portion 2.1.5

Python data structure and operations for intervals

portion - data structure and operations for intervals

The portion library (formerly distributed as python-intervals) provides data structure and operations for intervals in Python 3.5+.

• Support intervals of any (comparable) objects.
• Closed or open, finite or (semi-)infinite intervals.
• Interval sets (union of atomic intervals) are supported.
• Automatic simplification of intervals.
• Support comparison, transformation, intersection, union, complement, difference and containment.
• Provide test for emptiness, atomicity, overlap and adjacency.
• Discrete iterations on the values of an interval.
• Dict-like structure to map intervals to data.
• Import and export intervals to strings and to Python built-in data types.
• Heavily tested with high code coverage.

Latest release:

• portion: 2.1.5 on 2021-02-28 (documentation, changes).
• python-intervals: 1.10.0 on 2019-09-26 (documentation, changes).

Note that python-intervals will no longer receive updates since it has been replaced by portion.

Table of contents

• Installation
• Documentation & usage
  • Interval creation
  • Interval bounds & attributes
  • Interval operations
  • Comparison operators
  • Interval transformation
  • Discrete iteration
  • Map intervals to data
  • Import & export intervals to strings
  • Import & export intervals to Python built-in data types
• Changelog
• Contributions
• License

Installation

You can use pip to install it, as usual: pip install portion.

This will install the latest available version from PyPI. Pre-releases are available from the master branch on GitHub and can be installed with pip install git+https://github.com/AlexandreDecan/portion.

The test environment can be installed with pip install portion[test] and relies on pytest.

portion is also available on conda-forge.

Documentation & usage

Interval creation

Assuming this library is imported using import portion as P, intervals can be easily created using one of the following helpers:

>>> P.open(1, 2)
(1,2)
>>> P.closed(1, 2)
[1,2]
>>> P.openclosed(1, 2)
(1,2]
>>> P.closedopen(1, 2)
[1,2)
>>> P.singleton(1)
[1]
>>> P.empty()
()

The bounds of an interval can be any arbitrary values, as long as they are comparable:

>>> P.closed(1.2, 2.4)
[1.2,2.4]
>>> P.closed('a', 'z')
['a','z']
>>> import datetime
>>> P.closed(datetime.date(2011, 3, 15), datetime.date(2013, 10, 10))
[datetime.date(2011, 3, 15),datetime.date(2013, 10, 10)]

Infinite and semi-infinite intervals are supported using P.inf and -P.inf as upper or lower bounds. These two objects support comparison with any other object. When infinities are used as a lower or upper bound, the corresponding boundary is automatically converted to an open one.

>>> P.inf > 'a', P.inf > 0, P.inf > True
(True, True, True)
>>> P.openclosed(-P.inf, 0)
(-inf,0]
>>> P.closed(-P.inf, P.inf)  # Automatically converted to an open interval
(-inf,+inf)

Empty intervals always resolve to (P.inf, -P.inf), regardless of the provided bounds:

>>> P.empty() == P.open(P.inf, -P.inf)
True
>>> P.closed(4, 3) == P.open(P.inf, -P.inf)
True
>>> P.openclosed('a', 'a') == P.open(P.inf, -P.inf)
True

Intervals created with this library are Interval instances. An Interval instance is a disjunction of atomic intervals each representing a single interval (e.g. [1,2]). Intervals can be iterated to access the underlying atomic intervals, sorted by their lower and upper bounds.

>>> list(P.open(10, 11) | P.closed(0, 1) | P.closed(20, 21))
[[0,1], (10,11), [20,21]]

Nested intervals can also be retrieved with a position or a slice:

>>> (P.open(10, 11) | P.closed(0, 1) | P.closed(20, 21))[0]
[0,1]
>>> (P.open(10, 11) | P.closed(0, 1) | P.closed(20, 21))[-2]
(10,11)
>>> (P.open(10, 11) | P.closed(0, 1) | P.closed(20, 21))[:2]
[0,1] | (10,11)

For convenience, intervals are automatically simplified:

>>> P.closed(0, 2) | P.closed(2, 4)
[0,4]
>>> P.closed(1, 2) | P.closed(3, 4) | P.closed(2, 3)
[1,4]
>>> P.empty() | P.closed(0, 1)
[0,1]
>>> P.closed(1, 2) | P.closed(2, 3) | P.closed(4, 5)
[1,3] | [4,5]

Note that simplification of discrete intervals is not supported by portion (but it can be simulated though, see #24). For example, combining [0,1] with [2,3] will not result in [0,3] even if there is no integer between 1 and 2.

