# Set theory operations (union, intersection etc.) for one dimensional integer intervals

For my first not-just-a-few-days-long Python project, I needed something to handle basic set theory operations (union, intersection etc.) for one dimensional integer intervals, so I came up with this module.

It would be cool if you could tell me if the code is OK or something you would frown upon if you had to continue my project, and if so, what I can do better.

"""intervals

Union, intersection, set difference and symmetric difference
of possibly overlapping or touching integer intervals.
Intervals are defined right-open. (1, 4) -> 1, 2, 3

e.g.
union([(1, 4), (7, 9)], (3, 5)) -> [(1, 5), (7, 9)]
intersection([(1, 4), (7, 9)], (3, 5)) -> [(3, 4)]
set_difference([(1, 4), (7, 9)], (3, 5)) -> [(1, 3), (7, 9)]
set_difference([(3, 5)], [(1, 4), (7, 9)]) -> [(4, 5)]
symmetric_difference([(1, 4), (7, 9)], (3, 5)) -> [(1, 3), (4, 5), (7, 9)]

see: http://en.wikipedia.org/wiki/Set_theory#Basic_concepts_and_notation
"""

import copy
from itertools import accumulate, chain, islice, repeat
from operator import itemgetter
import unittest

class Intervals():
"""Holds a non overlapping list of intervals.
One single interval is just a pair.
Overlapping or touching intervals are automatically merged.
"""

def __init__(self, interval_list=[]):
"""Raises a ValueError if the length of one of the
intervals in the list is negative.
"""
if any(begin > end for begin, end in interval_list):
raise ValueError('Invalid interval')
self._interval_list = _merge_interval_lists(
interval_list, [])

def __repr__(self):
"""Just write out all included intervals.
"""
return 'Intervals ' + str(self._interval_list)

def get(self, copy_content=True):
"""Return the list of intervals.
"""
return copy.copy(self._interval_list) if copy_content\
else self._interval_list

def union(a, b):
"""Merge a and b (union).
"""
return Intervals(_merge_interval_lists(
a.get(False), b.get(False)))

def intersections(a, b):
"""Intersects a and b.
"""
return Intervals(_merge_interval_lists(
a.get(False), b.get(False), merge_type='intersections'))

def set_difference(a, b):
"""Removes b from a.
Set difference is not commutative.
"""
return Intervals(_merge_interval_lists(
a.get(False), b.get(False), merge_type='set difference'))

def symmetric_difference(a, b):
"""Symmetric difference of a and b.
"""
return Intervals(_merge_interval_lists(
a.get(False), b.get(False), merge_type='symmetric difference'))

# class Intervals makes sure, by always building the union first,
# that no invalid a's or b's are fed here.
def _merge_interval_lists(a, b, merge_type='union'):
"""Merges two lists of intervals in O(n*log(n)).
Overlapping or touching intervals are simplified to one.

Arguments:
a and b -- The interval lists to merge.
merge_type -- Can be:
'union',
'intersections',
'symmetric difference', or
'set difference'.

Return the sorted result as a list.
"""

# If we want to calculate the set difference
# we invert the second interval list,
# i.e. swap begin and end.
if merge_type == 'set difference':
b = map(lambda p: (p[1], p[0]), b)

# Separately sort begins and ends
# and pair them with the implied change
# of the count of currently open intervals.
# e.g. (1, 4), (7, 9), (3, 5) ->
#     begins = [(1, 1), (3, 1), (7, 1)]
#     ends = [(4, -1), (5, -1), (9, -1)]
both = list(chain(a, b))
begins = zip(sorted(map(itemgetter(0), both)),
repeat(1))
ends = zip(sorted(map(itemgetter(1), both)),
repeat(-1))

# Sort begins and ends together.
# If the value is the same, begins come before ends
# to ensure touching intervals being merged to one.
# In our example above this means:
# edges = [(1, 4), (3, 1), (4, -1), (5, -1), (7, 1), (9, -1)]
edges = sorted(chain(begins, ends), key=lambda x: (x[0], -x[1]))

