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How to make sure that a list of generated numbers follow a uniform distribution

I have a list of 150 numbers from 0 to 149. I would like to use a for loop with 150 iterations in order to generate 150 lists of 6 numbers such that,t in each iteration k, the number k is included as well as 5 different random numbers. For example:
S0 = [0, r1, r2, r3, r4, r5] # r1, r2,..., r5 are random numbers between 0 and 150
S1 = [1, r1', r2', r3', r4', r5'] # r1', r2',..., r5' are new random numbers between 0 and 150
...
S149 = [149, r1'', r2'', r3'', r4'', r5'']
In addition, the numbers in each list have to be different and with a minimum distance of 5. This is the code I am using:
import random
import numpy as np
final_list = []
for k in range(150):
S = [k]
for it in range(5):
domain = [ele for ele in range(150) if ele not in S]
d = 0
x = k
while d < 5:
d = np.Infinity
x = random.sample(domain, 1)[0]
for ch in S:
if np.abs(ch - x) < d:
d = np.abs(ch - x)
S.append(x)
final_list.append(S)
Output:
[[0, 149, 32, 52, 39, 126],
[1, 63, 16, 50, 141, 79],
[2, 62, 21, 42, 35, 71],
...
[147, 73, 38, 115, 82, 47],
[148, 5, 78, 115, 140, 43],
[149, 36, 3, 15, 99, 23]]
Now, the code is working but I would like to know if it's possible to force that number of repetitions that each number has through all the iterations is approximately the same. For example, after using the previous code, this plot indicates how many times each number has appeared in the generated lists:
As you can see, there are numbers that have appeared more than 10 times while there are others that have appeared only 2 times. Is it possible to reduce this level of variation so that this plot can be approximated as a uniform distribution? Thanks.

First, I am not sure that your assertion that the current results are not uniformly distributed is necessarily correct. It would seem prudent to me to try and examine the histogram over several repetitions of the process, rather than just one.
I am not a statistician, but when I want to approximate uniform distribution (and assuming that the functions in random provide uniform distribution), what I try to do is to simply accept all results returned by random functions. For that, I need to limit the choices given to these functions ahead of calling them. This is how I would go about your task:
import random
import numpy as np
N = 150
def random_subset(n):
result = []
cands = set(range(N))
for i in range(6):
result.append(n) # Initially, n is the number that must appear in the result
cands -= set(range(n - 4, n + 5)) # Remove candidates less than 5 away
n = random.choice(list(cands)) # Select next number
return result
result = np.array([random_subset(n) for n in range(N)])
print(result)
Simply put, whenever I add a number n to the result set, I take out of the selection candidates, an environment of the proper size, to ensure no number of a distance of less than 5 can be selected in the future.
The code is not optimized (multiple set to list conversions) but it works (as per my uderstanding).

You can force it to be precisely uniform, if you so desire.
Apologies for the mix of globals and locals, this seemed the most readable. You would want to rewrite according to how variable your constants are =)
import random
SIZE = 150
SAMPLES = 5
def get_samples():
pool = list(range(SIZE)) * SAMPLES
random.shuffle(pool)
items = []
for i in range(SIZE):
selection, pool = pool[:SAMPLES], pool[SAMPLES:]
item = [i] + selection
items.append(item)
return items
Then you will have exactly 5 of each (and one more in the leading position, which is a weird data structure).
>>> set(collections.Counter(vv for v in get_samples() for vv in v).values())
{6}
The method above does not guarantee the last 5 numbers are unique, in fact, you would expect ~10/150 to have a duplicate. If that is important, you need to filter your distribution a little more and decide how well you value tight uniformity, duplicates, etc.
If your numbers are approximately what you gave above, you also can patch up the results (fairly) and hope to avoid long search times (not the case for SAMPLES sizes closer to OPTIONS size)
def get_samples():
pool = list(range(SIZE)) * SAMPLES
random.shuffle(pool)
i = 0
while i < len(pool):
if i % SAMPLES == 0:
seen = set()
v = pool[i]
if v in seen: # swap
dst = random.choice(range(SIZE))
pool[dst], pool[i] = pool[i], pool[dst]
i = dst - dst % SAMPLES # Restart from swapped segment
else:
seen.add(v)
i += 1
items = []
for i in range(SIZE):
selection, pool = pool[:SAMPLES], pool[SAMPLES:]
assert len(set(selection)) == SAMPLES, selection
item = [i] + selection
items.append(item)
return items
This will typically take less than 5 passes through to clean up any duplicates, and should leave all arrangements satisfying your conditions equally likely.

