the associated overhead

An exception will be raised if the necessary memory cannot be allo-
cated (see the description of the method ""new()"" above for possible
causes) or if the given string cannot be converted successfully (see
the description of the method ""from_Enum()"" further below for
details).

In the latter case, the memory occupied by the new bit vector is
released first (i.e., "free"d) before the exception is actually
raised.

o "\$vector = Bit::Vector->Concat_List(@vectors);"

This method creates a new vector containing all bit vectors from the
argument list in concatenated form.

The argument list may contain any number of arguments (including
zero); the only condition is that all arguments must be bit vectors.

There is no condition concerning the length (in number of bits) of
these arguments.

The vectors from the argument list are not changed in any way.

If the argument list is empty or if all arguments have length zero,
the resulting bit vector will also have length zero.

Note that the RIGHTMOST bit vector from the argument list will become
the LEAST significant part of the resulting bit vector, and the LEFT-
MOST bit vector from the argument list will become the MOST signifi-
cant part of the resulting bit vector.

OBJECT METHODS

o "\$vec2 = \$vec1->new(\$bits);"

"@veclist = \$vec->new(\$bits);"

This is an alternative way of calling the bit vector constructor
method.

Vector "\$vec1" (or "\$vec") is not affected by this, it just serves as
an anchor for the method invocation mechanism.

In fact ALL class methods in this module can be called this way, even
though this is probably considered to be "politically incorrect" by

So even if you are too lazy to type ""Bit::Vector->"" instead of
""\$vec1->"" (and even though laziness is - allegedly - a programmer's
virtue ":-)"), maybe it is better not to use this feature if you
don't want to get booed at. ;-)

Creates a NEW bit vector "\$vec2" of the SAME SIZE as "\$vec1" but
which is EMPTY.

Just like a shadow that has the same shape as the object it origi-
nates from, but is flat and has no volume, i.e., contains nothing.

o "\$vec2 = \$vec1->Clone();"

Creates a NEW bit vector "\$vec2" of the SAME SIZE as "\$vec1" which is
an EXACT COPY of "\$vec1".

o "\$vector = \$vec1->Concat(\$vec2);"

This method returns a new bit vector object which is the result of
the concatenation of the contents of "\$vec1" and "\$vec2".

Note that the contents of "\$vec1" become the MOST significant part of
the resulting bit vector, and "\$vec2" the LEAST significant part.

If both bit vector arguments have length zero, the resulting bit vec-
tor will also have length zero.

o "\$vector = \$vec1->Concat_List(\$vec2,\$vec3,...);"

This is an alternative way of calling this (class) method as an
object method.

The method returns a new bit vector object which is the result of the
concatenation of the contents of "\$vec1 . \$vec2 . \$vec3 . ..."

See the section "class methods" above for a detailed description of
this method.

Note that the argument list may be empty and that all arguments must
be bit vectors if it isn't.

o "\$bits = \$vector->Size();"

Returns the size (number of bits) the given vector was created with
(or ""Resize()""d to).

o "\$vector->Resize(\$bits);"

Changes the size of the given vector to the specified number of bits.

This method allows you to change the size of an existing bit vector,
preserving as many bits from the old vector as will fit into the new
one (i.e., all bits with indices smaller than the minimum of the
sizes of both vectors, old and new).

If the number of machine words needed to store the new vector is
smaller than or equal to the number of words needed to store the old
vector, the memory allocated for the old vector is reused for the new
one, and only the relevant book-keeping information is adjusted
accordingly.

This means that even if the number of bits increases, new memory is
not necessarily being allocated (i.e., if the old and the new number
of bits fit into the same number of machine words).

If the number of machine words needed to store the new vector is
greater than the number of words needed to store the old vector, new
memory is allocated for the new vector, the old vector is copied to
the new one, the remaining bits in the new vector are cleared (turned
off) and the old vector is deleted, i.e., the memory that was allo-
cated for it is released.

(An exception will be raised if the method is unable to allocate the
necessary memory for the new vector.)

As a consequence, if you decrease the size of a given vector so that
it will use fewer machine words, and increase it again later so that
it will use more words than immediately before but still less than
the original vector, new memory will be allocated anyway because the
information about the size of the original vector is lost whenever
you resize it.

Note also that if you specify a negative number for "\$bits" it will
be interpreted as a large positive number due to its internal two's
complement binary representation.

In such a case, "Resize()" will obediently attempt to create a bit
vector of that size, probably resulting in an exception, as explained
above.

Finally, note that - in contrast to previous versions - resizing a
bit vector to a size of zero bits (length zero) is now permitted.

o "\$vec2->Copy(\$vec1);"

Copies the contents of bit vector "\$vec1" to bit vector "\$vec2".

The previous contents of bit vector "\$vec2" get overwritten, i.e.,
are lost.

Both vectors must exist beforehand, i.e., this method does not CREATE
any new bit vector object.

The two vectors may be of any size.

If the source bit vector is larger than the target, this method will
copy as much of the least significant bits of the source vector as
will fit into the target vector, thereby discarding any extraneous
most significant bits.

BEWARE that this causes a brutal cutoff in the middle of your data,
and it will also leave you with an almost unpredictable sign if sub-
sequently the number in the target vector is going to be interpreted
as a number! (You have been warned!)

If the target bit vector is larger than the source, this method fills
up the remaining most significant bits in the target bit vector with
either 0's or 1's, depending on the sign (= the most significant bit)
of the source bit vector. This is also known as "sign extension".

This makes it possible to copy numbers from a smaller bit vector into
a larger one while preserving the number's absolute value as well as
its sign (due to the two's complement binary representation of num-
bers).

o "\$vector->Empty();"

Clears all bits in the given vector.

o "\$vector->Fill();"

Sets all bits in the given vector.

o "\$vector->Flip();"

Flips (i.e., complements) all bits in the given vector.

o "\$vector->Primes();"

Clears the given bit vector and sets all bits whose indices are prime
numbers.

This method uses the algorithm known as the "Sieve of Erathostenes"
internally.

o "\$vec2->Reverse(\$vec1);"

This method copies the given vector "\$vec1" to the vector "\$vec2",
thereby reversing the order of all bits.

I.e., the least significant bit of "\$vec1" becomes the most signifi-
cant bit of "\$vec2", whereas the most significant bit of "\$vec1"
becomes the least significant bit of "\$vec2", and so forth for all
bits in between.

Note that in-place processing is also possible, i.e., "\$vec1" and
"\$vec2" may be identical.

(Internally, this is the same as "\$vec1->Inter-
val_Reverse(0,\$vec1->Size()-1);".)

o "\$vector->Interval_Empty(\$min,\$max);"

Clears all bits in the interval "[\$min..\$max]" (including both lim-
its) in the given vector.

"\$min" and "\$max" may have the same value; this is the same as clear-
ing a single bit with ""Bit_Off()"" (but less efficient).

Note that "\$vector->Interval_Empty(0,\$vector->Size()-1);" is the same
as "\$vector->Empty();" (but less efficient).

o "\$vector->Interval_Fill(\$min,\$max);"

Sets all bits in the interval "[\$min..\$max]" (including both limits)
in the given vector.

"\$min" and "\$max" may have the same value; this is the same as set-
ting a single bit with ""Bit_On()"" (but less efficient).

Note that "\$vector->Interval_Fill(0,\$vector->Size()-1);" is the same
as "\$vector->Fill();" (but less efficient).

o "\$vector->Interval_Flip(\$min,\$max);"

Flips (i.e., complements) all bits in the interval "[\$min..\$max]"
(including both limits) in the given vector.

"\$min" and "\$max" may have the same value; this is the same as flip-
ping a single bit with ""bit_flip()"" (but less efficient).

Note that "\$vector->Interval_Flip(0,\$vector->Size()-1);" is the same
as "\$vector->Flip();" and "\$vector->Complement(\$vector);" (but less
efficient).

o "\$vector->Interval_Reverse(\$min,\$max);"

Reverses the order of all bits in the interval "[\$min..\$max]"
(including both limits) in the given vector.

I.e., bits "\$min" and "\$max" swap places, and so forth for all bits
in between.

"\$min" and "\$max" may have the same value; this has no effect whatso-
ever, though.

o "if ((\$min,\$max) = \$vector->Interval_Scan_inc(\$start))"

Returns the minimum and maximum indices of the next contiguous block
of set bits (i.e., bits in the "on" state).

The search starts at index "\$start" (i.e., "\$min" >= "\$start") and
proceeds upwards (i.e., "\$max" >= "\$min"), thus repeatedly increments
the search pointer "\$start" (internally).

Note though that the contents of the variable (or scalar literal
value) "\$start" is NOT altered. I.e., you have to set it to the
desired value yourself prior to each call to ""Interval_Scan_inc()""

Actually, the bit vector is not searched bit by bit, but one machine
word at a time, in order to speed up execution (which means that this
method is quite efficient).

An empty list is returned if no such block can be found.

Note that a single set bit (surrounded by cleared bits) is a valid
block by this definition. In that case the return values for "\$min"
and "\$max" are the same.

Typical use:

\$start = 0;
while ((\$start < \$vector->Size()) &&
((\$min,\$max) = \$vector->Interval_Scan_inc(\$start)))
{
\$start = \$max + 2;

# do something with \$min and \$max
}

o "if ((\$min,\$max) = \$vector->Interval_Scan_dec(\$start))"

Returns the minimum and maximum indices of the next contiguous block
of set bits (i.e., bits in the "on" state).

