Set::IntSpan(3) Manages sets of integers

SYNOPSIS


# BEGIN { $Set::IntSpan::integer = 1 }
use Set::IntSpan qw(grep_set map_set grep_spans map_spans);
# $Set::IntSpan::Empty_String = '-'; # or '';
$set = new Set::IntSpan $set_spec;
$set = new Set::IntSpan @set_specs;
$valid = valid Set::IntSpan $run_list;
$set = copy $set $set_spec;
$run_list = run_list $set;
@elements = elements $set;
@sets = sets $set;
@spans = spans $set;
$u_set = union $set $set_spec;
$i_set = intersect $set $set_spec;
$x_set = xor $set $set_spec;
$d_set = diff $set $set_spec;
$c_set = complement $set;
$set->U($set_spec); # Union
$set->I($set_spec); # Intersect
$set->X($set_spec); # Xor
$set->D($set_spec); # Diff
$set->C; # Complement
equal $set $set_spec
equivalent $set $set_spec
superset $set $set_spec
subset $set $set_spec
$n = cardinality $set;
$n = size $set;
empty $set
finite $set
neg_inf $set
pos_inf $set
infinite $set
universal $set
member $set $n;
insert $set $n;
remove $set $n;
$min = min $set;
$max = max $set;
$holes = holes $set;
$cover = cover $set;
$inset = inset $set $n;
$smaller = trim $set $n;
$bigger = pad $set $n;
$subset = grep_set { ... } $set;
$mapset = map_set { ... } $set;
$subset = grep_spans { ... } $set;
$mapset = map_spans { ... } $set;
for ($element=$set->first; defined $element; $element=$set->next) { ... }
for ($element=$set->last ; defined $element; $element=$set->prev) { ... }
$element = $set->start($n);
$element = $set->current;
$n = $set->at($i);
$slice = $set->slice($from, $to);
$i = $set->ord($n);
$i = $set->span_ord($n);

Operator overloads

  $u_set =  $set + $set_spec;   # union
  $i_set =  $set * $set_spec;   # intersect
  $x_set =  $set ^ $set_spec;   # xor
  $d_set =  $set - $set_spec;   # diff
  $c_set = ~$set;               # complement
  $set += $set_spec;            # union
  $set *= $set_spec;            # intersect
  $set ^= $set_spec;            # xor
  $set -= $set_spec;            # diff
  $set eq $set_spec             # equal
  $set ne $set_spec             # not equal
  $set le $set_spec             # subset
  $set lt $set_spec             # proper subset
  $set ge $set_spec             # superset
  $set gt $set_spec             # proper superset
  # compare sets by cardinality
  $set1 ==  $set2
  $set1 !=  $set2
  $set1 <=  $set2
  $set1 <   $set2
  $set1 >=  $set2
  $set1 >   $set2
  $set1 <=> $set2
  # compare cardinality of set to an integer
  $set1 ==  $n
  $set1 !=  $n
  $set1 <=  $n
  $set1 <   $n
  $set1 >=  $n
  $set1 >   $n
  $set1 <=> $n
  @sorted = sort @sets;         # sort sets by cardinality
  if ($set) { ... }             # true if $set is not empty
  print "$set\n";               # stringizes to the run list

EXPORTS

@EXPORT

Nothing

@EXPORT_OK

"grep_set", "map_set", "grep_spans", "map_spans"

DESCRIPTION

"Set::IntSpan" manages sets of integers. It is optimized for sets that have long runs of consecutive integers. These arise, for example, in .newsrc files, which maintain lists of articles:

  alt.foo: 1-21,28,31
  alt.bar: 1-14192,14194,14196-14221

A run of consecutive integers is also called a span.

Sets are stored internally in a run-length coded form. This provides for both compact storage and efficient computation. In particular, set operations can be performed directly on the encoded representation.

"Set::IntSpan" is designed to manage finite sets. However, it can also represent some simple infinite sets, such as { x | x>n }. This allows operations involving complements to be carried out consistently, without having to worry about the actual value of INT_MAX on your machine.

SPANS

A span is a run of consecutive integers. A span may be represented by an array reference, in any of 5 forms:

Finite forms

    Span                Set
  [ $n,    $n    ]      { n }
  [ $a,    $b    ]      { x | a<=x && x<=b}

Infinite forms

    Span                Set
  [ undef, $b    ]      { x | x<=b }
  [ $a   , undef ]      { x | x>=a }
  [ undef, undef ]      The set of all integers

Some methods operate directly on spans.