Find the value of an index and the n-closest neighbors in a NumPy array

I wondered if anyone could tell me how to find the index of a number and the indexes of the n-closest neighbors in a NumPy array.
For example, In this array, I would like to find the index of the value 87, and its four closest neighbors 86, 88 to the left and 78, 43 to the right.
a = np.random.randint(1,101,15)
array([79, 86, 88, 87, 78, 43, 57])

If you want to change the values from time to time, although this would be expensive for large arrays, should do the trick:
a = np.array([79, 86, 88, 87, 78, 43, 57])
number = 87
n_nearest = 4
index = np.where(a == number)[0][0] # np.where returns (array([3]),)
a = a[max(0, index - n_nearest // 2):index + n_nearest // 2 + 1]
nearests = np.delete(a, n_nearest // 2)
print(nearests)
Output: [86 88 78 43]
First, find the index of the value you get the neighbors (may not work with duplicate values though).
You should do max(0, index - 2) in case of the value you want may be at the beginning of array (position 0 or 1).
Then, delete the number from the result. The rest of it will be the neighbors you want.

I had a go at it, with the caveat that I'm not hugely experienced with python or numpy - only a couple of months
(...so I'd also look for someone else to chip in a much cleaner/simpler/better method!)
from functools import reduce
import operator
a = np.array([5, 10, 15, 12, 88, 86, 5, 87, 1,2,3, 87,1,2,3])
look_for = 87
# find indicies where a == 87:
np.nonzero(a == look_for)
# get as interable
np.nonzero(a == look_for)[0]
# put into list comprehension with the delta in indicies you want and the values
# from above inside 'range' to generate a range of values b/w index-delta:index+delta,
# then wrap it into a list to generate the list from the range iterator:
delta = 2
[list(range(i-delta,i+delta+1)) for i in np.nonzero(a==87)[0]]
# above gives you a list of lists, need to flatten into a single list
reduce(operator.concat, [list(range(i-delta,i+delta+1)) for i in np.nonzero(a==87)[0]])
# want only unique values, so one way is to turn above into a set
set(reduce(operator.concat, [list(range(i-delta,i+delta+1)) for i in np.nonzero(a==87)[0]]))
# the above gives you a set with all the indicies, with only unique values.
# one remaning problem is it still could have values < 0 or > a.size, so
# you'd now want to wrap it all into another list comprehension to get rid of
# any values < 0 or > a.size

How to find highest power of 2 less than n in a list?

I have a list likes
lst = [20, 40, 110]
I want to find the highest power of 2 in the list satisfied as
For the first number, the highest power of 2 will get the first element of the list as input. So the result is 16 (closest to 20)
For the next numbers, it will get the summation of previous result (i.e 16) and current number (.i.e 40) so the closest number will be 32 (closest 40 +16)
So the output what I expect is
lst_pow2 = [16, 32, 128]
This is my current code to find the highest number of a number, but for my problem it should change something because my input is list. Any suggestion? Thanks
# Python3 program to find highest
# power of 2 smaller than or
# equal to n.
import math
def highestPowerof2(n):
p = int(math.log(n, 2));
return int(pow(2, p));
So what I tried but it does not do the summation
lst_power2 = [highestPowerof2(lst[i]) for i in range(len(lst))]

You can perhaps use the following :
lst_power2 = [highestPowerof2(lst[i]+((i>0) and highestPowerof2(lst[i-1]))) for i in range(len(lst))]
instead of
lst_power2 = [highestPowerof2(lst[i]) for i in range(len(lst))]

You may want to modify your approach thus:
Modify your function to take 2 integers. prev_power and curr_num (this was n in your code)
Calculate the power of 2 for the first number and add to a result list
Now pass this number and the next number in the list to your highestPowerof2 function

Use an extra variable that keeps track of the value to be added, and build your logic while iterating.
lst = [20, 40, 110]
import math
def highestPowerof2(n):
p = int(math.log(n, 2)) #you do not need semi colons in python
return int(pow(2, p))
acc = 0 #to keep track of what was the last highest* power
result = []
for n in lst:
result.append(highestPowerof2(n + acc))
acc = result[-1]
print(result)
#Output:
[16, 32, 128]

This question has an accepted answer but I thought this would be a good problem that could also be solved by using a generator. The accepted answer is definitely compact but I though it would be fun to give this solution as well.
lst = [20, 40, 110]
import math
def highestPowerof2(lst):
last = 0
for element in lst:
p = int(math.log(element + last, 2))
last = int(pow(2, p)) # Remember the last value
yield last
lst_power2 = [i for i in highestPowerof2(lst)]
print(lst_power2)