# Depending on the operation carried out,
# the criteria for interval begins and ends in the result differ.
# E.g:
#      a = | - - - - |      | - - - |
#      b =     | - - - - |
# counts = 1   2     1   0  1       0   (union, intersection, sym diff)
# counts = 1   0    -1   0  1       0   (set diff)
# union  = | - - - - - - |  | - - - |
# inter  =     | - - - - |
# sym d  =           | - |
# set d  = | - |            | - - - |
#
# One can see that union begins if the count changes from 0 to 1
# and ends if the count changes from 1 to 0
# An intersection begins at a change from 1 to 2 and ends with 2 to 1.
# A symmetric difference begins at every change to one
# and ends at every change away from one.
# The conditions for the set difference are the same as for the union.
check_begin = {'union': lambda change: change == (0, 1),
'intersections': lambda change: change == (1, 2),
'symmetric difference': lambda change: change[1] == 1,
'set difference': lambda change: change == (0, 1)
}[merge_type]

check_end = {'union': lambda change: change == (1, 0),
'intersections': lambda change: change == (2, 1),
'symmetric difference': lambda change: change[1] != 1,
'set difference': lambda change: change == (1, 0)
}[merge_type]

# The number of opened intervals after each edge.
counts = list(accumulate(map(itemgetter(1), edges)))
# The changes of opened intervals at each edge.
changes = zip(chain([0], counts), counts)
# Just the x positions of the edges.
xs = map(itemgetter(0), edges)
xs_and_changes = list(zip(xs, changes))

# Now we filter out the begins and ends from the changes
# and get their x positions.
res_begins = map(itemgetter(0),
starfilter(lambda x, change: check_begin(change),
xs_and_changes))
res_ends = map(itemgetter(0),
starfilter(lambda x, change: check_end(change),
xs_and_changes))

# The result is then just pairing up the sorted begins and ends.
result = pairwise(sorted(chain(res_begins, res_ends)), False)

# No empty intervals in the result.
def length_greater_than_zero(interval):
return interval[0] < interval[1]
return list(filter(length_greater_than_zero, result))

class TestIntervals(unittest.TestCase):

def test_ctor(self):
# Check ctors sanity check.
self.assertRaises(ValueError, Intervals, [(2, 4), (3, 1)])

# Check adding right of the last interval.
intervals = Intervals([(0, 2)])
intervals = union(intervals, Intervals([(3, 4)]))
self.assertEqual(intervals.get(), [(0, 2), (3, 4)])

# Check adding left to the first interval.
intervals = Intervals([(3, 4)])
intervals = union(intervals, Intervals([(1, 2)]))
self.assertEqual(intervals.get(), [(1, 2), (3, 4)])

# Check adding between two intervals.
intervals = Intervals([(1, 2)])
intervals = union(intervals, Intervals([(6, 9)]))
intervals = union(intervals, Intervals([(3, 5)]))
self.assertEqual(intervals.get(), [(1, 2), (3, 5), (6, 9)])

# Check adding a interval touching an existing one.
intervals = Intervals([(1, 3)])
intervals = union(intervals, Intervals([(3, 5)]))
self.assertEqual(intervals.get(), [(1, 5)])

# Check adding a interval overlapping an existing one.
intervals = Intervals([(1, 4)])
intervals = union(intervals, Intervals([(3, 5)]))
self.assertEqual(intervals.get(), [(1, 5)])

# Check adding a interval overlapping multiple existing ones.
intervals = Intervals([(1, 4)])
intervals = union(intervals, Intervals([(5, 7)]))
intervals = union(intervals, Intervals([(8, 10)]))
intervals = union(intervals, Intervals([(3, 9)]))
self.assertEqual(intervals.get(), [(1, 10)])

# Check adding a interval completely covering an existing one.
intervals = Intervals([(2, 3)])
intervals = union(intervals, Intervals([(1, 4)]))
self.assertEqual(intervals.get(), [(1, 4)])

def test_sub(self):
# Check removing an interval
intervals = Intervals([(0, 3)])
intervals = union(intervals, Intervals([(5, 7)]))
intervals = set_difference(intervals, Intervals([(2, 6)]))
self.assertEqual(intervals.get(), [(0, 2), (6, 7)])

def test_intersections(self):
# Check adding right of the last interval.
intervals = Intervals([(0, 3)])
intervals = union(intervals, Intervals([(5, 7)]))
intervals = intersections(intervals, Intervals([(2, 6)]))
self.assertEqual(intervals.get(), [(2, 3), (5, 6)])

def test_symmetric_difference(self):
# Check symmetric difference
intervals = Intervals([(0, 3)])
intervals = union(intervals, Intervals([(5, 7)]))
intervals = symmetric_difference(intervals, Intervals([(2, 6)]))
self.assertEqual(intervals.get(), [(0, 2), (3, 5), (6, 7)])

def tuple_wise(iterable, size, step):
"""Tuples up the elements of iterable.