SET SPECIFICATIONS

Many of the methods take a set specification. There are four kinds of set specifications.

Empty

If a set specification is omitted, then the empty set is assumed. Thus,

  $set = new Set::IntSpan;

creates a new, empty set. Similarly,

  copy $set;

removes all elements from $set.

Object reference

If an object reference is given, it is taken to be a "Set::IntSpan" object.

Run list

If a string is given, it is taken to be a run list. A run list specifies a set using a syntax similar to that in newsrc files.

A run list is a comma-separated list of runs. Each run specifies a set of consecutive integers. The set is the union of all the runs.

Runs may be written in any of 5 forms.

Finite forms

n
{ n }
a-b
{ x | a<=x && x<=b }

Infinite forms

(-n
{ x | x<=n }
n-)
{ x | x>=n }
(-)
The set of all integers

Empty forms

The empty set is consistently written as '' (the null string). It is also denoted by the special form '-' (a single dash).

Restrictions

The runs in a run list must be disjoint, and must be listed in increasing order.

Valid characters in a run list are 0-9, '(', ')', '-' and ','. White space and underscore (_) are ignored. Other characters are not allowed.

Examples

  Run list          Set
  "-"               { }
  "1"               { 1 }
  "1-2"             { 1, 2 }
  "-5--1"           { -5, -4, -3, -2, -1 }
  "(-)"             the integers
  "(--1"            the negative integers
  "1-3, 4, 18-21"   { 1, 2, 3, 4, 18, 19, 20, 21 }

Array reference

If an array reference is given, then the elements of the array specify the elements of the set. The array may contain
  • integers
  • spans

The set is the union of all the integers and spans in the array. The integers and spans need not be disjoint. The integers and spans may be in any order.

Examples

  Array ref                         Set
  [ ]                               { }
  [ 1, 1 ]                          { 1 }
  [ 1, 3, 2 ]                       { 1, 2, 3 }
  [ 1, [ 5, 8 ], 5, [ 7, 9 ], 2 ]   { 1, 2, 5, 6, 7, 8, 9 }
  [ undef, undef ]                  the integers
  [ undef, -1 ]                     the negative integers

ITERATORS

Each set has a single iterator, which is shared by all calls to "first", "last", "start", "next", "prev", and "current". At all times, the iterator is either an element of the set, or "undef".

"first", "last", and "start" set the iterator; "next", and "prev" move it; and "current" returns it. Calls to these methods may be freely intermixed.

Using "next" and "prev", a single loop can move both forwards and backwards through a set. Using "start", a loop can iterate over portions of an infinite set.

METHODS

Creation

$set = "new" "Set::IntSpan" $set_spec
$set = "new" "Set::IntSpan" @set_specs
Creates and returns a "Set::IntSpan" object.

The initial contents of the set are given by $set_spec, or by the union of all the @set_specs.

$ok = "valid" "Set::IntSpan" $run_list
Returns true if $run_list is a valid run list. Otherwise, returns false and leaves an error message in [email protected]
$set = "copy" $set $set_spec
Copies $set_spec into $set. The previous contents of $set are lost. For convenience, "copy" returns $set.
$run_list = "run_list" $set
Returns a run list that represents $set. The run list will not contain white space. $set is not affected.

By default, the empty set is formatted as '-'; a different string may be specified in $Set::IntSpan::Empty_String.

@elements = "elements" $set
Returns an array containing the elements of $set. The elements will be sorted in numerical order. In scalar context, returns an array reference. $set is not affected.
@sets = "sets" $set
Returns the runs in $set, as a list of "Set::IntSpan" objects. The sets in the list are in order.
@spans = "spans" $set
Returns the runs in $set, as a list of the form

  ([$a1, $b1],
   [$a2, $b2],
   ...
   [$aN, $bN])

If a run contains only a single integer, then the upper and lower bounds of the corresponding span will be equal.

If the set has no lower bound, then $a1 will be "undef". Similarly, if the set has no upper bound, then $bN will be "undef".

The runs in the list are in order.

Set operations

For these operations, a new "Set::IntSpan" object is created and returned. The operands are not affected.
$u_set = "union" $set $set_spec
Returns the set of integers in either $set or $set_spec.
$i_set = "intersect" $set $set_spec
Returns the set of integers in both $set and $set_spec.
$x_set = "xor" $set $set_spec
Returns the set of integers in $set or $set_spec, but not both.
$d_set = "diff" $set $set_spec
Returns the set of integers in $set but not in $set_spec.
$c_set = "complement" $set
Returns the set of integers that are not in $set.