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Interval bounds & attributes

An Interval defines the following properties:

• i.empty is True if and only if the interval is empty.

  >>> P.closed(0, 1).empty
  False
  >>> P.closed(0, 0).empty
  False
  >>> P.openclosed(0, 0).empty
  True
  >>> P.empty().empty
  True
  

• i.atomic is True if and only if the interval is a disjunction of a single (possibly empty) interval.

  >>> P.closed(0, 2).atomic
  True
  >>> (P.closed(0, 1) | P.closed(1, 2)).atomic
  True
  >>> (P.closed(0, 1) | P.closed(2, 3)).atomic
  False
  

• i.enclosure refers to the smallest atomic interval that includes the current one.

  >>> (P.closed(0, 1) | P.open(2, 3)).enclosure
  [0,3)
  

The left and right boundaries, and the lower and upper bounds of an interval can be respectively accessed with its left, right, lower and upper attributes. The left and right bounds are either P.CLOSED or P.OPEN. By definition, P.CLOSED == ~P.OPEN and vice-versa.

>> P.CLOSED, P.OPEN
CLOSED, OPEN
>>> x = P.closedopen(0, 1)
>>> x.left, x.lower, x.upper, x.right
(CLOSED, 0, 1, OPEN)

If the interval is not atomic, then left and lower refer to the lower bound of its enclosure, while right and upper refer to the upper bound of its enclosure:

>>> x = P.open(0, 1) | P.closed(3, 4)
>>> x.left, x.lower, x.upper, x.right
(OPEN, 0, 4, CLOSED)

One can easily check for some interval properties based on the bounds of an interval:

>>> x = P.openclosed(-P.inf, 0)
>>> # Check that interval is left/right closed
>>> x.left == P.CLOSED, x.right == P.CLOSED
(False, True)
>>> # Check that interval is left/right bounded
>>> x.lower == -P.inf, x.upper == P.inf
(True, False)
>>> # Check for singleton
>>> x.lower == x.upper
False

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Interval operations

Interval instances support the following operations:

• i.intersection(other) and i & other return the intersection of two intervals.

  >>> P.closed(0, 2) & P.closed(1, 3)
  [1,2]
  >>> P.closed(0, 4) & P.open(2, 3)
  (2,3)
  >>> P.closed(0, 2) & P.closed(2, 3)
  [2]
  >>> P.closed(0, 2) & P.closed(3, 4)
  ()
  

• i.union(other) and i | other return the union of two intervals.

  >>> P.closed(0, 1) | P.closed(1, 2)
  [0,2]
  >>> P.closed(0, 1) | P.closed(2, 3)
  [0,1] | [2,3]
  

• i.complement(other) and ~i return the complement of the interval.

  >>> ~P.closed(0, 1)
  (-inf,0) | (1,+inf)
  >>> ~(P.open(-P.inf, 0) | P.open(1, P.inf))
  [0,1]
  >>> ~P.open(-P.inf, P.inf)
  ()
  

• i.difference(other) and i - other return the difference between i and other.

  >>> P.closed(0,2) - P.closed(1,2)
  [0,1)
  >>> P.closed(0, 4) - P.closed(1, 2)
  [0,1) | (2,4]
  

• i.contains(other) and other in i hold if given item is contained in the interval. It supports intervals and arbitrary comparable values.

  >>> 2 in P.closed(0, 2)
  True
  >>> 2 in P.open(0, 2)
  False
  >>> P.open(0, 1) in P.closed(0, 2)
  True
  

• i.adjacent(other) tests if the two intervals are adjacent, i.e., if they do not overlap and their union form a single atomic interval. While this definition corresponds to the usual notion of adjacency for atomic intervals, it has stronger requirements for non-atomic ones since it requires all underlying atomic intervals to be adjacent (i.e. that one interval fills the gaps between the atomic intervals of the other one).

