def count_ways_to_obtain_largest_subpopulation(n, m):
"""Return dict of number of ways to obtain largest subpopulation.
Inputs
:n: total number (e.g., total number of highest scoring results)
:m: number of non-negative integers to sum to n (e.g., number of
workers)
Output
:ways: dictionary whose keys are the maximum value of a multiset
and whose values are the sum of each distinct ordering of
results, corresponding to an arrangement of a multiset,
computed over all arrangements of all multisets sharing
a maximum value.
Implementation
Although Multiset.uniq_msets() returns tuples in lexicographical
order, this implementation would function regardless of order.
"""
mset = Multiset(n)
ways = defaultdict(int)
for grp in mset.uniq_msets(n, m):
ways[max(grp)] += (mset.multinomial_coeff(grp) *
mset.number_of_arrangements(grp))
return ways
def run_example():
"""Demonstrate sample outputs.
::
>> run_example() # ADD > to re-activate doctest (runs in ~20 sec)
Short example, involving 108 multisets
Printing the probability of missing 1 or more results from the top 20
results, given 4 workers, as a function of the number of top results
requested per worker.
Probability of 5 or more of top 20 from one of 4 sets is 1.0000e+00.
Probability of 6 or more of top 20 from one of 4 sets is 9.8933e-01.
Probability of 7 or more of top 20 from one of 4 sets is 7.5516e-01.
Probability of 8 or more of top 20 from one of 4 sets is 3.9874e-01.
Probability of 9 or more of top 20 from one of 4 sets is 1.6346e-01.
Probability of 10 or more of top 20 from one of 4 sets is 5.5457e-02.
Probability of 11 or more of top 20 from one of 4 sets is 1.5769e-02.
Probability of 12 or more of top 20 from one of 4 sets is 3.7416e-03.
Probability of 13 or more of top 20 from one of 4 sets is 7.3482e-04.
Probability of 14 or more of top 20 from one of 4 sets is 1.1805e-04.
Probability of 15 or more of top 20 from one of 4 sets is 1.5252e-05.
Probability of 16 or more of top 20 from one of 4 sets is 1.5461e-06.
Probability of 17 or more of top 20 from one of 4 sets is 1.1842e-07.
Probability of 18 or more of top 20 from one of 4 sets is 6.4429e-09.
Probability of 19 or more of top 20 from one of 4 sets is 2.2192e-10.
Probability of 20 or more of top 20 from one of 4 sets is 3.6380e-12.
computing longer example, involving 6292069 multisets ...
Longer example
Chance of omitting documents from top 100 when returning 20 results
from each of 10 workers is 8.0721981476e-03
"""
mset = Multiset()
n1, m1 = 20, 4
print """Short example, involving %d multisets
Printing the probability of missing 1 or more results from the top %d
results, given %d workers, as a function of the number of top results
requested per worker.""" % (mset.num_uniq_msets(total=n1, length=m1),
n1, m1)
print_cumulative_prob(n=n1, m=m1)
n2, m2 = 100, 10
num_docs = 20
num_ms = mset.num_uniq_msets(total=n2, length=m2)
print 'computing longer example, involving %d multisets ...' % num_ms
# add one because result is omitted only when set size exceeds request
for stats in compute_probabilities(n=n2, m=m2, t=num_docs + 1):
if stats['count'] == num_docs + 1:
print ' '.join(['Longer example\nChance of omitting documents',
'from top %d when returning %d results\nfrom each of',
'%d workers is %0.10e']) % (n2, num_docs, m2, stats['p'])
def compute_probabilities(n, m, t=()):
"""Compute probability that a result is missed.
Inputs
:n: total number (e.g., total number of highest scoring results)
:m: number of non-negative integers to sum to n (e.g., number of
workers, each returning an integer number of results)
:t: optional threshold to short-circuit computation
* integer t is the maximum number of results to return per worker
Output
:stats: dict containing fields:
* count is the the number of results returned per worker
* n is the total number of highest scoring results
* m is the number of workers
* p is the cumulative probability that a result is missed
"""
if not is_nonneg_int(t):
t = ()
numerator = m ** n
denominator = float(numerator)
stats = {'n': n, 'm': m, 'count': 0, 'p': 0}
mset = Multiset(n)
for (cnt, ways) in mset.num_ways(n, m):
stats['count'] = cnt
stats['p'] = numerator / denominator
if cnt < t:
yield stats.copy()
elif cnt == t:
yield stats.copy()
raise StopIteration
else:
raise StopIteration
numerator -= ways
def setUp(self):
self.mset = Multiset()
