defmodule Interval do
@moduledoc """
An interval represents the points between two endpoints.
The interval can be empty.
The empty interval is never contained in any other interval,
and contains itself no points.
It can also be left and/or right unbounded, in which case
it contains all points in the unbounded direction.
A fully unbounded interval contains all other intervals, except
the empty interval.
"""
alias Interval.Point
alias Interval.Endpoint
defstruct left: nil, right: nil
@typedoc """
The `Interval` struct, representing all points between
two endpoints.
The struct has two fields: `left` and `right`,
representing the left (lower) and right (upper) points
in the interval.
The endpoints are stored as an `t:Interval.Endpoint.t/0` or
the atom `:unbounded`.
A special case exists for the empty interval,
which is represented by both `left` and `right` being
set to the atom `:empty`
"""
@type t() :: %__MODULE__{
# Left endpoint
left: :empty | :unbounded | Interval.Endpoint.t(),
# Right endpoint
right: :empty | :unbounded | Interval.Endpoint.t()
}
@doc """
Create a new Interval containing a single point.
"""
def single(point) when not is_list(point) do
# assert that Point is implemented for given variable
true = Point.type(point) in [:discrete, :continuous]
endpoint = Endpoint.inclusive(point)
from_endpoints(endpoint, endpoint)
end
@doc """
Create a new unbounded interval
"""
def new(opts \\ [])
def new(opts) when is_list(opts) do
left = Keyword.get(opts, :left, nil)
right = Keyword.get(opts, :right, nil)
bounds = Keyword.get(opts, :bounds, "[)")
{left_bound, right_bound} = unpack_bounds(bounds)
left_endpoint =
case {left, left_bound} do
{nil, _} -> :unbounded
{_, :unbounded} -> :unbounded
{_, :inclusive} -> Endpoint.inclusive(left)
{_, :exclusive} -> Endpoint.exclusive(left)
end
right_endpoint =
case {right, right_bound} do
{nil, _} -> :unbounded
{_, :unbounded} -> :unbounded
{_, :inclusive} -> Endpoint.inclusive(right)
{_, :exclusive} -> Endpoint.exclusive(right)
end
from_endpoints(left_endpoint, right_endpoint)
end
def from_endpoints(left, right)
when (left == :unbounded or is_struct(left, Endpoint)) and
(right == :unbounded or is_struct(right, Endpoint)) do
%__MODULE__{left: left, right: right}
|> normalize()
end
@doc """
Normalize an `Interval` struct
"""
# lef and right endpoints set to :empty, special case for normalized empty interval
def normalize(%__MODULE__{left: :empty, right: :empty} = self), do: self
# non-empty non-unbounded Interval:
def normalize(%__MODULE__{left: %Endpoint{} = left, right: %Endpoint{} = right} = original) do
left_point_impl = Point.impl_for(left.point)
right_point_impl = Point.impl_for(right.point)
if left_point_impl != right_point_impl do
raise """
The Interval.Point implementation for the left and right side
of the interval must be identical, but got
left=#{left_point_impl}, right=#{right_point_impl}
"""
end
type = Point.type(left.point)
comp = Point.compare(left.point, right.point)
inclusive_left = Endpoint.inclusive?(left)
inclusive_right = Endpoint.inclusive?(right)
case {type, comp, inclusive_left, inclusive_right} do
# left > right is an error:
{_, :gt, _, _} ->
raise "left > right which is invalid"
# intervals given as either (p,p), [p,p) or (p,p]
# are all normalized to empty.
# (If you want a single point in an interval, give it as [p,p])
{_, :eq, false, false} ->
into_empty(original)
{_, :eq, true, false} ->
into_empty(original)
{_, :eq, false, true} ->
into_empty(original)