The search starts at index "\$start" (i.e., "\$max" <= "\$start") and
proceeds downwards (i.e., "\$min" <= "\$max"), thus repeatedly decre-
ments the search pointer "\$start" (internally).

Note though that the contents of the variable (or scalar literal
value) "\$start" is NOT altered. I.e., you have to set it to the
desired value yourself prior to each call to ""Interval_Scan_dec()""

Actually, the bit vector is not searched bit by bit, but one machine
word at a time, in order to speed up execution (which means that this
method is quite efficient).

An empty list is returned if no such block can be found.

Note that a single set bit (surrounded by cleared bits) is a valid
block by this definition. In that case the return values for "\$min"
and "\$max" are the same.

Typical use:

\$start = \$vector->Size() - 1;
while ((\$start >= 0) &&
((\$min,\$max) = \$vector->Interval_Scan_dec(\$start)))
{
\$start = \$min - 2;

# do something with \$min and \$max
}

o "\$vec2->Interval_Copy(\$vec1,\$offset2,\$offset1,\$length);"

This method allows you to copy a stretch of contiguous bits (starting
at any position "\$offset1" you choose, with a length of "\$length"
bits) from a given "source" bit vector "\$vec1" to another position
"\$offset2" in a "target" bit vector "\$vec2".

Note that the two bit vectors "\$vec1" and "\$vec2" do NOT need to have
the same (matching) size!

Consequently, any of the two terms ""\$offset1 + \$length"" and ""\$off-
set2 + \$length"" (or both) may exceed the actual length of its corre-
sponding bit vector (""\$vec1->Size()"" and ""\$vec2->Size()"", respec-
tively).

In such a case, the "\$length" parameter is automatically reduced
internally so that both terms above are bounded by the number of bits
of their corresponding bit vector.

This may even result in a length of zero, in which case nothing is
copied at all.

(Of course the value of the "\$length" parameter, supplied by you in
the initial method call, may also be zero right from the start!)

Note also that "\$offset1" and "\$offset2" must lie within the range
"0" and, respectively, ""\$vec1->Size()-1"" or ""\$vec2->Size()-1"", or
a fatal "offset out of range" error will occur.

Note further that "\$vec1" and "\$vec2" may be identical, i.e., you may
copy a stretch of contiguous bits from one part of a given bit vector
to another part.

The source and the target interval may even overlap, in which case
the copying is automatically performed in ascending or descending
order (depending on the direction of the copy - downwards or upwards
in the bit vector, respectively) to handle this situation correctly,
i.e., so that no bits are being overwritten before they have been
copied themselves.

o "\$vec2->Interval_Substitute(\$vec1,\$off2,\$len2,\$off1,\$len1);"

This method is (roughly) the same for bit vectors (i.e., arrays of
booleans) as what the "splice" function in Perl is for lists (i.e.,
arrays of scalars).

The method allows you to substitute a stretch of contiguous bits
(defined by a position (offset) "\$off1" and a length of "\$len1" bits)
from a given "source" bit vector "\$vec1" for a different stretch of
contiguous bits (defined by a position (offset) "\$off2" and a length
of "\$len2" bits) in another, "target" bit vector "\$vec2".

Note that the two bit vectors "\$vec1" and "\$vec2" do NOT need to have
the same (matching) size!

Note further that "\$off1" and "\$off2" must lie within the range "0"
and, respectively, ""\$vec1->Size()"" or ""\$vec2->Size()"", or a fatal
"offset out of range" error will occur.

""\$vec1->Size()-1"" and ""\$vec2->Size()-1"", as they would be for any
other method in this module, but that these offsets may actually
point to one position PAST THE END of the corresponding bit vector.

This is necessary in order to make it possible to APPEND a given
stretch of bits to the target bit vector instead of REPLACING some-
thing in it.

For reasons of symmetry and generality, the same applies to the off-
set in the source bit vector, even though such an offset (one posi-
tion past the end of the bit vector) does not serve any practical
purpose there (but does not cause any harm either).

(Actually this saves you from the need of testing for this special
case, in certain circumstances.)

Note that whenever the term ""\$off1 + \$len1"" exceeds the size
""\$vec1->Size()"" of bit vector "\$vec1" (or if ""\$off2 + \$len2""
exceeds ""\$vec2->Size()""), the corresponding length ("\$len1" or
"\$len2", respectively) is automatically reduced internally so that
""\$off1 + \$len1 <= \$vec1->Size()"" (and ""\$off2 + \$len2 <=
\$vec2->Size()"") holds.

(Note that this does NOT alter the intended result, even though this
may seem counter-intuitive at first!)

This may even result in a length ("\$len1" or "\$len2") of zero.

A length of zero for the interval in the SOURCE bit vector (""\$len1
== 0"") means that the indicated stretch of bits in the target bit
vector (starting at position "\$off2") is to be replaced by NOTHING,
i.e., is to be DELETED.

A length of zero for the interval in the TARGET bit vector ("\$len2 ==
0") means that NOTHING is replaced, and that the stretch of bits from
the source bit vector is simply INSERTED into the target bit vector
at the indicated position ("\$off2").

If both length parameters are zero, nothing is done at all.

Note that in contrast to any other method in this module (especially
""Interval_Copy()"", ""Insert()"" and ""Delete()""), this method
IMPLICITLY and AUTOMATICALLY adapts the length of the resulting bit
vector as needed, as given by

\$size = \$vec2->Size();   #  before
\$size += \$len1 - \$len2;  #  after

(The only other method in this module that changes the size of a bit
vector is the method ""Resize()"".)

In other words, replacing a given interval of bits in the target bit
vector with a longer or shorter stretch of bits from the source bit
vector, or simply inserting (""\$len2 == 0"") a stretch of bits into
or deleting (""\$len1 == 0"") an interval of bits from the target bit
vector will automatically increase or decrease, respectively, the
size of the target bit vector accordingly.

For the sake of generality, this may even result in a bit vector with
a size of zero (containing no bits at all).

This is also the reason why bit vectors of length zero are permitted
in this module in the first place, starting with version 5.0.

Finally, note that "\$vec1" and "\$vec2" may be identical, i.e., in-
place processing is possible.

(If you think about that for a while or if you look at the code, you
will see that this is far from trivial!)

o "if (\$vector->is_empty())"

Tests whether the given bit vector is empty, i.e., whether ALL of its
bits are cleared (in the "off" state).

In "big integer" arithmetic, this is equivalent to testing whether
the number stored in the bit vector is zero ("0").

Returns "true" ("1") if the bit vector is empty and "false" ("0")
otherwise.

Note that this method also returns "true" ("1") if the given bit vec-
tor has a length of zero, i.e., if it contains no bits at all.

o "if (\$vector->is_full())"

Tests whether the given bit vector is full, i.e., whether ALL of its
bits are set (in the "on" state).

In "big integer" arithmetic, this is equivalent to testing whether
the number stored in the bit vector is minus one ("-1").

Returns "true" ("1") if the bit vector is full and "false" ("0") oth-
erwise.

If the given bit vector has a length of zero (i.e., if it contains no
bits at all), this method returns "false" ("0").

o "if (\$vec1->equal(\$vec2))"

Tests the two given bit vectors for equality.

Returns "true" ("1") if the two bit vectors are exact copies of one
another and "false" ("0") otherwise.

o "\$cmp = \$vec1->Lexicompare(\$vec2);"

Compares the two given bit vectors, which are regarded as UNSIGNED
numbers in binary representation.

The method returns ""-1"" if the first bit vector is smaller than the
second bit vector, "0" if the two bit vectors are exact copies of one
another and "1" if the first bit vector is greater than the second
bit vector.

o "\$cmp = \$vec1->Compare(\$vec2);"

Compares the two given bit vectors, which are regarded as SIGNED num-
bers in binary representation.

The method returns ""-1"" if the first bit vector is smaller than the
second bit vector, "0" if the two bit vectors are exact copies of one
another and "1" if the first bit vector is greater than the second
bit vector.

o "\$string = \$vector->to_Hex();"

Returns a hexadecimal string representing the given bit vector.

Note that this representation is quite compact, in that it only needs
at most twice the number of bytes needed to store the bit vector
itself, internally.

Note also that since a hexadecimal digit is always worth four bits,
the length of the resulting string is always a multiple of four bits,
regardless of the true length (in bits) of the given bit vector.

Finally, note that the LEAST significant hexadecimal digit is located
at the RIGHT end of the resulting string, and the MOST significant
digit at the LEFT end.

o "\$vector->from_Hex(\$string);"

Allows to read in the contents of a bit vector from a hexadecimal
string, such as returned by the method ""to_Hex()"" (see above).

Remember that the least significant bits are always to the right of a
hexadecimal string, and the most significant bits to the left. There-
fore, the string is actually read in from right to left while the bit
vector is filled accordingly, 4 bits at a time, starting with the
least significant bits and going upward to the most significant bits.

If the given string contains less hexadecimal digits than are needed
to completely fill the given bit vector, the remaining (most signifi-
cant) bits are all cleared.

This also means that, even if the given string does not contain
enough digits to completely fill the given bit vector, the previous
contents of the bit vector are erased completely.

If the given string is longer than it needs to fill the given bit
vector, the superfluous characters are simply ignored.

(In fact they are ignored completely - they are not even checked for

This behaviour is intentional so that you may read in the string rep-
resenting one bit vector into another bit vector of different size,
i.e., as much of it as will fit.