You could use reduce() too:
functools.reduce(lambda res,n:res+[highestPowerof2(n+res[-1])],lst,[0])[1:]
which is short, just the [1:] is ugly at the end
Or as:
functools.reduce(lambda res,n:res+[highestPowerof2(n+(len(res) and res[-1]))],lst,[])
which does not need the slicing, but it is less readable inside.
Full example:
import math,functools
def highestPowerof2(n):
p = int(math.log(n, 2))
return int(pow(2, p))
lst = [20, 40, 110]
print(functools.reduce(lambda res,n:res+[highestPowerof2(n+res[-1])],lst,[0])[1:])
print(functools.reduce(lambda res,n:res+[highestPowerof2(n+(len(res) and res[-1]))],lst,[]))

Faster way of searching than nested loops in python

I'm trying to find the optimal order for least expected cost for an array.
The input is:
input = [[390, 185, 624], [686, 351, 947], [276, 1023, 1024], [199, 148, 250]]
This is an array of four choices, the first number being a cost and the second two being the probability of getting the result, the first ([i][1]) of which is the numerator and the second ([i][2]) is the denominator.
The goal is to find the optimal order of these value/probability pairs that will provide the result at the least total cost.
def answer(input):
from itertools import permutations
length = len(input)
best_total = 999
for combination in permutations(input):
# print combination
total = 0.0
for i in range(0, length):
current_value = 1.0
for j in range(0, i):
current_value = current_value * (1.0 - \
(float(combination[j][1]) / float(combination[j][2])))
total = total + (float(combination[i][0]) * current_value)
if total > best_total:
i = length
# print total
if total <= best_total:
best_total = total
best_combination = combination
answer = map(input.index, best_combination)
return answer
Running:
print answer(input)
should return
[2, 3, 0, 1]
for the given input.
This is obviously an exhaustive search, which becomes very slow very quickly with more than four choices. I've considered binary search trees as the input for those is very similar, however I can't figure out how to implement it.
I've been working on this for four days and can't seem to come up with fast version that works for any input (assuming positive costs and probabilities).
This isn't for homework or anything, just a puzzle I've been trying to figure out.

I would determine the value of each case in the original array, store these values, and then sort the list. This is in python 3 so I don't know if that affects you.
Determining the value of each case in the original array and storing them:
inputA = [[390, 185, 624], [686, 351, 947], [276, 1023, 1024], [199, 148, 250]]
results = []
for idx,val in enumerate(inputA):
results.append((val[0]*val[1]/val[2], idx))
Sorting the list, extracting positions:
l = lambda t:t[1]
print(list(map(l,sorted(results,reverse=True))))
Iterating over the list is O(n), and the sort is O(nlogn). Map/list/print iterates over it again for O(n) so performance should be O(nlogn).

Python: simple list merging based on intersections

Consider there are some lists of integers as:
#--------------------------------------
0 [0,1,3]
1 [1,0,3,4,5,10,...]
2 [2,8]
3 [3,1,0,...]
...
n []
#--------------------------------------
The question is to merge lists having at least one common element. So the results only for the given part will be as follows:
#--------------------------------------
0 [0,1,3,4,5,10,...]
2 [2,8]
#--------------------------------------
What is the most efficient way to do this on large data (elements are just numbers)?
Is tree structure something to think about?
I do the job now by converting lists to sets and iterating for intersections, but it is slow! Furthermore I have a feeling that is so-elementary! In addition, the implementation lacks something (unknown) because some lists remain unmerged sometime! Having said that, if you were proposing self-implementation please be generous and provide a simple sample code [apparently Python is my favoriate :)] or pesudo-code.
Update 1:
Here is the code I was using:
#--------------------------------------
lsts = [[0,1,3],
[1,0,3,4,5,10,11],
[2,8],
[3,1,0,16]];
#--------------------------------------
The function is (buggy!!):
#--------------------------------------
def merge(lsts):
sts = [set(l) for l in lsts]
i = 0
while i < len(sts):
j = i+1
while j < len(sts):
if len(sts[i].intersection(sts[j])) > 0:
sts[i] = sts[i].union(sts[j])
sts.pop(j)
else: j += 1 #---corrected
i += 1
lst = [list(s) for s in sts]
return lst
#--------------------------------------
The result is:
#--------------------------------------
>>> merge(lsts)
>>> [0, 1, 3, 4, 5, 10, 11, 16], [8, 2]]
#--------------------------------------
Update 2:
To my experience the code given by Niklas Baumstark below showed to be a bit faster for the simple cases. Not tested the method given by "Hooked" yet, since it is completely different approach (by the way it seems interesting).
The testing procedure for all of these could be really hard or impossible to be ensured of the results. The real data set I will use is so large and complex, so it is impossible to trace any error just by repeating. That is I need to be 100% satisfied of the reliability of the method before pushing it in its place within a large code as a module. Simply for now Niklas's method is faster and the answer for simple sets is correct of course.
However how can I be sure that it works well for real large data set? Since I will not be able to trace the errors visually!
Update 3:
Note that reliability of the method is much more important than speed for this problem. I will be hopefully able to translate the Python code to Fortran for the maximum performance finally.
Update 4:
There are many interesting points in this post and generously given answers, constructive comments. I would recommend reading all thoroughly. Please accept my appreciation for the development of the question, amazing answers and constructive comments and discussion.