  >>> P.closed(0, 1).adjacent(P.openclosed(1, 2))
  True
  >>> P.closed(0, 1).adjacent(P.closed(1, 2))
  False
  >>> (P.closed(0, 1) | P.closed(2, 3)).adjacent(P.open(1, 2) | P.open(3, 4))
  True
  >>> (P.closed(0, 1) | P.closed(2, 3)).adjacent(P.open(3, 4))
  False
  >>> P.closed(0, 1).adjacent(P.open(1, 2) | P.open(3, 4))
  False
  

• i.overlaps(other) tests if there is an overlap between two intervals.

  >>> P.closed(1, 2).overlaps(P.closed(2, 3))
  True
  >>> P.closed(1, 2).overlaps(P.open(2, 3))
  False
  

Finally, intervals are hashable as long as their bounds are hashable (and we have defined a hash value for P.inf and -P.inf).

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Comparison operators

Equality between intervals can be checked with the classical == operator:

>>> P.closed(0, 2) == P.closed(0, 1) | P.closed(1, 2)
True
>>> P.closed(0, 2) == P.open(0, 2)
False

Moreover, intervals are comparable using >, >=, < or <=. These comparison operators have a different behaviour than the usual ones. For instance, a < b holds if a is entirely on the left of the lower bound of b and a > b holds if a is entirely on the right of the upper bound of b.

>>> P.closed(0, 1) < P.closed(2, 3)
True
>>> P.closed(0, 1) < P.closed(1, 2)
False

Similarly, a <= b holds if a is entirely on the left of the upper bound of b, and a >= b holds if a is entirely on the right of the lower bound of b.

>>> P.closed(0, 1) <= P.closed(2, 3)
True
>>> P.closed(0, 2) <= P.closed(1, 3)
True
>>> P.closed(0, 3) <= P.closed(1, 2)
False

Intervals can also be compared with single values. If i is an interval and x a value, then x < i holds if x is on the left of the lower bound of i and x <= i holds if x is on the left of the upper bound of i.

>>> 5 < P.closed(0, 10)
False
>>> 5 <= P.closed(0, 10)
True
>>> P.closed(0, 10) < 5
False
>>> P.closed(0, 10) <= 5
True

This behaviour is similar to the one that could be obtained by first converting x to a singleton interval (except for infinities since they resolve to empty intervals).

Note that all these semantics differ from classical comparison operators. As a consequence, some intervals are never comparable in the classical sense, as illustrated hereafter:

>>> P.closed(0, 4) <= P.closed(1, 2) or P.closed(0, 4) >= P.closed(1, 2)
False
>>> P.closed(0, 4) < P.closed(1, 2) or P.closed(0, 4) > P.closed(1, 2)
False
>>> P.empty() < P.empty()
True
>>> P.empty() < P.closed(0, 1) and P.empty() > P.closed(0, 1)
True

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Interval transformation

Intervals are immutable but provide a replace method to create a new interval based on the current one. This method accepts four optional parameters left, lower, upper, and right:

>>> i = P.closed(0, 2)
>>> i.replace(P.OPEN, -1, 3, P.CLOSED)
(-1,3]
>>> i.replace(lower=1, right=P.OPEN)
[1,2)

Functions can be passed instead of values. If a function is passed, it is called with the current corresponding value.

>>> P.closed(0, 2).replace(upper=lambda x: 2 * x)
[0,4]

The provided function won't be called on infinities, unless ignore_inf is set to False.

>>> i = P.closedopen(0, P.inf)
>>> i.replace(upper=lambda x: 10)  # No change, infinity is ignored
[0,+inf)
>>> i.replace(upper=lambda x: 10, ignore_inf=False)  # Infinity is not ignored
[0,10)

When replace is applied on an interval that is not atomic, it is extended and/or restricted such that its enclosure satisfies the new bounds.

>>> i = P.openclosed(0, 1) | P.closed(5, 10)
>>> i.replace(P.CLOSED, -1, 8, P.OPEN)
[-1,1] | [5,8)
>>> i.replace(lower=4)
(4,10]

To apply arbitrary transformations on the underlying atomic intervals, intervals expose an apply method that acts like map. This method accepts a function that will be applied on each of the underlying atomic intervals to perform the desired transformation. The provided function is expected to return either an Interval, or a 4-uple (left, lower, upper, right).