If during the process of reading the given string any character is
encountered which is not a hexadecimal digit, a fatal syntax error
ensues ("input string syntax error").

o "\$string = \$vector->to_Bin();"

Returns a binary string representing the given bit vector.

Example:

\$vector = Bit::Vector->new(8);
\$vector->Primes();
\$string = \$vector->to_Bin();
print "'\$string'\n";

This prints:

'10101100'

(Bits #7, #5, #3 and #2 are set.)

Note that the LEAST significant bit is located at the RIGHT end of
the resulting string, and the MOST significant bit at the LEFT end.

o "\$vector->from_Bin(\$string);"

This method allows you to read in the contents of a bit vector from a
binary string, such as returned by the method ""to_Bin()"" (see
above).

Note that this method assumes that the LEAST significant bit is
located at the RIGHT end of the binary string, and the MOST signifi-
cant bit at the LEFT end. Therefore, the string is actually read in
from right to left while the bit vector is filled accordingly, one
bit at a time, starting with the least significant bit and going
upward to the most significant bit.

If the given string contains less binary digits ("0" and "1") than
are needed to completely fill the given bit vector, the remaining
(most significant) bits are all cleared.

This also means that, even if the given string does not contain
enough digits to completely fill the given bit vector, the previous
contents of the bit vector are erased completely.

If the given string is longer than it needs to fill the given bit
vector, the superfluous characters are simply ignored.

(In fact they are ignored completely - they are not even checked for

This behaviour is intentional so that you may read in the string rep-
resenting one bit vector into another bit vector of different size,
i.e., as much of it as will fit.

If during the process of reading the given string any character is
encountered which is not either "0" or "1", a fatal syntax error
ensues ("input string syntax error").

o "\$string = \$vector->to_Dec();"

This method returns a string representing the contents of the given
bit vector converted to decimal (base 10).

Note that this method assumes the given bit vector to be SIGNED (and
to contain a number in two's complement binary representation).

Consequently, whenever the most significant bit of the given bit vec-
tor is set, the number stored in it is regarded as being NEGATIVE.

The resulting string can be fed into ""from_Dec()"" (see below) in
order to copy the contents of this bit vector to another one (or to
restore the contents of this one). This is not advisable, though,
since this would be very inefficient (there are much more efficient
methods for storing and copying bit vectors in this module).

Note that such conversion from binary to decimal is inherently slow
since the bit vector has to be repeatedly divided by 10 with remain-
der until the quotient becomes 0 (each remainder in turn represents a
single decimal digit of the resulting string).

This is also true for the implementation of this method in this mod-
ule, even though a considerable effort has been made to speed it up:
instead of repeatedly dividing by 10, the bit vector is repeatedly
divided by the largest power of 10 that will fit into a machine word.
The remainder is then repeatedly divided by 10 using only machine
word arithmetics, which is much faster than dividing the whole bit
vector ("divide and rule" principle).

According to my own measurements, this resulted in an 8-fold speed
increase over the straightforward approach.

Still, conversion to decimal should be used only where absolutely
necessary.

Keep the resulting string stored in some variable if you need it
again, instead of converting the bit vector all over again.

Beware that if you set the configuration for overloaded operators to
"output=decimal", this method will be called for every bit vector
enclosed in double quotes!

o "\$vector->from_Dec(\$string);"

This method allows you to convert a given decimal number, which may
be positive or negative, into two's complement binary representation,
which is then stored in the given bit vector.

The decimal number should always be provided as a string, to avoid
possible truncation (due to the limited precision of integers in
Perl) or formatting (due to Perl's use of scientific notation for
large numbers), which would lead to errors.

If the binary representation of the given decimal number is too big
to fit into the given bit vector (if the given bit vector does not
contain enough bits to hold it), a fatal "numeric overflow error"
occurs.

If the input string contains other characters than decimal digits
("0-9") and an optional leading sign (""+"" or ""-""), a fatal "input
string syntax error" occurs.

Beware that large positive numbers which cause the most significant
bit to be set (e.g. "255" in a bit vector with 8 bits) will be
printed as negative numbers when converted back to decimal using the
method "to_Dec()" (e.g.  "-1", in our example), because numbers with
the most significant bit set are considered to be negative in two's
complement binary representation.

Note also that while it is possible to thusly enter negative numbers
as large positive numbers (e.g. "255" for "-1" in a bit vector with 8
bits), the contrary isn't, i.e., you cannot enter "-255" for "+1", in
our example.  A fatal "numeric overflow error" will occur if you try
to do so.

If possible program abortion is unwanted or intolerable, use
""eval"", like this:

eval { \$vector->from_Dec("1152921504606846976"); };
if (\$@)
{
# an error occurred
}

There are four possible error messages:

if (\$@ =~ /item is not a string/)

if (\$@ =~ /input string syntax error/)

if (\$@ =~ /numeric overflow error/)

if (\$@ =~ /unable to allocate memory/)

Note that the conversion from decimal to binary is costly in terms of
processing time, since a lot of multiplications have to be carried
out (in principle, each decimal digit must be multiplied with the
binary representation of the power of 10 corresponding to its posi-
tion in the decimal number, i.e., 1, 10, 100, 1000, 10000 and so on).

This is not as time consuming as the opposite conversion, from binary
to decimal (where successive divisions have to be carried out, which
are even more expensive than multiplications), but still noticeable.

Again (as in the case of ""to_Dec()""), the implementation of this
method in this module uses the principle of "divide and rule" in
order to speed up the conversion, i.e., as many decimal digits as
possible are first accumulated (converted) in a machine word and only
then stored in the given bit vector.

Even so, use this method only where absolutely necessary if speed is
an important consideration in your application.

Beware that if you set the configuration for overloaded operators to
"input=decimal", this method will be called for every scalar operand
you use!

o "\$string = \$vector->to_Enum();"

Converts the given bit vector or set into an enumeration of single
indices and ranges of indices (".newsrc" style), representing the
bits that are set ("1") in the bit vector.

Example:

\$vector = Bit::Vector->new(20);
\$vector->Bit_On(2);
\$vector->Bit_On(3);
\$vector->Bit_On(11);
\$vector->Interval_Fill(5,7);
\$vector->Interval_Fill(13,19);
print "'", \$vector->to_Enum(), "'\n";

which prints

'2,3,5-7,11,13-19'

If the given bit vector is empty, the resulting string will also be
empty.

Note, by the way, that the above example can also be written a little
handier, perhaps, as follows:

Bit::Vector->Configuration("out=enum");
\$vector = Bit::Vector->new(20);
\$vector->Index_List_Store(2,3,5,6,7,11,13,14,15,16,17,18,19);
print "'\$vector'\n";

o "\$vector->from_Enum(\$string);"

This method first empties the given bit vector and then tries to set
the bits and ranges of bits specified in the given string.

The string "\$string" must only contain unsigned integers or ranges of
integers (two unsigned integers separated by a dash "-"), separated
by commas (",").

All other characters are disallowed (including white space!)  and
will lead to a fatal "input string syntax error".

In each range, the first integer (the lower limit of the range) must
always be less than or equal to the second integer (the upper limit),
or a fatal "minimum > maximum index" error occurs.

All integers must lie in the permitted range for the given bit vec-
tor, i.e., they must lie between "0" and ""\$vector->Size()-1"".

If this condition is not met, a fatal "index out of range" error
occurs.

If possible program abortion is unwanted or intolerable, use
""eval"", like this:

eval { \$vector->from_Enum("2,3,5-7,11,13-19"); };
if (\$@)
{
# an error occurred
}

There are four possible error messages:

if (\$@ =~ /item is not a string/)

if (\$@ =~ /input string syntax error/)

if (\$@ =~ /index out of range/)

if (\$@ =~ /minimum > maximum index/)

Note that the order of the indices and ranges is irrelevant, i.e.,

eval { \$vector->from_Enum("11,5-7,3,13-19,2"); };

results in the same vector as in the example above.

Ranges and indices may also overlap.

This is because each (single) index in the string is passed to the
method ""Bit_On()"", internally, and each range to the method
""Interval_Fill()"".

This means that the resulting bit vector is just the union of all the
indices and ranges specified in the given string.

o "\$vector->Bit_Off(\$index);"

Clears the bit with index "\$index" in the given vector.

o "\$vector->Bit_On(\$index);"

Sets the bit with index "\$index" in the given vector.

o "\$vector->bit_flip(\$index)"

Flips (i.e., complements) the bit with index "\$index" in the given
vector.

Moreover, this method returns the NEW state of the bit in question,
i.e., it returns "0" if the bit is cleared or "1" if the bit is set
(AFTER flipping it).

o "if (\$vector->bit_test(\$index))"

"if (\$vector->contains(\$index))"

Returns the current state of the bit with index "\$index" in the given
vector, i.e., returns "0" if it is cleared (in the "off" state) or
"1" if it is set (in the "on" state).

o "\$vector->Bit_Copy(\$index,\$bit);"

Sets the bit with index "\$index" in the given vector either to "0" or
"1" depending on the boolean value "\$bit".

o "\$vector->LSB(\$bit);"

Allows you to set the least significant bit in the given bit vector
to the value given by the boolean parameter "\$bit".

This is a (faster) shortcut for ""\$vector->Bit_Copy(0,\$bit);"".

o "\$vector->MSB(\$bit);"

Allows you to set the most significant bit in the given bit vector to
the value given by the boolean parameter "\$bit".