My attempt:
def merge(lsts):
sets = [set(lst) for lst in lsts if lst]
merged = True
while merged:
merged = False
results = []
while sets:
common, rest = sets[0], sets[1:]
sets = []
for x in rest:
if x.isdisjoint(common):
sets.append(x)
else:
merged = True
common |= x
results.append(common)
sets = results
return sets
lst = [[65, 17, 5, 30, 79, 56, 48, 62],
[6, 97, 32, 93, 55, 14, 70, 32],
[75, 37, 83, 34, 9, 19, 14, 64],
[43, 71],
[],
[89, 49, 1, 30, 28, 3, 63],
[35, 21, 68, 94, 57, 94, 9, 3],
[16],
[29, 9, 97, 43],
[17, 63, 24]]
print merge(lst)
Benchmark:
import random
# adapt parameters to your own usage scenario
class_count = 50
class_size = 1000
list_count_per_class = 100
large_list_sizes = list(range(100, 1000))
small_list_sizes = list(range(0, 100))
large_list_probability = 0.5
if False: # change to true to generate the test data file (takes a while)
with open("/tmp/test.txt", "w") as f:
lists = []
classes = [
range(class_size * i, class_size * (i + 1)) for i in range(class_count)
]
for c in classes:
# distribute each class across ~300 lists
for i in xrange(list_count_per_class):
lst = []
if random.random() < large_list_probability:
size = random.choice(large_list_sizes)
else:
size = random.choice(small_list_sizes)
nums = set(c)
for j in xrange(size):
x = random.choice(list(nums))
lst.append(x)
nums.remove(x)
random.shuffle(lst)
lists.append(lst)
random.shuffle(lists)
for lst in lists:
f.write(" ".join(str(x) for x in lst) + "\n")
setup = """
# Niklas'
def merge_niklas(lsts):
sets = [set(lst) for lst in lsts if lst]
merged = 1
while merged:
merged = 0
results = []
while sets:
common, rest = sets[0], sets[1:]
sets = []
for x in rest:
if x.isdisjoint(common):
sets.append(x)
else:
merged = 1
common |= x
results.append(common)
sets = results
return sets
# Rik's
def merge_rik(data):
sets = (set(e) for e in data if e)
results = [next(sets)]
for e_set in sets:
to_update = []
for i, res in enumerate(results):
if not e_set.isdisjoint(res):
to_update.insert(0, i)
if not to_update:
results.append(e_set)
else:
last = results[to_update.pop(-1)]
for i in to_update:
last |= results[i]
del results[i]
last |= e_set
return results
# katrielalex's
def pairs(lst):
i = iter(lst)
first = prev = item = i.next()
for item in i:
yield prev, item
prev = item
yield item, first
import networkx
def merge_katrielalex(lsts):
g = networkx.Graph()
for lst in lsts:
for edge in pairs(lst):
g.add_edge(*edge)
return networkx.connected_components(g)
# agf's (optimized)
from collections import deque
def merge_agf_optimized(lists):
sets = deque(set(lst) for lst in lists if lst)
results = []
disjoint = 0
current = sets.pop()
while True:
merged = False
newsets = deque()
for _ in xrange(disjoint, len(sets)):
this = sets.pop()
if not current.isdisjoint(this):
current.update(this)
merged = True
disjoint = 0
else:
newsets.append(this)
disjoint += 1
if sets:
newsets.extendleft(sets)
if not merged:
results.append(current)
try:
current = newsets.pop()
except IndexError:
break
disjoint = 0
sets = newsets
return results
# agf's (simple)
def merge_agf_simple(lists):
newsets, sets = [set(lst) for lst in lists if lst], []
while len(sets) != len(newsets):
sets, newsets = newsets, []
for aset in sets:
for eachset in newsets:
if not aset.isdisjoint(eachset):
eachset.update(aset)
break
else:
newsets.append(aset)
return newsets
# alexis'
def merge_alexis(data):
bins = range(len(data)) # Initialize each bin[n] == n
nums = dict()
data = [set(m) for m in data] # Convert to sets
for r, row in enumerate(data):
for num in row:
if num not in nums:
# New number: tag it with a pointer to this row's bin
nums[num] = r
continue
else:
dest = locatebin(bins, nums[num])
if dest == r:
continue # already in the same bin
if dest > r:
dest, r = r, dest # always merge into the smallest bin