>>> i = P.closed(2, 3) | P.open(4, 5)
>>> # Increment bound values
>>> i.apply(lambda x: (x.left, x.lower + 1, x.upper + 1, x.right))
[3,4] | (5,6)
>>> # Invert bounds
>>> i.apply(lambda x: (~x.left, x.lower, x.upper, ~x.right))
(2,3) | [4,5]

The apply method is very powerful when used in combination with replace. Because the latter allows functions to be passed as parameters and ignores infinities by default, it can be conveniently used to transform (disjunction of) intervals in presence of infinities.

>>> i = P.openclosed(-P.inf, 0) | P.closed(3, 4) | P.closedopen(8, P.inf)
>>> # Increment bound values
>>> i.apply(lambda x: x.replace(upper=lambda v: v + 1))
(-inf,1] | [3,5] | [8,+inf)
>>> # Intervals are still automatically simplified
>>> i.apply(lambda x: x.replace(lower=lambda v: v * 2))
(-inf,0] | [16,+inf)
>>> # Invert bounds
>>> i.apply(lambda x: x.replace(left=lambda v: ~v, right=lambda v: ~v))
(-inf,0) | (3,4) | (8,+inf)
>>> # Replace infinities with -10 and 10
>>> conv = lambda v: -10 if v == -P.inf else (10 if v == P.inf else v)
>>> i.apply(lambda x: x.replace(lower=conv, upper=conv, ignore_inf=False))
(-10,0] | [3,4] | [8,10)

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Discrete iteration

The iterate function takes an interval, and returns a generator to iterate over the values of an interval. Obviously, as intervals are continuous, it is required to specify the step between consecutive values. The iteration then starts from the lower bound and ends on the upper one. Only values contained by the interval are returned this way.

>>> list(P.iterate(P.closed(0, 3), step=1))
[0, 1, 2, 3]
>>> list(P.iterate(P.closed(0, 3), step=2))
[0, 2]
>>> list(P.iterate(P.open(0, 3), step=2))
[2]

When an interval is not atomic, iterate consecutively iterates on all underlying atomic intervals, starting from each lower bound and ending on each upper one:

>>> list(P.iterate(P.singleton(0) | P.singleton(3) | P.singleton(5), step=2))  # Won't be [0]
[0, 3, 5]
>>> list(P.iterate(P.closed(0, 2) | P.closed(4, 6), step=3))  # Won't be [0, 6]
[0, 4]

By default, the iteration always starts on the lower bound of each underlying atomic interval. The base parameter can be used to change this behaviour, by specifying how the initial value to start the iteration from must be computed. This parameter accepts a callable that is called with the lower bound of each underlying atomic interval, and that returns the initial value to start the iteration from. It can be helpful to deal with (semi-)infinite intervals, or to align the generated values of the iterator:

>>> # Align on integers
>>> list(P.iterate(P.closed(0.3, 4.9), step=1, base=int))
[1, 2, 3, 4]
>>> # Restrict values of a (semi-)infinite interval
>>> list(P.iterate(P.openclosed(-P.inf, 2), step=1, base=lambda x: max(0, x)))
[0, 1, 2]

The base parameter can be used to change how iterate applies on unions of atomic interval, by specifying a function that returns a single value, as illustrated next:

>>> base = lambda x: 0
>>> list(P.iterate(P.singleton(0) | P.singleton(3) | P.singleton(5), step=2, base=base))
[0]
>>> list(P.iterate(P.closed(0, 2) | P.closed(4, 6), step=3, base=base))
[0, 6]

Notice that defining base such that it returns a single value can be extremely inefficient in terms of performance when the intervals are "far apart" each other (i.e., when the gaps between atomic intervals are large).

Finally, iteration can be performed in reverse order by specifying reverse=True.

>>> list(P.iterate(P.closed(0, 3), step=-1, reverse=True))  # Mind step=-1
[3, 2, 1, 0]
>>> list(P.iterate(P.closed(0, 3), step=-2, reverse=True))  # Mind step=-2
[3, 1]

Again, this library does not make any assumption about the objects being used in an interval, as long as they are comparable. However, it is not always possible to provide a meaningful value for step (e.g., what would be the step between two consecutive characters?). In these cases, a callable can be passed instead of a value. This callable will be called with the current value, and is expected to return the next possible value.