# otherwise, if the point type is continuous, the the orignal
# interval was already normalized form:
{:continuous, _, _, _} ->
original
## Discrete types:
# if discrete type, we want to always normalize to bounds == [)
# because it makes life a bit easier elsewhere.
# if both bounds are exclusive, we also need to check for empty, because
# we could still have an empty interval like (1,2)
{:discrete, _, false, false} ->
case Point.compare(Point.next(left.point), right.point) do
:eq ->
into_empty(original)
:lt ->
%__MODULE__{original | left: normalize_left_endpoint(left)}
end
# Remaining bound combinations are:
# [], (], [)
# we don't need to touch [), so we only need to deal with
# the ones that are upper-inclusive. We want to perform the following
# transformations:
# [a,b] -> [a, b+1)
# (a,b] -> [a+1, b+1)
{:discrete, _, true, true} ->
%__MODULE__{
original
| right: normalize_right_endpoint(right)
}
{:discrete, _, false, true} ->
%__MODULE__{
original
| left: normalize_left_endpoint(left),
right: normalize_right_endpoint(right)
}
# Finally, if we have an [) interval, then the original was
# valid:
{:discrete, :lt, true, false} ->
original
end
end
# Either left or right or both must be unbounded
def normalize(%__MODULE__{left: left, right: right} = original) do
%{
original
| left: normalize_left_endpoint(left),
right: normalize_right_endpoint(right)
}
end
defp normalize_right_endpoint(:unbounded), do: :unbounded
defp normalize_right_endpoint(right) do
case {Point.type(right.point), Endpoint.inclusive?(right)} do
{:discrete, true} -> Endpoint.exclusive(Point.next(right.point))
{_, _} -> right
end
end
defp normalize_left_endpoint(:unbounded), do: :unbounded
defp normalize_left_endpoint(left) do
case {Point.type(left.point), Endpoint.inclusive?(left)} do
{:discrete, false} -> Endpoint.inclusive(Point.next(left.point))
{_, _} -> left
end
end
@doc """
Is the interval empty?
An empty interval is an interval that represents no points.
Any interval interval containing no points is considered empty.
## Examples
iex> empty?(new(left: 0, right: 0))
true
iex> empty?(single(1.0))
false
iex> empty?(new(left: 1, right: 2))
false
"""
def empty?(%__MODULE__{left: :empty, right: :empty}), do: true
def empty?(%__MODULE__{}), do: false
@doc """
Check if the interval is left-unbounded.
The interval is left-unbounded if all points
left of the right bound is included in this interval.
## Examples
iex> unbounded_left?(new())
true
iex> unbounded_left?(new(right: 2))
true
iex> unbounded_left?(new(left: 1, right: 2))
false
"""
def unbounded_left?(%__MODULE__{left: :unbounded}), do: true
def unbounded_left?(%__MODULE__{}), do: false
@doc """
Check if the interval is right-unbounded.
The interval is right-unbounded if all points
right of the left bound is included in this interval.
## Examples
iex> unbounded_right?(new(right: 1))
false
iex> unbounded_right?(new())
true
iex> unbounded_right?(new(left: 1))
true
"""
def unbounded_right?(%__MODULE__{right: :unbounded}), do: true
def unbounded_right?(%__MODULE__{}), do: false
@doc """
Is the interval left-inclusive?
The interval is left-inclusive if the left endpoint
value is included in the interval.
> #### Note {: .info}
> Discrete intervals (like integers and dates) are always normalized
> to be left-inclusive right-exclusive (`[)`) which this function reflects.
iex> inclusive_left?(new(left: 1.0, right: 2.0, bounds: "[]"))
true
iex> inclusive_left?(new(left: 1.0, right: 2.0, bounds: "[)"))
true
iex> inclusive_left?(new(left: 1.0, right: 2.0, bounds: "()"))
false
"""
def inclusive_left?(%__MODULE__{left: %Endpoint{} = left}), do: Endpoint.inclusive?(left)
def inclusive_left?(%__MODULE__{}), do: false
@doc """
Is the interval right-inclusive?
The interval is right-inclusive if the right endpoint
value is included in the interval.
> #### Note {: .info}
> Discrete intervals (like integers and dates) are always normalized
> to be left-inclusive right-exclusive (`[)`) which this function reflects.
iex> inclusive_right?(new(left: 1.0, right: 2.0, bounds: "[]"))
true
iex> inclusive_right?(new(left: 1.0, right: 2.0, bounds: "[)"))
false
iex> inclusive_right?(new(left: 1.0, right: 2.0, bounds: "()"))
false
"""
def inclusive_right?(%__MODULE__{right: %Endpoint{} = right}), do: Endpoint.inclusive?(right)
def inclusive_right?(%__MODULE__{}), do: false
@doc """
Is `a` strictly left of `b`.
`a` is strictly left of `b` if no point in `a` is in `b`,
and all points in `a` is left (<) of all points in `b`.