This is a (faster) shortcut for ""\$vector->Bit_Copy(\$vec-
tor->Size()-1,\$bit);"".

o "\$bit = \$vector->lsb();"

Returns the least significant bit of the given bit vector.

This is a (faster) shortcut for ""\$bit = \$vector->bit_test(0);"".

o "\$bit = \$vector->msb();"

Returns the most significant bit of the given bit vector.

This is a (faster) shortcut for ""\$bit = \$vector->bit_test(\$vec-
tor->Size()-1);"".

o "\$carry_out = \$vector->rotate_left();"

carry             MSB           vector:           LSB
out:
+---+            +---+---+---+---     ---+---+---+---+
|   |  <---+---  |   |   |   |    ...    |   |   |   |  <---+
+---+      |     +---+---+---+---     ---+---+---+---+      |
|                                                |
+------------------------------------------------+

The least significant bit (LSB) is the bit with index "0", the most
significant bit (MSB) is the bit with index ""\$vector->Size()-1"".

o "\$carry_out = \$vector->rotate_right();"

MSB           vector:           LSB            carry
out:
+---+---+---+---     ---+---+---+---+           +---+
+--->  |   |   |   |    ...    |   |   |   |  ---+---> |   |
|      +---+---+---+---     ---+---+---+---+     |     +---+
|                                                |
+------------------------------------------------+

The least significant bit (LSB) is the bit with index "0", the most
significant bit (MSB) is the bit with index ""\$vector->Size()-1"".

o "\$carry_out = \$vector->shift_left(\$carry_in);"

carry         MSB           vector:           LSB         carry
out:                                                      in:
+---+        +---+---+---+---     ---+---+---+---+        +---+
|   |  <---  |   |   |   |    ...    |   |   |   |  <---  |   |
+---+        +---+---+---+---     ---+---+---+---+        +---+

The least significant bit (LSB) is the bit with index "0", the most
significant bit (MSB) is the bit with index ""\$vector->Size()-1"".

o "\$carry_out = \$vector->shift_right(\$carry_in);"

carry         MSB           vector:           LSB         carry
in:                                                       out:
+---+        +---+---+---+---     ---+---+---+---+        +---+
|   |  --->  |   |   |   |    ...    |   |   |   |  --->  |   |
+---+        +---+---+---+---     ---+---+---+---+        +---+

The least significant bit (LSB) is the bit with index "0", the most
significant bit (MSB) is the bit with index ""\$vector->Size()-1"".

o "\$vector->Move_Left(\$bits);"

Shifts the given bit vector left by "\$bits" bits, i.e., inserts
"\$bits" new bits at the lower end (least significant bit) of the bit
vector, moving all other bits up by "\$bits" places, thereby losing
the "\$bits" most significant bits.

The inserted new bits are all cleared (set to the "off" state).

This method does nothing if "\$bits" is equal to zero.

Beware that the whole bit vector is cleared WITHOUT WARNING if
"\$bits" is greater than or equal to the size of the given bit vector!

In fact this method is equivalent to

for ( \$i = 0; \$i < \$bits; \$i++ ) { \$vector->shift_left(0); }

except that it is much more efficient (for "\$bits" greater than or
equal to the number of bits in a machine word on your system) than
this straightforward approach.

o "\$vector->Move_Right(\$bits);"

Shifts the given bit vector right by "\$bits" bits, i.e., deletes the
"\$bits" least significant bits of the bit vector, moving all other
bits down by "\$bits" places, thereby creating "\$bits" new bits at the
upper end (most significant bit) of the bit vector.

These new bits are all cleared (set to the "off" state).

This method does nothing if "\$bits" is equal to zero.

Beware that the whole bit vector is cleared WITHOUT WARNING if
"\$bits" is greater than or equal to the size of the given bit vector!

In fact this method is equivalent to

for ( \$i = 0; \$i < \$bits; \$i++ ) { \$vector->shift_right(0); }

except that it is much more efficient (for "\$bits" greater than or
equal to the number of bits in a machine word on your system) than
this straightforward approach.

o "\$vector->Insert(\$offset,\$bits);"

This method inserts "\$bits" fresh new bits at position "\$offset" in
the given bit vector.

The "\$bits" most significant bits are lost, and all bits starting
with bit number "\$offset" up to and including bit number ""\$vec-
tor->Size()-\$bits-1"" are moved up by "\$bits" places.

The now vacant "\$bits" bits starting at bit number "\$offset" (up to
and including bit number ""\$offset+\$bits-1"") are then set to zero
(cleared).

Note that this method does NOT increase the size of the given bit
vector, i.e., the bit vector is NOT extended at its upper end to
"rescue" the "\$bits" uppermost (most significant) bits - instead,
these bits are lost forever.

If you don't want this to happen, you have to increase the size of
the given bit vector EXPLICITLY and BEFORE you perform the "Insert"
operation, with a statement such as the following:

\$vector->Resize(\$vector->Size() + \$bits);

Or use the method ""Interval_Substitute()"" instead of ""Insert()"",
which performs automatic growing and shrinking of its target bit vec-
tor.

Note also that "\$offset" must lie in the permitted range between "0"
and ""\$vector->Size()-1"", or a fatal "offset out of range" error
will occur.

If the term ""\$offset + \$bits"" exceeds ""\$vector->Size()-1"", all
the bits starting with bit number "\$offset" up to bit number ""\$vec-
tor->Size()-1"" are simply cleared.

o "\$vector->Delete(\$offset,\$bits);"

This method deletes, i.e., removes the bits starting at position
"\$offset" up to and including bit number ""\$offset+\$bits-1"" from the
given bit vector.

The remaining uppermost bits (starting at position ""\$offset+\$bits""
up to and including bit number ""\$vector->Size()-1"") are moved down
by "\$bits" places.

The now vacant uppermost (most significant) "\$bits" bits are then set
to zero (cleared).

Note that this method does NOT decrease the size of the given bit
vector, i.e., the bit vector is NOT clipped at its upper end to "get
rid of" the vacant "\$bits" uppermost bits.

If you don't want this, i.e., if you want the bit vector to shrink
accordingly, you have to do so EXPLICITLY and AFTER the "Delete"
operation, with a couple of statements such as these:

\$size = \$vector->Size();
if (\$bits > \$size) { \$bits = \$size; }
\$vector->Resize(\$size - \$bits);

Or use the method ""Interval_Substitute()"" instead of ""Delete()"",
which performs automatic growing and shrinking of its target bit vec-
tor.

Note also that "\$offset" must lie in the permitted range between "0"
and ""\$vector->Size()-1"", or a fatal "offset out of range" error
will occur.

If the term ""\$offset + \$bits"" exceeds ""\$vector->Size()-1"", all
the bits starting with bit number "\$offset" up to bit number ""\$vec-
tor->Size()-1"" are simply cleared.

o "\$carry = \$vector->increment();"

This method increments the given bit vector.

Note that this method regards bit vectors as being unsigned, i.e.,
the largest possible positive number is directly followed by the
smallest possible (or greatest possible, speaking in absolute terms)
negative number:

before:  2 ^ (b-1) - 1    (= "0111...1111")
after:   2 ^ (b-1)        (= "1000...0000")

where ""b"" is the number of bits of the given bit vector.

The method returns "false" ("0") in all cases except when a carry
over occurs (in which case it returns "true", i.e., "1"), which hap-
pens when the number "1111...1111" is incremented, which gives
"0000...0000" plus a carry over to the next higher (binary) digit.

This can be used for the terminating condition of a "while" loop, for
instance, in order to cycle through all possible values the bit vec-
tor can assume.

o "\$carry = \$vector->decrement();"

This method decrements the given bit vector.

Note that this method regards bit vectors as being unsigned, i.e.,
the smallest possible (or greatest possible, speaking in absolute
terms) negative number is directly followed by the largest possible
positive number:

before:  2 ^ (b-1)        (= "1000...0000")
after:   2 ^ (b-1) - 1    (= "0111...1111")

where ""b"" is the number of bits of the given bit vector.

The method returns "false" ("0") in all cases except when a carry
over occurs (in which case it returns "true", i.e., "1"), which hap-
pens when the number "0000...0000" is decremented, which gives
"1111...1111" minus a carry over to the next higher (binary) digit.

This can be used for the terminating condition of a "while" loop, for
instance, in order to cycle through all possible values the bit vec-
tor can assume.

o "\$overflow = \$vec2->inc(\$vec1);"

This method copies the contents of bit vector "\$vec1" to bit vector
"\$vec2" and increments the copy (not the original).

If by incrementing the number its sign becomes invalid, the return
value ("overflow" flag) will be true ("1"), or false ("0") if not.
(See the description of the method "add()" below for a more in-depth
explanation of what "overflow" means).

Note that in-place operation is also possible, i.e., "\$vec1" and
"\$vec2" may be identical.

o "\$overflow = \$vec2->dec(\$vec1);"

This method copies the contents of bit vector "\$vec1" to bit vector
"\$vec2" and decrements the copy (not the original).

If by decrementing the number its sign becomes invalid, the return
value ("overflow" flag) will be true ("1"), or false ("0") if not.
(See the description of the method "subtract()" below for a more in-
depth explanation of what "overflow" means).

Note that in-place operation is also possible, i.e., "\$vec1" and
"\$vec2" may be identical.

This method adds the two numbers contained in bit vector "\$vec1" and
"\$vec2" with carry "\$carry" and stores the result in bit vector
"\$vec3".