>>> list(P.iterate(P.closed('a', 'd'), step=lambda d: chr(ord(d) + 1)))
['a', 'b', 'c', 'd']
>>> # Since we reversed the order, we changed plus to minus in step.
>>> list(P.iterate(P.closed('a', 'd'), step=lambda d: chr(ord(d) - 1), reverse=True))
['d', 'c', 'b', 'a']

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Map intervals to data

The library provides an IntervalDict class, a dict-like data structure to store and query data along with intervals. Any value can be stored in such data structure as long as it supports equality.

>>> d = P.IntervalDict()
>>> d[P.closed(0, 3)] = 'banana'
>>> d[4] = 'apple'
>>> d
{[0,3]: 'banana', [4]: 'apple'}

When a value is defined for an interval that overlaps an existing one, it is automatically updated to take the new value into account:

>>> d[P.closed(2, 4)] = 'orange'
>>> d
{[0,2): 'banana', [2,4]: 'orange'}

An IntervalDict can be queried using single values or intervals. If a single value is used as a key, its behaviour corresponds to the one of a classical dict:

>>> d[2]
'orange'
>>> d[5]  # Key does not exist
Traceback (most recent call last):
 ...
KeyError: 5
>>> d.get(5, default=0)
0

When the key is an interval, a new IntervalDict containing the values for the specified key is returned:

>>> d[~P.empty()]  # Get all values, similar to d.copy()
{[0,2): 'banana', [2,4]: 'orange'}
>>> d[P.closed(1, 3)]
{[1,2): 'banana', [2,3]: 'orange'}
>>> d[P.closed(-2, 1)]
{[0,1]: 'banana'}
>>> d[P.closed(-2, -1)]
{}

By using .get, a default value (defaulting to None) can be specified. This value is used to "fill the gaps" if the queried interval is not completely covered by the IntervalDict:

>>> d.get(P.closed(-2, 1), default='peach')
{[-2,0): 'peach', [0,1]: 'banana'}
>>> d.get(P.closed(-2, -1), default='peach')
{[-2,-1]: 'peach'}
>>> d.get(P.singleton(1), default='peach')  # Key is covered, default is not used
{[1]: 'banana'}

For convenience, an IntervalDict provides a way to look for specific data values. The .find method always returns a (possibly empty) Interval instance for which given value is defined:

>>> d.find('banana')
[0,2)
>>> d.find('orange')
[2,4]
>>> d.find('carrot')
()

The active domain of an IntervalDict can be retrieved with its .domain method. This method always returns a single Interval instance, where .keys returns a sorted view of disjoint intervals.

>>> d.domain()
[0,4]
>>> list(d.keys())
[[0,2), [2,4]]
>>> list(d.values())
['banana', 'orange']
>>> list(d.items())
[([0,2), 'banana'), ([2,4], 'orange')]

The .keys, .values and .items methods return exactly one element for each stored value (i.e., if two intervals share a value, they are merged into a disjunction), as illustrated next. See #44 to know how to obtained a sorted list of atomic intervals instead.

>>> d = P.IntervalDict()
>>> d[P.closed(0, 1)] = d[P.closed(2, 3)] = 'peach'
>>> list(d.items())
[([0,1] | [2,3], 'peach')]

Two IntervalDict instances can be combined together using the .combine method. This method returns a new IntervalDict whose keys and values are taken from the two source IntervalDict. Values corresponding to non-intersecting keys are simply copied, while values corresponding to intersecting keys are combined together using the provided function, as illustrated hereafter:

>>> d1 = P.IntervalDict({P.closed(0, 2): 'banana'})
>>> d2 = P.IntervalDict({P.closed(1, 3): 'orange'})
>>> concat = lambda x, y: x + '/' + y
>>> d1.combine(d2, how=concat)
{[0,1): 'banana', [1,2]: 'banana/orange', (2,3]: 'orange'}

Resulting keys always correspond to an outer join. Other joins can be easily simulated by querying the resulting IntervalDict as follows:

>>> d = d1.combine(d2, how=concat)
>>> d[d1.domain()]  # Left join
{[0,1): 'banana', [1,2]: 'banana/orange'}
>>> d[d2.domain()]  # Right join
{[1,2]: 'banana/orange', (2,3]: 'orange'}
>>> d[d1.domain() & d2.domain()]  # Inner join
{[1,2]: 'banana/orange'}

Finally, similarly to a dict, an IntervalDict also supports len, in and del, and defines .clear, .copy, .update, .pop, .popitem, and .setdefault. For convenience, one can export the content of an IntervalDict to a classical Python dict using the as_dict method.