Mutators

By popular demand, "Set::IntSpan" now has mutating forms of the binary set operations. These methods alter the object on which they are called.
$set->"U"($set_spec)
Makes $set the union of $set and $set_spec. Returns $set.
$set->"I"($set_spec)
Makes $set the intersection of $set and $set_spec. Returns $set.
$set->"X"($set_spec)
Makes $set the symmetric difference of $set and $set_spec. Returns $set.
$set->"D"($set_spec)
Makes $set the difference of $set and $set_spec. Returns $set.
$set->"C"
Converts $set to its own complement. Returns $set.

Comparison

"equal" $set $set_spec
Returns true iff $set and $set_spec contain the same elements.
"equivalent" $set $set_spec
Returns true iff $set and $set_spec contain the same number of elements. All infinite sets are equivalent.
"superset" $set $set_spec
Returns true iff $set is a superset of $set_spec.
"subset" $set $set_spec
Returns true iff $set is a subset of $set_spec.

Cardinality

$n = "cardinality" $set
$n = "size" $set
Returns the number of elements in $set. Returns -1 for infinite sets. "size" is provided as an alias for "cardinality".
"empty" $set
Returns true iff $set is empty.
"finite" $set
Returns true iff $set is finite.
"neg_inf" $set
Returns true iff $set contains {x | x<n} for some n.
"pos_inf" $set
Returns true iff $set contains {x | x>n} for some n.
"infinite" $set
Returns true iff $set is infinite.
"universal" $set
Returns true iff $set contains all integers.

Membership

"member" $set $n
Returns true iff the integer $n is a member of $set.
"insert" $set $n
Inserts the integer $n into $set. Does nothing if $n is already a member of $set.
"remove" $set $n
Removes the integer $n from $set. Does nothing if $n is not a member of $set.

Extrema

"min" $set
Returns the smallest element of $set, or "undef" if there is none.
"max" $set
Returns the largest element of $set, or "undef" if there is none.

Spans

$holes = "holes" $set
Returns a set containing all the holes in $set, that is, all the integers that are in-between spans of $set.

"holes" is always a finite set.

$cover = "cover" $set
Returns a set consisting of a single span from $set->"min" to $set->"max". This is the same as

  union $set $set->holes
$inset = "inset" $set $n
$smaller = "trim" $set $n
$bigger = "pad" $set $n
"inset" returns a set constructed by removing $n integers from each end of each span of $set. If $n is negative, then -$n integers are added to each end of each span.

In the first case, spans may vanish from the set; in the second case, holes may vanish.

"trim" is provided as a synonym for "inset".

"pad" $set $n is the same as "inset" $set -$n.

Iterators

$set->"first"
Sets the iterator for $set to the smallest element of $set. If there is no smallest element, sets the iterator to "undef". Returns the iterator.
$set->"last"
Sets the iterator for $set to the largest element of $set. If there is no largest element, sets the iterator to "undef". Returns the iterator.
$set->"start"($n)
Sets the iterator for $set to $n. If $n is not an element of $set, sets the iterator to "undef". Returns the iterator.
$set->"next"
Sets the iterator for $set to the next element of $set. If there is no next element, sets the iterator to "undef". Returns the iterator.

"next" will return "undef" only once; the next call to "next" will reset the iterator to the smallest element of $set.

$set->"prev"
Sets the iterator for $set to the previous element of $set. If there is no previous element, sets the iterator to "undef". Returns the iterator.

"prev" will return "undef" only once; the next call to "prev" will reset the iterator to the largest element of $set.

$set->"current"
Returns the iterator for $set.

Indexing

The elements of a set are kept in numerical order. These methods index into the set based on this ordering.
$n = $set->"at"($i)
Returns the $ith element of $set, or "undef" if there is no $ith element. Negative indices count backwards from the end of the set.

Dies if

  • $i is non-negative and $set is "neg_inf"
  • $i is negative and $set is "pos_inf"
$slice = $set->"slice"($from, $to)
Returns a "Set::IntSpan" object containing the elements of $set at indices $from..$to. Negative indices count backwards from the end of the set.

Dies if

  • $from is non-negative and $set is "neg_inf"
  • $from is negative and $set is "pos_inf"
$i = $set->"ord"($n)
The inverse of "at".