Arguments:
iterable -- source data
size -- size of the destination tuples
step -- step to do in iterable per destination tuple

tuple_wise(s, 3, 1): "s -> (s0,s1,s2), (s1,s2,s3), (s3,s4,s5), ...
tuple_wise(s, 2, 4): "s -> (s0,s1), (s4,s5), (s8,s9), ...
"""
return zip(
*(islice(iterable, start, None, step)
for start in range(size)))

def pairwise(iterable, overlapping):
"""Pairs up the elements of iterable.
overlapping: "s -> (s0,s1), (s2,s3), (s4,s5), ...
not overlapping: "s -> (s0,s1), (s1,s2), (s2,s3), ...
"""
return tuple_wise(iterable, 2, 1 if overlapping else 2)

def starfilter(function, iterable):
"""starfilter <--> filter  ==  starmap <--> map"""
return (item for item in iterable if function(*item))

if __name__ == '__main__':
unittest.main(verbosity=2)


IMHO, having a function which accepts the operation as a string is an atrocity, and I would disallow it in a code review.

Why not have separate functions? That’s much cleaner and makes the code easier to read. If you’re concerned about code duplication, take a look at the strategy pattern. You could for instance pass (named) sets of callbacks rather than strings. Like so:

id = lambda x: x
OP_UNION = (
id, # Only needed for symmetric difference
lambda change: change == (0, 1),
lambda change: change == (1, 0))
…


Or alternatively using namedtuple:

set_op = collections.namedtuple('set_op', ['transform', 'begin', 'end'])

# And here’s a handy helper to avoid lots of repetitive lambdas
def check_for(cond):
return lambda change: change == cond

OP_UNION = set_op(
transform = id,
begin = check_for((0, 1)),
end = check_for((1, 0)))
…


This would also shorten your – otherwise quite nice – _merge_interval_lists implementation (here without comments for brevity, though your comments were very good):

# Removing the default for merge_type – why have this?
def _merge_interval_lists(a, b, merge_type):
b = merge_type.transform(b)

both = list(chain(a, b))
begins = zip(sorted(map(itemgetter(0), both)),
repeat(1))
ends = zip(sorted(map(itemgetter(1), both)),
repeat(-1))

edges = sorted(chain(begins, ends), key=lambda x: (x[0], -x[1]))

counts = list(accumulate(map(itemgetter(1), edges)))
changes = zip(chain([0], counts), counts)
xs = map(itemgetter(0), edges)
xs_and_changes = list(zip(xs, changes))

res_begins = map(itemgetter(0),
starfilter(lambda x, change: merge_type.begin(change),
xs_and_changes))
res_ends = map(itemgetter(0),
starfilter(lambda x, change: merge_type.end(change),
xs_and_changes))

result = pairwise(sorted(chain(res_begins, res_ends)), False)

def length_greater_than_zero(interval):
return interval[0] < interval[1]

return list(filter(length_greater_than_zero, result))

• @Dobi Yes, id = identity. An indispensable tool in the functional programmer’s tool belt. ;-) And nice use of compose in your code! – Konrad Rudolph Jun 28 '13 at 11:45
• Wow, thanks for this awesome feedback! Yes, I knew about the strategy pattern from my c++ projects, but with first class functions here it is even more elegant. I like your solution and use it now: ideone.com/YBNZ2p (I guess with id you meant the identity (id in e.g. Haskell). ^_-) btw: Is it welcomed here to edit the question for posting the improved code? Or should I add an answer to this thread or just do nothing? :) – Tobias Hermann Jun 28 '13 at 11:46
• @Dobi I think it’s preferred not to edit the code in the question – it makes the answers hard to follow for other people. If you think you’ve received substantial improvement, post it as an own answer. If you have follow-up questions about the improved code, make a new question. Just my opinion, though. – Konrad Rudolph Jun 28 '13 at 11:47

First off, Intervals should be a new style class. There's no real reason to ever use old style classes nowadays. If you want to be fancy, you could throw in a set-like ABC, but you should at least inherit from object.

The default argument interval_list should ideally be immutable. Since you only require a sequence there anyway, you may as well just make it a tuple.

For b = map(lambda p: (p[1], p[0]), b) I would use zip instead, though this may just be personal preference.

The use of get seems potentially problematic. For instance, if you accidentally pass a dict into your union functions, it may silently work.

• Great, thanks for the feedback. I use Python 3, so the classes are new style by default, but explicitly inheriting from object surely makes it cleaner. I also changed interval_list to be immutable, so my current version looks like this: ideone.com/JhYhNC How can I elegantly swap the elements of the pairs in list with zip? And do you have a suggestion for the problem with get? – Tobias Hermann Jun 28 '13 at 9:47