## Examples
a: [---)
b: [---)
r: true
a: [---)
b: [---)
r: true
a: [---)
b: [---)
r: false (overlaps)
iex> strictly_left_of?(new(left: 1, right: 2), new(left: 3, right: 4))
true
iex> strictly_left_of?(new(left: 1, right: 3), new(left: 2, right: 4))
false
iex> strictly_left_of?(new(left: 3, right: 4), new(left: 1, right: 2))
false
"""
@spec strictly_left_of?(t(), t()) :: boolean()
def strictly_left_of?(a, b) do
not unbounded_right?(a) and
not unbounded_left?(b) and
not empty?(a) and
not empty?(b) and
case Point.compare(a.right.point, b.left.point) do
:lt -> true
:eq -> not inclusive_right?(a) or not inclusive_left?(b)
:gt -> false
end
end
@doc """
Is `a` strictly right of `b`.
`a` is strictly right of `b` if no point in `a` is in `b`,
and all points in `a` is right (>) of all points in `b`.
## Examples
a: [---)
b: [---)
r: true
a: [---)
b: [---)
r: true
a: [---)
b: [---)
r: false (overlaps)
iex> strictly_right_of?(new(left: 1, right: 2), new(left: 3, right: 4))
false
iex> strictly_right_of?(new(left: 1, right: 3), new(left: 2, right: 4))
false
iex> strictly_right_of?(new(left: 3, right: 4), new(left: 1, right: 2))
true
"""
@spec strictly_right_of?(t(), t()) :: boolean()
def strictly_right_of?(a, b) do
not unbounded_left?(a) and
not unbounded_right?(b) and
not empty?(a) and
not empty?(b) and
case Point.compare(a.left.point, b.right.point) do
:lt -> false
:eq -> not inclusive_left?(a) or not inclusive_right?(b)
:gt -> true
end
end
@doc """
Is the interval `a` adjacent to `b`, to the left of `b`.
`a` is adjacent to `b` left of `b`, if `a` and `b` do _not_ overlap,
and there are no points between `a.right` and `b.left`.
a: [---)
b: [---)
r: true
a: [---]
b: [---]
r: false (overlaps)
a: (---)
b: (---)
r: false (points exist between a.right and b.left)
## Examples
iex> adjacent_left_of?(new(left: 1, right: 2), new(left: 2, right: 3))
true
iex> adjacent_left_of?(new(left: 1, right: 3), new(left: 2, right: 4))
false
iex> adjacent_left_of?(new(left: 3, right: 4), new(left: 1, right: 2))
false
iex> adjacent_left_of?(new(right: 2, bounds: "[]"), new(left: 3))
true
"""
@spec adjacent_left_of?(t(), t()) :: boolean()
def adjacent_left_of?(a, b) do
prerequisite =
not unbounded_right?(a) and
not unbounded_left?(b) and
not empty?(a) and
not empty?(b)
with true <- prerequisite do
case Point.type(a.right.point) do
:discrete ->
check =
inclusive_right?(a) != inclusive_left?(b) and
Point.compare(a.right.point, b.left.point) == :eq
# NOTE: Don't think this is needed when we also
# normalize discrete values to [)
next_check =
inclusive_right?(a) and inclusive_left?(b) and
Point.compare(Point.next(a.right.point), b.left.point) == :eq
check or next_check
:continuous ->
inclusive_right?(a) != inclusive_left?(b) and
Point.compare(a.right.point, b.left.point) == :eq
end
end
end
@doc """
Is the interval `a` adjacent to `b`, to the right of `b`.
`a` is adjacent to `b` right of `b`, if `a` and `b` do _not_ overlap,
and there are no points between `a.left` and `b.right`.
a: [---)
b: [---)
r: true
a: [---)
b: [---]
r: false (overlaps)
a: (---)
b: (---)
r: false (points exist between a.left and b.right)
## Examples
iex> adjacent_right_of?(new(left: 2, right: 3), new(left: 1, right: 2))
true
iex> adjacent_right_of?(new(left: 1, right: 3), new(left: 2, right: 4))
false
iex> adjacent_right_of?(new(left: 1, right: 2), new(left: 3, right: 4))
false
iex> adjacent_right_of?(new(left: 3), new(right: 2, bounds: "]"))
true
"""
@spec adjacent_right_of?(t(), t()) :: boolean()
def adjacent_right_of?(a, b) do
prerequisite =
not unbounded_left?(a) and
not unbounded_right?(b) and
not empty?(a) and
not empty?(b)
with true <- prerequisite do
case Point.type(a.left.point) do
:discrete ->
check =
inclusive_left?(a) != inclusive_right?(b) and
Point.compare(a.left.point, b.right.point) == :eq
# NOTE: Don't think this is needed when we also
# normalize discrete values to [)
next_check =
inclusive_left?(a) and inclusive_right?(b) and
Point.compare(Point.previous(a.left.point), b.right.point) == :eq
check or next_check
:continuous ->
Point.compare(a.left.point, b.right.point) == :eq and
inclusive_left?(a) != inclusive_right?(b)
end
end
end
@doc """
Does `a` overlap with `b`?