Returns the index $i of the integer $n in $set, or "undef" if $n if not an element of $set.

Dies if $set is "neg_inf".

$i = $set->"span_ord"($n)
Returns the index $i of the span containing the integer $n, or "undef" if $n if not an element of $set.

To recover the span containing $n, write

  ($set->spans)[$i]

OPERATOR OVERLOADS

For convenience, some operators are overloaded on "Set::IntSpan" objects.

set operations

One operand must be a "Set::IntSpan" object. The other operand may be a "Set::IntSpan" object or a set specification.

  $u_set =  $set + $set_spec;   # union
  $i_set =  $set * $set_spec;   # intersect
  $x_set =  $set ^ $set_spec;   # xor
  $d_set =  $set - $set_spec;   # diff
  $c_set = ~$set;               # complement
  $set += $set_spec;            # union
  $set *= $set_spec;            # intersect
  $set ^= $set_spec;            # xor
  $set -= $set_spec;            # diff

equality

The string comparison operations are overloaded to compare sets for equality and containment. One operand must be a "Set::IntSpan" object. The other operand may be a "Set::IntSpan" object or a set specification.

  $set eq $set_spec             # equal
  $set ne $set_spec             # not equal
  $set le $set_spec             # subset
  $set lt $set_spec             # proper subset
  $set ge $set_spec             # superset
  $set gt $set_spec             # proper superset

equivalence

The numerical comparison operations are overloaded to compare sets by cardinality. One operand must be a "Set::IntSpan" object. The other operand may be a "Set::IntSpan" object or an integer.

  $set1 ==  $set2
  $set1 !=  $set2
  $set1 <=  $set2
  $set1 <   $set2
  $set1 >=  $set2
  $set1 >   $set2
  $set1 <=> $set2
  $set1 cmp $set2
  $set1 ==  $n
  $set1 !=  $n
  $set1 <=  $n
  $set1 <   $n
  $set1 >=  $n
  $set1 >   $n
  $set1 <=> $n
  $set1 cmp $n

N.B. The "cmp" operator is overloaded to compare sets by cardinality, not containment. This is done so that

  sort @sets

will sort a list of sets by cardinality.

conversion

In boolean context, a "Set::IntSpan" object evaluates to true if it is not empty.

A "Set::IntSpan" object stringizes to its run list.

FUNCTIONS

$sub_set = "grep_set" { ... } $set
Evaluates the BLOCK for each integer in $set (locally setting $_ to each integer) and returns a "Set::IntSpan" object containing those integers for which the BLOCK returns TRUE.

Returns "undef" if $set is infinite.

$map_set = "map_set" { ... } $set
Evaluates the BLOCK for each integer in $set (locally setting $_ to each integer) and returns a "Set::IntSpan" object containing all the integers returned as results of all those evaluations.

Evaluates the BLOCK in list context, so each element of $set may produce zero, one, or more elements in the returned set. The elements may be returned in any order, and need not be disjoint.

Returns "undef" if $set is infinite.

$sub_set = "grep_spans" { ... } $set
Evaluates the BLOCK for each span in $set and returns a "Set::IntSpan" object containing those spans for which the BLOCK returns TRUE.

Within BLOCK, $_ is locally set to an array ref of the form

  [ $lower, $upper ]

where $lower and $upper are the bounds of the span. If the span contains only one integer, then $lower and $upper will be equal. If the span is unbounded, then the corresponding element(s) of the array will be "undef".

$map_set = "map_spans" { ... } $set
Evaluates the BLOCK for each span in $set, and returns a "Set::IntSpan" object consisting of the union of all the spans returned as results of all those evaluations.

Within BLOCK, $_ is locally set to an array ref of the form

  [ $lower, $upper ]

as described above for "grep_spans". Each evaluation of BLOCK must return a list of spans. Each returned list may contain zero, one, or more spans. Spans may be returned in any order, and need not be disjoint. However, for each bounded span, the constraint

  $lower <= $upper

must hold.

CLASS VARIABLES

$Set::IntSpan::Empty_String
$Set::IntSpan::Empty_String contains the string that is returned when "run_list" is called on the empty set. $Empty_String is initially '-'; alternatively, it may be set to ''. Other values should be avoided, to ensure that "run_list" always returns a valid run list.

"run_list" accesses $Empty_String through a reference stored in $set->{"empty_string"}. Subclasses that wish to override the value of $Empty_String can reassign this reference.