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Import & export intervals to strings

Intervals can be exported to string, either using repr (as illustrated above) or with the to_string function.

>>> P.to_string(P.closedopen(0, 1))
'[0,1)'

The way string representations are built can be easily parametrized using the various parameters supported by to_string:

>>> params = {
...   'disj': ' or ',
...   'sep': ' - ',
...   'left_closed': '<',
...   'right_closed': '>',
...   'left_open': '..',
...   'right_open': '..',
...   'pinf': '+oo',
...   'ninf': '-oo',
...   'conv': lambda v: '"{}"'.format(v),
... }
>>> x = P.openclosed(0, 1) | P.closed(2, P.inf)
>>> P.to_string(x, **params)
'.."0" - "1"> or <"2" - +oo..'

Similarly, intervals can be created from a string using the from_string function. A conversion function (conv parameter) has to be provided to convert a bound (as string) to a value.

>>> P.from_string('[0, 1]', conv=int) == P.closed(0, 1)
True
>>> P.from_string('[1.2]', conv=float) == P.singleton(1.2)
True
>>> converter = lambda s: datetime.datetime.strptime(s, '%Y/%m/%d')
>>> P.from_string('[2011/03/15, 2013/10/10]', conv=converter)
[datetime.datetime(2011, 3, 15, 0, 0),datetime.datetime(2013, 10, 10, 0, 0)]

Similarly to to_string, function from_string can be parametrized to deal with more elaborated inputs. Notice that as from_string expects regular expression patterns, we need to escape some characters.

>>> s = '.."0" - "1"> or <"2" - +oo..'
>>> params = {
...   'disj': ' or ',
...   'sep': ' - ',
...   'left_closed': '<',
...   'right_closed': '>',
...   'left_open': r'\.\.',  # from_string expects regular expression patterns
...   'right_open': r'\.\.',  # from_string expects regular expression patterns
...   'pinf': r'\+oo',  # from_string expects regular expression patterns
...   'ninf': '-oo',
...   'conv': lambda v: int(v[1:-1]),
... }
>>> P.from_string(s, **params)
(0,1] | [2,+inf)

When a bound contains a comma or has a representation that cannot be automatically parsed with from_string, the bound parameter can be used to specify the regular expression that should be used to match its representation.

>>> s = '[(0, 1), (2, 3)]'  # Bounds are expected to be tuples
>>> P.from_string(s, conv=eval, bound=r'\(.+?\)')
[(0, 1),(2, 3)]

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Import & export intervals to Python built-in data types

Intervals can also be exported to a list of 4-uples with to_data, e.g., to support JSON serialization. P.CLOSED and P.OPEN are represented by Boolean values True (inclusive) and False (exclusive).

>>> P.to_data(P.openclosed(0, 2))
[(False, 0, 2, True)]

The values used to represent positive and negative infinities can be specified with pinf and ninf. They default to float('inf') and float('-inf') respectively.

>>> x = P.openclosed(0, 1) | P.closedopen(2, P.inf)
>>> P.to_data(x)
[(False, 0, 1, True), (True, 2, inf, False)]

The function to convert bounds can be specified with the conv parameter.

>>> x = P.closedopen(datetime.date(2011, 3, 15), datetime.date(2013, 10, 10))
>>> P.to_data(x, conv=lambda v: (v.year, v.month, v.day))
[(True, (2011, 3, 15), (2013, 10, 10), False)]

Intervals can be imported from such a list of 4-tuples with from_data. The same set of parameters can be used to specify how bounds and infinities are converted.

>>> x = [(True, (2011, 3, 15), (2013, 10, 10), False)]
>>> P.from_data(x, conv=lambda v: datetime.date(*v))
[datetime.date(2011, 3, 15),datetime.date(2013, 10, 10))

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Changelog

This library adheres to a semantic versioning scheme. See CHANGELOG.md for the list of changes.

Contributions

Contributions are very welcome! Feel free to report bugs or suggest new features using GitHub issues and/or pull requests.

License

Distributed under LGPLv3 - GNU Lesser General Public License, version 3.

You can refer to this library using:

@software{portion,
  author = {Decan, Alexandre},
  title = {portion: Python data structure and operations for intervals},
  url = {https://github.com/AlexandreDecan/portion},
}