I.e.,
\$vec3 = \$vec1 + \$vec2 + \$carry

Note that the "\$carry" parameter is a boolean value, i.e., only its
least significant bit is taken into account. (Think of it as though
""\$carry &= 1;"" was always executed internally.)

In scalar context, the method returns a boolean value which indicates
if a carry over (to the next higher bit position) has occured. In
list context, the method returns the carry and the overflow flag (in
this order).

The overflow flag is true ("1") if the sign (i.e., the most signifi-
cant bit) of the result is wrong. This can happen when adding two
very large positive numbers or when adding two (by their absolute

The carry in- and output is needed mainly for cascading, i.e., to add
numbers that are fragmented into several pieces.

Example:

# initialize

for ( \$i = 0; \$i < \$n; \$i++ )
{
\$a[\$i] = Bit::Vector->new(\$bits);
\$b[\$i] = Bit::Vector->new(\$bits);
\$c[\$i] = Bit::Vector->new(\$bits);
}

# fill @a and @b

# \$a[  0 ] is low order part,
# \$a[\$n-1] is high order part,
# and same for @b

\$carry = 0;
for ( \$i = 0; \$i < \$n; \$i++ )
{
}

Note that it makes no difference to this method whether the numbers
in "\$vec1" and "\$vec2" are unsigned or signed (i.e., in two's comple-
ment binary representation).

Note however that the return value (carry flag) is not meaningful
when the numbers are SIGNED.

Moreover, when the numbers are signed, a special type of error can
occur which is commonly called an "overflow error".

An overflow error occurs when the sign of the result (its most sig-
nificant bit) is flipped (i.e., falsified) by a carry over from the
next-lower bit position ("MSB-1").

In fact matters are a bit more complicated than that: the overflow
flag is set to "true" whenever there is a carry over from bit posi-
tion MSB-1 to the most significant bit (MSB) but no carry over from
the MSB to the output carry flag, or vice-versa, i.e., when there is
no carry over from bit position MSB-1 to the most significant bit
(MSB) but a carry over to the output carry flag.

Thus the overflow flag is the result of an exclusive-or operation
between incoming and outgoing carry over at the most significant bit
position.

o "\$carry = \$vec3->subtract(\$vec1,\$vec2,\$carry);"

"(\$carry,\$overflow) = \$vec3->subtract(\$vec1,\$vec2,\$carry);"

This method subtracts the two numbers contained in bit vector "\$vec1"
and "\$vec2" with carry "\$carry" and stores the result in bit vector
"\$vec3".

I.e.,
\$vec3 = \$vec1 - \$vec2 - \$carry

Note that the "\$carry" parameter is a boolean value, i.e., only its
least significant bit is taken into account. (Think of it as though
""\$carry &= 1;"" was always executed internally.)

In scalar context, the method returns a boolean value which indicates
if a carry over (to the next higher bit position) has occured. In
list context, the method returns the carry and the overflow flag (in
this order).

The overflow flag is true ("1") if the sign (i.e., the most signifi-
cant bit) of the result is wrong. This can happen when subtracting a
very large negative number from a very large positive number or

The carry in- and output is needed mainly for cascading, i.e., to
subtract numbers that are fragmented into several pieces.

Example:

# initialize

for ( \$i = 0; \$i < \$n; \$i++ )
{
\$a[\$i] = Bit::Vector->new(\$bits);
\$b[\$i] = Bit::Vector->new(\$bits);
\$c[\$i] = Bit::Vector->new(\$bits);
}

# fill @a and @b

# \$a[  0 ] is low order part,
# \$a[\$n-1] is high order part,
# and same for @b

# subtract

\$carry = 0;
for ( \$i = 0; \$i < \$n; \$i++ )
{
\$carry = \$c[\$i]->subtract(\$a[\$i],\$b[\$i],\$carry);
}

Note that it makes no difference to this method whether the numbers
in "\$vec1" and "\$vec2" are unsigned or signed (i.e., in two's comple-
ment binary representation).

Note however that the return value (carry flag) is not meaningful
when the numbers are SIGNED.

Moreover, when the numbers are signed, a special type of error can
occur which is commonly called an "overflow error".

An overflow error occurs when the sign of the result (its most sig-
nificant bit) is flipped (i.e., falsified) by a carry over from the
next-lower bit position ("MSB-1").

In fact matters are a bit more complicated than that: the overflow
flag is set to "true" whenever there is a carry over from bit posi-
tion MSB-1 to the most significant bit (MSB) but no carry over from
the MSB to the output carry flag, or vice-versa, i.e., when there is
no carry over from bit position MSB-1 to the most significant bit
(MSB) but a carry over to the output carry flag.

Thus the overflow flag is the result of an exclusive-or operation
between incoming and outgoing carry over at the most significant bit
position.

o "\$vec2->Neg(\$vec1);"

"\$vec2->Negate(\$vec1);"

This method calculates the two's complement of the number in bit vec-
tor "\$vec1" and stores the result in bit vector "\$vec2".

Calculating the two's complement of a given number in binary repre-
sentation consists of inverting all bits and incrementing the result
by one.

This is the same as changing the sign of the given number from ""+""
to ""-"" or vice-versa. In other words, applying this method twice on
a given number yields the original number again.

Note that in-place processing is also possible, i.e., "\$vec1" and
"\$vec2" may be identical.

Most importantly, beware that this method produces a counter-intu-
itive result if the number contained in bit vector "\$vec1" is "2 ^
(n-1)" (i.e., "1000...0000"), where ""n"" is the number of bits the
given bit vector contains: The negated value of this number is the
number itself!

o "\$vec2->Abs(\$vec1);"

"\$vec2->Absolute(\$vec1);"

Depending on the sign (i.e., the most significant bit) of the number
in bit vector "\$vec1", the contents of bit vector "\$vec1" are copied
to bit vector "\$vec2" either with the method ""Copy()"" (if the num-
ber in bit vector "\$vec1" is positive), or with ""Negate()"" (if the
number in bit vector "\$vec1" is negative).

In other words, this method calculates the absolute value of the num-
ber in bit vector "\$vec1" and stores the result in bit vector
"\$vec2".

Note that in-place processing is also possible, i.e., "\$vec1" and
"\$vec2" may be identical.

Most importantly, beware that this method produces a counter-intu-
itive result if the number contained in bit vector "\$vec1" is "2 ^
(n-1)" (i.e., "1000...0000"), where ""n"" is the number of bits the
given bit vector contains: The absolute value of this number is the
number itself, even though this number is still negative by defini-
tion (the most significant bit is still set)!

o "\$sign = \$vector->Sign();"

This method returns "0" if all bits in the given bit vector are
cleared, i.e., if the given bit vector contains the number "0", or if
the given bit vector has a length of zero (contains no bits at all).

If not all bits are cleared, this method returns ""-1"" if the most
significant bit is set (i.e., if the bit vector contains a negative
number), or "1" otherwise (i.e., if the bit vector contains a posi-
tive number).

o "\$vec3->Multiply(\$vec1,\$vec2);"

This method multiplies the two numbers contained in bit vector
"\$vec1" and "\$vec2" and stores the result in bit vector "\$vec3".

Note that this method regards its arguments as SIGNED.

If you want to make sure that a large number can never be treated as
being negative by mistake, make your bit vectors at least one bit
longer than the largest number you wish to represent, right from the
start, or proceed as follows:

\$msb1 = \$vec1->msb();
\$msb2 = \$vec2->msb();
\$vec1->Resize(\$vec1->Size()+1);
\$vec2->Resize(\$vec2->Size()+1);
\$vec3->Resize(\$vec3->Size()+1);
\$vec1->MSB(\$msb1);
\$vec2->MSB(\$msb2);
\$vec3->Multiply(\$vec1,\$vec2);

Note also that all three bit vector arguments must in principle obey
the rule of matching sizes, but that the bit vector "\$vec3" may be
larger than the two factors "\$vec1" and "\$vec2".

In fact multiplying two binary numbers with ""n"" bits may yield a
result which is at most ""2n"" bits long.

Therefore, it is usually a good idea to let bit vector "\$vec3" have
twice the size of bit vector "\$vec1" and "\$vec2", unless you are
absolutely sure that the result will fit into a bit vector of the
same size as the two factors.

If you are wrong, a fatal "numeric overflow error" will occur.

Finally, note that in-place processing is possible, i.e., "\$vec3" may
be identical with "\$vec1" or "\$vec2", or both.

o "\$quot->Divide(\$vec1,\$vec2,\$rest);"

This method divides the two numbers contained in bit vector "\$vec1"
and "\$vec2" and stores the quotient in bit vector "\$quot" and the
remainder in bit vector "\$rest".

I.e.,
\$quot = \$vec1 / \$vec2;  #  div
\$rest = \$vec1 % \$vec2;  #  mod

Therefore, "\$quot" and "\$rest" must be two DISTINCT bit vectors, or a
fatal "result vector(s) must be distinct" error will occur.

Note also that a fatal "division by zero error" will occur if "\$vec2"
is equal to zero.

Note further that this method regards its arguments as SIGNED.