`a` overlaps with `b` if any point in `a` is also in `b`.
a: [---)
b: [---)
r: true
a: [---)
b: [---)
r: false
a: [---]
b: [---]
r: true
a: (---)
b: (---)
r: false
a: [---)
b: [---)
r: false
## Examples
[--a--)
[--b--)
iex> overlaps?(new(left: 1, right: 3), new(left: 2, right: 4))
true
[--a--)
[--b--)
iex> overlaps?(new(left: 1, right: 3), new(left: 3, right: 5))
false
[--a--]
[--b--]
iex> overlaps?(new(left: 1, right: 3), new(left: 2, right: 4))
true
(--a--)
(--b--)
iex> overlaps?(new(left: 1, right: 3), new(left: 3, right: 5))
false
[--a--)
[--b--)
iex> overlaps?(new(left: 1, right: 2), new(left: 3, right: 4))
false
"""
@spec overlaps?(t(), t()) :: boolean()
def overlaps?(a, b) do
not empty?(a) and
not empty?(b) and
not strictly_left_of?(a, b) and
not strictly_right_of?(a, b)
end
@doc """
Does `a` contain `b`?
`a` contains `b` of all points in `b` is also in `a`.
For an interval `a` to contain an interval `b`, all points
in `b` must be contained in `a`:
a: [-------]
b: [---]
r: true
a: [---]
b: [---]
r: true
a: [---]
b: (---)
r: true
a: (---)
b: [---]
r: false
a: [---]
b: [-------]
r: false
This means that `a.left` is less than `b.left` (or unbounded), and `a.right` is greater than
`b.right` (or unbounded)
If `a` and `b`'s point match, then `b` is "in" `a` if `a` and `b` share bound types.
E.g. if `a.left` and `b.left` matches, then `a` contains `b` if both `a` and `b`'s
`left` is inclusive or exclusive.
If either of `b` endpoints are unbounded, then `a` only contains `b`
if the corresponding endpoint in `a` is also unbounded.
## Examples
iex> contains?(new(left: 1, right: 2), new(left: 1, right: 2))
true
iex> contains?(new(left: 1, right: 3), new(left: 2, right: 3))
true
iex> contains?(new(left: 2, right: 3), new(left: 1, right: 4))
false
iex> contains?(new(left: 1, right: 3), new(left: 1, right: 2))
true
iex> contains?(new(left: 1, right: 2, bounds: "()"), new(left: 1, right: 3))
false
iex> contains?(new(right: 1), new(left: 0, right: 1))
true
"""
@spec contains?(t(), t()) :: boolean()
def contains?(%__MODULE__{} = a, %__MODULE__{} = b) do
# Neither A or B must be empty, so that's a prerequisite for
# even checking anything.
prerequisite = not (empty?(a) or empty?(b))
with true <- prerequisite do
# check that a.left.point is less than or equal to (if inclusive) b.left.point:
contains_left =
unbounded_left?(a) or
(not unbounded_left?(b) and
case Point.compare(a.left.point, b.left.point) do
:gt -> false
:eq -> inclusive_left?(a) == inclusive_left?(b)
:lt -> true
end)
# check that a.right.point is greater than or equal to (if inclusive) b.right.point:
contains_right =
unbounded_right?(a) or
(not unbounded_right?(b) and
case Point.compare(a.right.point, b.right.point) do
:gt -> true
:eq -> inclusive_right?(a) == inclusive_right?(b)
:lt -> false
end)
# a contains b if both the left check and right check passes:
contains_left and contains_right
end
end
@doc """
Computes the union of `a` and `b`.
The union contains all of the points that are either in `a` or `b`.