$Set::IntSpan::integer
Up until version 1.16, "Set::IntSpan" specified "use integer", because they were sets of...you know...integers. As of 2012, users are reporting newsgroups with article numbers above 0x7fffffff, which break "Set::IntSpan" on 32-bit processors.

Version 1.17 removes "use integer" by default. This extends the usable range of "Set::IntSpan" to the number of bits in the mantissa of your floating point representation. For IEEE 754 doubles, this is 53 bits, or around 9e15.

I benchmarked "Set::IntSpan" on a Pentium 4, and it looks like "use integer" provides a 2% to 4% speedup, depending on the application.

If you want "use integer" back, either for performance, or because you are somehow dependent on its semantics, write

  BEGIN { $Set::IntSpan::integer = 1 }
  use Set::IntSpan;

DIAGNOSTICS

Any method (except "valid") will "die" if it is passed an invalid run list.
"Set::IntSpan::_copy_run_list: Bad syntax:" $runList
(F) $run_list has bad syntax
"Set::IntSpan::_copy_run_list: Bad order:" $runList
(F) $run_list has overlapping runs or runs that are out of order.
"Set::IntSpan::elements: infinite set"
(F) An infinite set was passed to "elements".
"Set::IntSpan::at: negative infinite set"
(F) "at" was called with a non-negative index on a negative infinite set.
"Set::IntSpan::at: positive infinite set"
(F) "at" was called with a negative index on a positive infinite set.
"Set::IntSpan::slice: negative infinite set"
(F) "slice" was called with $from non-negative on a negative infinite set.
"Set::IntSpan::slice: positive infinite set"
(F) "slice" was called with $from negative on a positive infinite set.
"Set::IntSpan::ord: negative infinite set"
(F) "ord" was called on a negative infinite set.
Out of memory!
(X) "elements" $set can generate an ``Out of memory!'' message on sufficiently large finite sets.

NOTES

Traps

Beware of forms like

  union $set [1..5];

This passes an element of @set to union, which is probably not what you want. To force interpretation of $set and [1..5] as separate arguments, use forms like

    union $set +[1..5];

or

    $set->union([1..5]);

grep_set and map_set

"grep_set" and "map_set" make it easy to construct sets for which the internal representation used by "Set::IntSpan" is not small. Consider:

  $billion = new Set::IntSpan '0-1_000_000_000';   # OK
  $odd     = grep_set { $_ & 1 } $billion;         # trouble
  $even    = map_set  { $_ * 2 } $billion;         # double trouble

Error handling

There are two common approaches to error handling: exceptions and return codes. There seems to be some religion on the topic, so "Set::IntSpan" provides support for both.

To catch exceptions, protect method calls with an eval:

    $run_list = <STDIN>;
    eval { $set = new Set::IntSpan $run_list };
    [email protected] and print "[email protected]: try again\n";

To check return codes, use an appropriate method call to validate arguments:

    $run_list = <STDIN>;
    if (valid Set::IntSpan $run_list)
       { $set = new Set::IntSpan $run_list }
    else
       { print "[email protected] try again\n" }

Similarly, use "finite" to protect calls to "elements":

    finite $set and @elements = elements $set;

Calling "elements" on a large, finite set can generate an ``Out of memory!'' message, which cannot (easily) be trapped. Applications that must retain control after an error can use "intersect" to protect calls to "elements":

    @elements = elements { intersect $set "-1_000_000 - 1_000_000" };

or check the size of $set first:

    finite $set and cardinality $set < 2_000_000 and @elements = elements $set;

Limitations

Although "Set::IntSpan" can represent some infinite sets, it does not perform infinite-precision arithmetic. Therefore, finite elements are restricted to the range of integers on your machine.

Extensions

Users report that you can construct Set::IntSpan objects on anything that behaves like an integer. For example:

    $x   = new Math::BigInt ...;
    $set = new Set::Intspan [ [ $x, $x+5 ] ];

I'm not documenting this as supported behavior, because I don't have the resources to test it, but I'll try not to break it. If anyone finds problems with it, let me know.

Roots

The sets implemented here are based on a Macintosh data structure called a region. See Inside Macintosh for more information.

"Set::IntSpan" was originally written to manage run lists for the "News::Newsrc" module.

AUTHOR

Steven McDougall <[email protected]>

ACKNOWLEDGMENTS

COPYRIGHT

Copyright (c) 1996-2013 by Steven McDougall. This module is free software; you can redistribute it and/or modify it under the same terms as Perl itself.