If you want to make sure that a large number can never be treated as
being negative by mistake, make your bit vectors at least one bit
longer than the largest number you wish to represent, right from the
start, or proceed as follows:

\$msb1 = \$vec1->msb();
\$msb2 = \$vec2->msb();
\$vec1->Resize(\$vec1->Size()+1);
\$vec2->Resize(\$vec2->Size()+1);
\$quot->Resize(\$quot->Size()+1);
\$rest->Resize(\$rest->Size()+1);
\$vec1->MSB(\$msb1);
\$vec2->MSB(\$msb2);
\$quot->Divide(\$vec1,\$vec2,\$rest);

Finally, note that in-place processing is possible, i.e., "\$quot" may
be identical with "\$vec1" or "\$vec2" or both, and "\$rest" may also be
identical with "\$vec1" or "\$vec2" or both, as long as "\$quot" and
"\$rest" are distinct. (!)

o "\$vecgcd->GCD(\$veca,\$vecb);"

This method calculates the "Greatest Common Divisor" of the two num-
bers contained in bit vector "\$veca" and "\$vecb" and stores the
result in bit vector "\$vecgcd".

The method uses Euklid's algorithm internally:

int GCD(int a, int b)
{
int t;

while (b != 0)
{
t = a % b; /* = remainder of (a div b) */
a = b;
b = t;
}
return(a);
}

Note that "GCD(z,0) == GCD(0,z) == z".

o "\$vecgcd->GCD(\$vecx,\$vecy,\$veca,\$vecb);"

This variant of the "GCD" method calculates the "Greatest Common
Divisor" of the two numbers contained in bit vector "\$veca" and
"\$vecb" and stores the result in bit vector "\$vecgcd".

Moreover, it determines the two factors which are necessary in order
to represent the greatest common divisor as a linear combination of
its two arguments, i.e., the two factors "x" and "y" so that
"GCD(a,b) == x * a + y * b", and stores them in bit vector "\$vecx"
and "\$vecy", respectively.

For example:

a = 2322
b =  654

GCD( 2322, 654 ) == 6

x =  20
y = -71

20 * 2322 - 71 * 654 == 6

explanation of how this extension of Euklid's algorithm works.

o "\$vec3->Power(\$vec1,\$vec2);"

This method calculates the exponentiation of base "\$vec1" elevated to
the "\$vec2" power, i.e., ""\$vec1 ** \$vec2"", and stores the result in
bit vector "\$vec3".

The method uses an efficient divide-and-conquer algorithm:

Suppose the exponent is (decimal) 13, for example. The binary repre-
sentation of this exponent is "1101".

This means we want to calculate

\$vec1 * \$vec1 * \$vec1 * \$vec1 * \$vec1 * \$vec1 * \$vec1 * \$vec1 *
\$vec1 * \$vec1 * \$vec1 * \$vec1 *
\$vec1

That is, "\$vec1" multiplied with itself 13 times. The grouping into
lines above is no coincidence. The first line comprises 8 factors,
the second contains 4, and the last line just one. This just happens
to be the binary representation of 13. ";-)"

We then calculate a series of squares (of squares of squares...) of
the base, i.e.,

\$power[0] = \$vec1;
\$power[1] = \$vec1 * \$vec1;
\$power[2] = \$power[1] * \$power[1];
\$power[3] = \$power[2] * \$power[2];
etc.

To calculate the power of our example, we simply initialize our
result with 1 and consecutively multiply it with the items of the
series of powers we just calculated, if the corresponding bit of the
binary representation of the exponent is set:

\$result = 1;
\$result *= \$power[0] if (\$vec2 & 1);
\$result *= \$power[1] if (\$vec2 & 2);
\$result *= \$power[2] if (\$vec2 & 4);
\$result *= \$power[3] if (\$vec2 & 8);
etc.

The bit vector "\$vec3" must be of the same size as the base "\$vec1"
or greater. "\$vec3" and "\$vec1" may be the same vector (i.e., in-
place calculation as in ""\$vec1 **= \$vec2;"" is possible), but
"\$vec3" and "\$vec2" must be distinct. Finally, the exponent "\$vec2"
must be positive. A fatal error occurs if any of these conditions is
not met.

o "\$vector->Block_Store(\$buffer);"

This method allows you to load the contents of a given bit vector in
one go.

This is useful when you store the contents of a bit vector in a file,
for instance (using method ""Block_Read()""), and when you want to
restore the previously saved bit vector.

For this, "\$buffer" MUST be a string (NO automatic conversion from
numeric to string is provided here as would normally in Perl!)  con-
taining the bit vector in "low order byte first" order.

If the given string is shorter than what is needed to completely fill
the given bit vector, the remaining (most significant) bytes of the
bit vector are filled with zeros, i.e., the previous contents of the
bit vector are always erased completely.

If the given string is longer than what is needed to completely fill
the given bit vector, the superfluous bytes are simply ignored.

See "sysread" in perlfunc for how to read in the contents of
"\$buffer" from a file prior to passing it to this method.

This method allows you to export the contents of a given bit vector
in one block.

This is useful when you want to save the contents of a bit vector for
later, for instance in a file.

The advantage of this method is that it allows you to do so in the
compactest possible format, in binary.

The method returns a Perl string which contains an exact copy of the
contents of the given bit vector in "low order byte first" order.

See "syswrite" in perlfunc for how to write the data from this string
to a file.

o "\$size = \$vector->Word_Size();"

Each bit vector is internally organized as an array of machine words.

The methods whose names begin with "Word_" allow you to access this
internal array of machine words.

Note that because the size of a machine word may vary from system to
system, these methods are inherently MACHINE-DEPENDENT!

Therefore, DO NOT USE these methods unless you are absolutely certain
that portability of your code is not an issue!

You have been warned!

To be machine-independent, use the methods whose names begin with
""Chunk_"" instead, with chunk sizes no greater than 32 bits.

The method ""Word_Size()"" returns the number of machine words that
the internal array of words of the given bit vector contains.

This is similar in function to the term ""scalar(@array)"" for a Perl
array.

o "\$vector->Word_Store(\$offset,\$word);"

This method allows you to store a given value "\$word" at a given
position "\$offset" in the internal array of words of the given bit
vector.

Note that "\$offset" must lie in the permitted range between "0" and
""\$vector->Word_Size()-1"", or a fatal "offset out of range" error
will occur.

This method is similar in function to the expression ""\$array[\$off-
set] = \$word;"" for a Perl array.

This method allows you to access the value of a given machine word at
position "\$offset" in the internal array of words of the given bit
vector.

Note that "\$offset" must lie in the permitted range between "0" and
""\$vector->Word_Size()-1"", or a fatal "offset out of range" error
will occur.

This method is similar in function to the expression ""\$word =
\$array[\$offset];"" for a Perl array.

o "\$vector->Word_List_Store(@words);"

This method allows you to store a list of values "@words" in the
internal array of machine words of the given bit vector.

Thereby the LEFTMOST value in the list ("\$words[0]") is stored in the
LEAST significant word of the internal array of words (the one with
offset "0"), the next value from the list ("\$words[1]") is stored in
the word with offset "1", and so on, as intuitively expected.

If the list "@words" contains fewer elements than the internal array
of words of the given bit vector contains machine words, the remain-
ing (most significant) words are filled with zeros.

If the list "@words" contains more elements than the internal array
of words of the given bit vector contains machine words, the super-
fluous values are simply ignored.

This method is comparable in function to the expression ""@array =
@words;"" for a Perl array.

This method allows you to retrieve the internal array of machine
words of the given bit vector all at once.

Thereby the LEFTMOST value in the returned list ("\$words[0]") is the
LEAST significant word from the given bit vector, and the RIGHTMOST
value in the returned list ("\$words[\$#words]") is the MOST signifi-
cant word of the given bit vector.

This method is similar in function to the expression ""@words =
@array;"" for a Perl array.

o "\$vector->Word_Insert(\$offset,\$count);"

This method inserts "\$count" empty new machine words at position
"\$offset" in the internal array of words of the given bit vector.

The "\$count" most significant words are lost, and all words starting
with word number "\$offset" up to and including word number ""\$vec-
tor->Word_Size()-\$count-1"" are moved up by "\$count" places.

The now vacant "\$count" words starting at word number "\$offset" (up
to and including word number ""\$offset+\$count-1"") are then set to
zero (cleared).

Note that this method does NOT increase the size of the given bit
vector, i.e., the bit vector is NOT extended at its upper end to
"rescue" the "\$count" uppermost (most significant) words - instead,
these words are lost forever.

If you don't want this to happen, you have to increase the size of
the given bit vector EXPLICITLY and BEFORE you perform the "Insert"
operation, with a statement such as the following:

\$vector->Resize(\$vector->Size() + \$count * Bit::Vector->Word_Bits());

Note also that "\$offset" must lie in the permitted range between "0"
and ""\$vector->Word_Size()-1"", or a fatal "offset out of range"
error will occur.

If the term ""\$offset + \$count"" exceeds ""\$vector->Word_Size()-1"",
all the words starting with word number "\$offset" up to word number
""\$vector->Word_Size()-1"" are simply cleared.

o "\$vector->Word_Delete(\$offset,\$count);"

This method deletes, i.e., removes the words starting at position
"\$offset" up to and including word number ""\$offset+\$count-1"" from
the internal array of machine words of the given bit vector.

The remaining uppermost words (starting at position ""\$off-
set+\$count"" up to and including word number ""\$vec-
tor->Word_Size()-1"") are moved down by "\$count" places.

The now vacant uppermost (most significant) "\$count" words are then
set to zero (cleared).

Note that this method does NOT decrease the size of the given bit
vector, i.e., the bit vector is NOT clipped at its upper end to "get
rid of" the vacant "\$count" uppermost words.

