I don't know about the implementation, but as an exercise, the set union is O(N) w.r.t. len(s)+len(t), and stays the same regardless of the degree of coincidence of s and t.

Check TimeComplexity for more details

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import timeit

# Time with varying length of s and t

setup = """ from random import random s = set(random() for i in xrange(%d)) t = set(random() for i in xrange(%d)) """

cmd = "u = s|t"

for len_s in xrange(1,101,10):

- for len_t in xrange(1,101,10):
- ans = timeit.Timer(cmd, setup=setup % (len_s, len_t)).timeit(200) print '%3d' % (len_s+len_t), '%6.4f' % (ans*1e4/(len_s+len_t))

# Time with varying % of elements in both sets

setup = """ from random import random n = 2000 k = int(n * %4.3f) u = set(random() for i in xrange(k)) s = set(random() for i in xrange(n-k)) | u t = set(random() for i in xrange(n-k)) | u """

cmd = """v = s|t"""

for i in xrange(0,101,5):

- k = 0.01*i ans = timeit.Timer(cmd, setup=setup % k).timeit(100) print '%3.2f' % k, '%6.1f' % (ans*1e4)