If either `a` or `b` are empty, the returned interval will be empty.
a: [---)
b: [---)
r: [-----)
## Examples
[--A--)
[--B--)
[----C----)
iex> union(new(left: 1, right: 3), new(left: 2, right: 4))
new(left: 1, right: 4)
[-A-)
[-B-)
[---C---)
iex> union(new(left: 1, right: 2), new(left: 2, right: 3))
new(left: 1, right: 3)
iex> union(new(left: 1, right: 2), new(left: 3, right: 4))
new(left: 0, right: 0)
"""
def union(a, b) do
cond do
# if either is empty, return the other
empty?(a) ->
b
empty?(b) ->
a
# if a and b overlap or are adjacent, we can union the intervals
overlaps?(a, b) or adjacent_left_of?(a, b) or adjacent_right_of?(a, b) ->
left = min_endpoint(a.left, b.left)
right = max_endpoint(a.right, b.right)
from_endpoints(left, right)
# fall-through, if neither A or B is empty,
# but there is also no overlap or adjacency,
# then the two intervals are either strictly left or strictly right,
# we return empty (A and B share an empty amount of points)
true ->
# TODO: remove this assertion.
# It should always be true, so no point in checking:
true == strictly_left_of?(a, b) or strictly_right_of?(a, b)
into_empty(a)
end
end
@doc """
Compute the intersection between `a` and `b`.
The intersection contains all of the points that are both in `a` and `b`.
If either `a` or `b` are empty, the returned interval will be empty.
a: [----]
b: [----]
r: [-]
a: (----)
b: (----)
r: (-)
a: [----)
b: [----)
r: [-)
## Examples:
Discrete:
a: [----)
b: [----)
c: [-)
iex> intersection(new(left: 1, right: 3), new(left: 2, right: 4))
new(left: 2, right: 3)
Continuous:
a: [----)
b: [----)
c: [-)
iex> intersection(new(left: 1.0, right: 3.0), new(left: 2.0, right: 4.0))
new(left: 2.0, right: 3.0)
a: (----)
b: (----)
c: (-)
iex> intersection(
...> new(left: 1.0, right: 3.0, bounds: "()"),
...> new(left: 2.0, right: 4.0, bounds: "()")
...> )
new(left: 2.0, right: 3.0, bounds: "()")
"""
def intersection(a, b) do
cond do
# if A is empty, we return A
empty?(a) ->
a
# if B is empty, we return B
empty?(b) ->
b
# if A and B doesn't overlap,
# then there can be no intersection
not overlaps?(a, b) ->
into_empty(a)
# otherwise, we can compute the intersection:
true ->
left = max_endpoint(a.left, b.left)
right = min_endpoint(a.right, b.right)
from_endpoints(left, right)
end
end
##
## Helpers
##
defp min_endpoint(:unbounded, _b), do: :unbounded
defp min_endpoint(_a, :unbounded), do: :unbounded
defp min_endpoint(left, right) do
case Point.compare(left.point, right.point) do
:gt ->
right
:eq ->
case {Endpoint.inclusive?(left), Endpoint.inclusive?(right)} do
{true, _} -> left
{_, true} -> right
_ -> left
end
:lt ->
left
end
end
defp max_endpoint(:unbounded, _b), do: :unbounded
defp max_endpoint(_a, :unbounded), do: :unbounded
defp max_endpoint(left, right) do
case Point.compare(left.point, right.point) do
:gt ->
left
:eq ->
case {Endpoint.inclusive?(left), Endpoint.inclusive?(right)} do
{true, _} -> left
{_, true} -> right
_ -> left
end
:lt ->
right
end
end
# completely unbounded:
defp unpack_bounds(""), do: {:unbounded, :unbounded}
# unbounded either left or right
defp unpack_bounds(")"), do: {:unbounded, :exclusive}
defp unpack_bounds("("), do: {:exclusive, :unbounded}
defp unpack_bounds("]"), do: {:unbounded, :inclusive}
defp unpack_bounds("["), do: {:inclusive, :unbounded}
# bounded both sides
defp unpack_bounds("()"), do: {:exclusive, :exclusive}
defp unpack_bounds("[]"), do: {:inclusive, :inclusive}
defp unpack_bounds("[)"), do: {:inclusive, :exclusive}
defp unpack_bounds("(]"), do: {:exclusive, :inclusive}
defp into_empty(interval) do
%{interval | left: :empty, right: :empty}
end
end