If you don't want this, i.e., if you want the bit vector to shrink
accordingly, you have to do so EXPLICITLY and AFTER the "Delete"
operation, with a couple of statements such as these:

\$bits = \$vector->Size();
\$count *= Bit::Vector->Word_Bits();
if (\$count > \$bits) { \$count = \$bits; }
\$vector->Resize(\$bits - \$count);

Note also that "\$offset" must lie in the permitted range between "0"
and ""\$vector->Word_Size()-1"", or a fatal "offset out of range"
error will occur.

If the term ""\$offset + \$count"" exceeds ""\$vector->Word_Size()-1"",
all the words starting with word number "\$offset" up to word number
""\$vector->Word_Size()-1"" are simply cleared.

o "\$vector->Chunk_Store(\$chunksize,\$offset,\$chunk);"

This method allows you to set more than one bit at a time with dif-
ferent values.

You can access chunks (i.e., ranges of contiguous bits) between one
and at most ""Bit::Vector->Long_Bits()"" bits wide.

In order to be portable, though, you should never use chunk sizes
larger than 32 bits.

If the given "\$chunksize" does not lie between "1" and ""Bit::Vec-
tor->Long_Bits()"", a fatal "chunk size out of range" error will
occur.

The method copies the "\$chunksize" least significant bits from the
value "\$chunk" to the given bit vector, starting at bit position
"\$offset" and proceeding upwards until bit number ""\$offset+\$chunk-
size-1"".

(I.e., bit number "0" of "\$chunk" becomes bit number "\$offset" in the
given bit vector, and bit number ""\$chunksize-1"" becomes bit number
""\$offset+\$chunksize-1"".)

If the term ""\$offset+\$chunksize-1"" exceeds ""\$vector->Size()-1"",
the corresponding superfluous (most significant) bits from "\$chunk"
are simply ignored.

Note that "\$offset" itself must lie in the permitted range between
"0" and ""\$vector->Size()-1"", or a fatal "offset out of range" error
will occur.

This method (as well as the other ""Chunk_"" methods) is useful, for
example, when you are reading in data in chunks of, say, 8 bits,
which you need to access later, say, using 16 bits at a time (like
audio CD wave files, for instance).

This method allows you to read the values of more than one bit at a
time.

You can read chunks (i.e., ranges of contiguous bits) between one and
at most ""Bit::Vector->Long_Bits()"" bits wide.

In order to be portable, though, you should never use chunk sizes
larger than 32 bits.

If the given "\$chunksize" does not lie between "1" and ""Bit::Vec-
tor->Long_Bits()"", a fatal "chunk size out of range" error will
occur.

The method returns the "\$chunksize" bits from the given bit vector
starting at bit position "\$offset" and proceeding upwards until bit
number ""\$offset+\$chunksize-1"".

(I.e., bit number "\$offset" of the given bit vector becomes bit num-
ber "0" of the returned value, and bit number ""\$offset+\$chunk-
size-1"" becomes bit number ""\$chunksize-1"".)

If the term ""\$offset+\$chunksize-1"" exceeds ""\$vector->Size()-1"",
the non-existent bits are simply not returned.

Note that "\$offset" itself must lie in the permitted range between
"0" and ""\$vector->Size()-1"", or a fatal "offset out of range" error
will occur.

o "\$vector->Chunk_List_Store(\$chunksize,@chunks);"

This method allows you to fill the given bit vector with a list of
data packets ("chunks") of any size ("\$chunksize") you like (within
certain limits).

In fact the given "\$chunksize" must lie in the range between "1" and
""Bit::Vector->Long_Bits()"", or a fatal "chunk size out of range"
error will occur.

In order to be portable, though, you should never use chunk sizes
larger than 32 bits.

The given bit vector is thereby filled in ascending order: The first
chunk from the list (i.e., "\$chunks[0]") fills the "\$chunksize" least
significant bits, the next chunk from the list ("\$chunks[1]") fills
the bits number "\$chunksize" to number ""2*\$chunksize-1"", the third
chunk ("\$chunks[2]") fills the bits number ""2*\$chunksize"", to num-
ber ""3*\$chunksize-1"", and so on.

If there a less chunks in the list than are needed to fill the entire
bit vector, the remaining (most significant) bits are cleared, i.e.,
the previous contents of the given bit vector are always erased com-
pletely.

If there are more chunks in the list than are needed to fill the
entire bit vector, and/or if a chunk extends beyond ""\$vec-
tor->Size()-1"" (which happens whenever ""\$vector->Size()"" is not a
multiple of "\$chunksize"), the superfluous chunks and/or bits are
simply ignored.

The method is useful, for example (and among many other applica-
tions), for the conversion of packet sizes in a data stream.

This method can also be used to store an octal string in a given bit
vector:

\$vector->Chunk_List_Store(3, split(//, reverse \$string));

Note however that unlike the conversion methods ""from_Hex()"",
""from_Bin()"", ""from_Dec()"" and ""from_Enum()"", this statement
does not include any syntax checking, i.e., it may fail silently,
without warning.

To perform syntax checking, add the following statements:

if (\$string =~ /^[0-7]+\$/)
{
# okay, go ahead with conversion as shown above
}
else
{
# error, string contains other than octal characters
}

Another application is to store a repetitive pattern in a given bit
vector:

\$length = 32;            # = length of \$pattern in bits
\$size = \$vector->Size();
\$factor = int(\$size / \$length);
if (\$size % \$length) { \$factor++; }
\$vector->Chunk_List_Store(\$length, (\$pattern) x \$factor);

This method allows you to access the contents of the given bit vector
in form of a list of data packets ("chunks") of a size ("\$chunksize")
of your choosing (within certain limits).

In fact the given "\$chunksize" must lie in the range between "1" and
""Bit::Vector->Long_Bits()"", or a fatal "chunk size out of range"
error will occur.

In order to be portable, though, you should never use chunk sizes
larger than 32 bits.

The given bit vector is thereby read in ascending order: The "\$chunk-
size" least significant bits (bits number "0" to ""\$chunksize-1"")
become the first chunk in the returned list (i.e., "\$chunks[0]"). The
bits number "\$chunksize" to ""2*\$chunksize-1"" become the next chunk
in the list ("\$chunks[1]"), and so on.

If ""\$vector->Size()"" is not a multiple of "\$chunksize", the last
chunk in the list will contain fewer bits than "\$chunksize".

BEWARE that for large bit vectors and/or small values of "\$chunk-
size", the number of returned list elements can be extremely large!
BE CAREFUL!

You could blow up your application with lack of memory (each list
element is a full-grown Perl scalar, internally, with an associated
able, more or less long-lasting "freeze" of your application!

Possible applications:

The method is especially useful in the conversion of packet sizes in
a data stream.

This method can also be used to convert a given bit vector to a
string of octal numbers:

o "\$vector->Index_List_Remove(@indices);"

This method allows you to specify a list of indices of bits which
should be turned off in the given bit vector.

In fact this method is a shortcut for

foreach \$index (@indices)
{
\$vector->Bit_Off(\$index);
}

In contrast to all other import methods in this module, this method
does NOT clear the given bit vector before processing its list of
arguments.

Instead, this method allows you to accumulate the results of various
consecutive calls.

(The same holds for the method ""Index_List_Store()"". As a conse-
quence, you can "wipe out" what you did using the method
""Index_List_Remove()"" by passing the identical argument list to the
method ""Index_List_Store()"".)

o "\$vector->Index_List_Store(@indices);"

This method allows you to specify a list of indices of bits which
should be turned on in the given bit vector.

In fact this method is a shortcut for

foreach \$index (@indices)
{
\$vector->Bit_On(\$index);
}

In contrast to all other import methods in this module, this method
does NOT clear the given bit vector before processing its list of
arguments.

Instead, this method allows you to accumulate the results of various
consecutive calls.

(The same holds for the method ""Index_List_Remove()"". As a conse-
quence, you can "wipe out" what you did using the method
""Index_List_Store()"" by passing the identical argument list to the
method ""Index_List_Remove()"".)

This method returns a list of Perl scalars.

The list contains one scalar for each set bit in the given bit vec-
tor.

BEWARE that for large bit vectors, this can result in a literally
overwhelming number of list elements! BE CAREFUL! You could run out
of memory or slow down your application considerably!

Each scalar contains the number of the index corresponding to the bit
in question.

These indices are always returned in ascending order.

If the given bit vector is empty (contains only cleared bits) or if
it has a length of zero (if it contains no bits at all), the method
returns an empty list.

This method can be useful, for instance, to obtain a list of prime
numbers:

\$limit = 1000; # or whatever
\$vector = Bit::Vector->new(\$limit+1);
\$vector->Primes();

o "\$vec3->Or(\$vec1,\$vec2);"

"\$set3->Union(\$set1,\$set2);"

This method calculates the union of "\$set1" and "\$set2" and stores
the result in "\$set3".

This is usually written as ""\$set3 = \$set1 u \$set2"" in set theory
(where "u" is the "cup" operator).

(On systems where the "cup" character is unavailable this operator is
often denoted by a plus sign "+".)

In-place calculation is also possible, i.e., "\$set3" may be identical
with "\$set1" or "\$set2" or both.

o "\$vec3->And(\$vec1,\$vec2);"

"\$set3->Intersection(\$set1,\$set2);"

This method calculates the intersection of "\$set1" and "\$set2" and
stores the result in "\$set3".

This is usually written as ""\$set3 = \$set1 n \$set2"" in set theory
(where "n" is the "cap" operator).

(On systems where the "cap" character is unavailable this operator is
often denoted by an asterisk "*".)

In-place calculation is also possible, i.e., "\$set3" may be identical
with "\$set1" or "\$set2" or both.

o "\$vec3->AndNot(\$vec1,\$vec2);"

"\$set3->Difference(\$set1,\$set2);"

This method calculates the difference of "\$set1" less "\$set2" and
stores the result in "\$set3".

This is usually written as ""\$set3 = \$set1 \ \$set2"" in set theory
(where "\" is the "less" operator).

In-place calculation is also possible, i.e., "\$set3" may be identical
with "\$set1" or "\$set2" or both.

o "\$vec3->Xor(\$vec1,\$vec2);"

"\$set3->ExclusiveOr(\$set1,\$set2);"

This method calculates the symmetric difference of "\$set1" and
"\$set2" and stores the result in "\$set3".

This can be written as ""\$set3 = (\$set1 u \$set2) \ (\$set1 n \$set2)""
in set theory (the union of the two sets less their intersection).

When sets are implemented as bit vectors then the above formula is
equivalent to the exclusive-or between corresponding bits of the two
bit vectors (hence the name of this method).

Note that this method is also much more efficient than evaluating the
above formula explicitly since it uses a built-in machine language
instruction internally.

In-place calculation is also possible, i.e., "\$set3" may be identical
with "\$set1" or "\$set2" or both.

o "\$vec2->Not(\$vec1);"

"\$set2->Complement(\$set1);"

This method calculates the complement of "\$set1" and stores the
result in "\$set2".

In "big integer" arithmetic, this is equivalent to calculating the
one's complement of the number stored in the bit vector "\$set1" in
binary representation.

In-place calculation is also possible, i.e., "\$set2" may be identical
with "\$set1".

o "if (\$set1->subset(\$set2))"

Returns "true" ("1") if "\$set1" is a subset of "\$set2" (i.e., com-
pletely contained in "\$set2") and "false" ("0") otherwise.

This means that any bit which is set ("1") in "\$set1" must also be
set in "\$set2", but "\$set2" may contain set bits which are not set in
"\$set1", in order for the condition of subset relationship to be true
between these two sets.

Note that by definition, if two sets are identical, they are also
subsets (and also supersets) of each other.

o "\$norm = \$set->Norm();"

Returns the norm (number of bits which are set) of the given vector.

This is equivalent to the number of elements contained in the given
set.

Uses a byte lookup table for calculating the number of set bits per
byte, and thus needs a time for evaluation (and a number of loops)
linearly proportional to the length of the given bit vector (in
bytes).

This should be the fastest algorithm on average.

o "\$norm = \$set->Norm2();"

Returns the norm (number of bits which are set) of the given vector.

This is equivalent to the number of elements contained in the given
set.

This does the same as the method ""Norm()"" above, only with a dif-
ferent algorithm:

This method counts the number of set and cleared bits at the same
time and will stop when either of them has been exhausted, thus need-
ing at most half as many loops per machine word as the total number
of bits in a machine word - in fact it will need a number of loops
equal to the minimum of the number of set bits and the number of
cleared bits.

This might be a faster algorithm than of the method ""Norm()"" above
on some systems, depending on the system's architecture and the com-
piler and optimisation used, for bit vectors with sparse set bits and
for bit vectors with sparse cleared bits (i.e., predominantly set
bits).

o "\$norm = \$set->Norm3();"

Returns the norm (number of bits which are set) of the given vector.

This is equivalent to the number of elements contained in the given
set.

This does the same as the two methods ""Norm()"" and ""Norm2()""
above, however with a different algorithm.

In fact this is the implementation of the method ""Norm()"" used in
previous versions of this module.

The method needs a number of loops per machine word equal to the num-
ber of set bits in that machine word.

Only for bit vectors with sparse set bits will this method be fast;
it will depend on a system's architecture and compiler whether the
method will be faster than any of the two methods above in such
cases.

On average however, this is probably the slowest method of the three.

o "\$min = \$set->Min();"

Returns the minimum of the given set, i.e., the minimum of all
indices of all set bits in the given bit vector "\$set".

If the set is empty (no set bits), plus infinity (represented by the
constant "MAX_LONG" on your system) is returned.

(This constant is usually 2 ^ (n-1) - 1, where ""n"" is the number of
bits of an unsigned long on your machine.)

o "\$max = \$set->Max();"

Returns the maximum of the given set, i.e., the maximum of all
indices of all set bits in the given bit vector "\$set".

If the set is empty (no set bits), minus infinity (represented by the
constant "MIN_LONG" on your system) is returned.

(This constant is usually -(2 ^ (n-1) - 1) or -(2 ^ (n-1)), where
""n"" is the number of bits of an unsigned long on your machine.)

o "\$m3->Multiplication(\$r3,\$c3,\$m1,\$r1,\$c1,\$m2,\$r2,\$c2);"

This method multiplies two boolean matrices (stored as bit vectors)
"\$m1" and "\$m2" and stores the result in matrix "\$m3".

The method uses the binary "xor" operation (""^"") as the boolean

An exception is raised if the product of the number of rows and col-
umns of any of the three matrices differs from the actual size of
their underlying bit vector.

An exception is also raised if the numbers of rows and columns of the
three matrices do not harmonize in the required manner:

rows3 == rows1
cols3 == cols2
cols1 == rows2

This method is used by the module "Math::MatrixBool".

See Math::MatrixBool(3) for details.

o "\$m3->Product(\$r3,\$c3,\$m1,\$r1,\$c1,\$m2,\$r2,\$c2);"

This method multiplies two boolean matrices (stored as bit vectors)
"\$m1" and "\$m2" and stores the result in matrix "\$m3".

This special method uses the binary "or" operation (""|"") as the

An exception is raised if the product of the number of rows and col-
umns of any of the three matrices differs from the actual size of
their underlying bit vector.

An exception is also raised if the numbers of rows and columns of the
three matrices do not harmonize in the required manner:

rows3 == rows1
cols3 == cols2
cols1 == rows2

This method is used by the module "Math::MatrixBool".

See Math::MatrixBool(3) for details.

o "\$matrix->Closure(\$rows,\$cols);"

This method calculates the reflexive transitive closure of the given
boolean matrix (stored as a bit vector) using Kleene's algoritm.

(See Math::Kleene(3) for a brief introduction into the theory behind
Kleene's algorithm.)

The reflexive transitive closure answers the question whether a path
exists between any two vertices of a graph whose edges are given as a
matrix:

If a (directed) edge exists going from vertex "i" to vertex "j", then
the element in the matrix with coordinates (i,j) is set to "1" (oth-
erwise it remains set to "0").

If the edges are undirected, the resulting matrix is symmetric, i.e.,
elements (i,j) and (j,i) always contain the same value.

The matrix representing the edges of the graph only answers the ques-
tion whether an EDGE exists between any two vertices of the graph or
not, whereas the reflexive transitive closure answers the question
whether a PATH (a series of adjacent edges) exists between any two
vertices of the graph!

Note that the contents of the given matrix are modified by this
method, so make a copy of the initial matrix in time if you are going
to need it again later.

An exception is raised if the given matrix is not quadratic, i.e., if
the number of rows and columns of the given matrix is not identical.

An exception is also raised if the product of the number of rows and
columns of the given matrix differs from the actual size of its
underlying bit vector.

This method is used by the module "Math::MatrixBool".

See Math::MatrixBool(3) for details.

o "\$matrix2->Transpose(\$rows2,\$cols2,\$matrix1,\$rows1,\$cols1);"

This method calculates the transpose of a boolean matrix "\$matrix1"
(stored as a bit vector) and stores the result in matrix "\$matrix2".

The transpose of a boolean matrix, representing the edges of a graph,
can be used for finding the strongly connected components of that
graph.

An exception is raised if the product of the number of rows and col-
umns of any of the two matrices differs from the actual size of its
underlying bit vector.

An exception is also raised if the following conditions are not met:

rows2 == cols1
cols2 == rows1

Note that in-place processing ("\$matrix1" and "\$matrix2" are identi-
cal) is only possible if the matrix is quadratic. Otherwise, a fatal
"matrix is not quadratic" error will occur.

This method is used by the module "Math::MatrixBool".

See Math::MatrixBool(3) for details.

```

```       Bit::Vector::Overload(3), Bit::Vector::String(3).

Set::IntRange(3), Math::MatrixBool(3), Math::MatrixReal(3),
DFA::Kleene(3), Math::Kleene(3), Graph::Kruskal(3).

```

## VERSION

```       This man page documents "Bit::Vector" version 6.4.

```

## AUTHOR

```         Steffen Beyer
mailto:sb@engelschall.com

```

```       Copyright (c) 1995 - 2004 by Steffen Beyer. All rights reserved.

```

```       This package is free software; you can redistribute it and/or modify it
under the same terms as Perl itself, i.e., under the terms of the

The C library at the core of this Perl module can additionally be
redistributed and/or modified under the terms of the "GNU Library Gen-

Please refer to the files "Artistic.txt", "GNU_GPL.txt" and
"GNU_LGPL.txt" in this distribution for details!

```

## DISCLAIMER

```       This package is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MER-
CHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

See the "GNU General Public License" for more details.

perl v5.8.8                       2004-10-03                         Vector(3)