data[dest].update(data[r])
data[r] = None
# Update our indices to reflect the move
bins[r] = dest
r = dest
# Filter out the empty bins
have = [m for m in data if m]
return have
def locatebin(bins, n):
while bins[n] != n:
n = bins[n]
return n
lsts = []
size = 0
num = 0
max = 0
for line in open("/tmp/test.txt", "r"):
lst = [int(x) for x in line.split()]
size += len(lst)
if len(lst) > max:
max = len(lst)
num += 1
lsts.append(lst)
"""
setup += """
print "%i lists, {class_count} equally distributed classes, average size %i, max size %i" % (num, size/num, max)
""".format(class_count=class_count)
import timeit
print "niklas"
print timeit.timeit("merge_niklas(lsts)", setup=setup, number=3)
print "rik"
print timeit.timeit("merge_rik(lsts)", setup=setup, number=3)
print "katrielalex"
print timeit.timeit("merge_katrielalex(lsts)", setup=setup, number=3)
print "agf (1)"
print timeit.timeit("merge_agf_optimized(lsts)", setup=setup, number=3)
print "agf (2)"
print timeit.timeit("merge_agf_simple(lsts)", setup=setup, number=3)
print "alexis"
print timeit.timeit("merge_alexis(lsts)", setup=setup, number=3)
These timings are obviously dependent on the specific parameters to the benchmark, like number of classes, number of lists, list size, etc. Adapt those parameters to your need to get more helpful results.
Below are some example outputs on my machine for different parameters. They show that all the algorithms have their strength and weaknesses, depending on the kind of input they get:
=====================
# many disjoint classes, large lists
class_count = 50
class_size = 1000
list_count_per_class = 100
large_list_sizes = list(range(100, 1000))
small_list_sizes = list(range(0, 100))
large_list_probability = 0.5
=====================
niklas
5000 lists, 50 equally distributed classes, average size 298, max size 999
4.80084705353
rik
5000 lists, 50 equally distributed classes, average size 298, max size 999
9.49251699448
katrielalex
5000 lists, 50 equally distributed classes, average size 298, max size 999
21.5317108631
agf (1)
5000 lists, 50 equally distributed classes, average size 298, max size 999
8.61671280861
agf (2)
5000 lists, 50 equally distributed classes, average size 298, max size 999
5.18117713928
=> alexis
=> 5000 lists, 50 equally distributed classes, average size 298, max size 999
=> 3.73504281044
===================
# less number of classes, large lists
class_count = 15
class_size = 1000
list_count_per_class = 300
large_list_sizes = list(range(100, 1000))
small_list_sizes = list(range(0, 100))
large_list_probability = 0.5
===================
niklas
4500 lists, 15 equally distributed classes, average size 296, max size 999
1.79993700981
rik
4500 lists, 15 equally distributed classes, average size 296, max size 999
2.58237695694
katrielalex
4500 lists, 15 equally distributed classes, average size 296, max size 999
19.5465381145
agf (1)
4500 lists, 15 equally distributed classes, average size 296, max size 999
2.75445604324
=> agf (2)
=> 4500 lists, 15 equally distributed classes, average size 296, max size 999
=> 1.77850699425
alexis
4500 lists, 15 equally distributed classes, average size 296, max size 999
3.23530197144
===================
# less number of classes, smaller lists
class_count = 15
class_size = 1000
list_count_per_class = 300
large_list_sizes = list(range(100, 1000))
small_list_sizes = list(range(0, 100))
large_list_probability = 0.1
===================
niklas
4500 lists, 15 equally distributed classes, average size 95, max size 997
0.773697137833
rik
4500 lists, 15 equally distributed classes, average size 95, max size 997
1.0523750782
katrielalex
4500 lists, 15 equally distributed classes, average size 95, max size 997
6.04466891289
agf (1)
4500 lists, 15 equally distributed classes, average size 95, max size 997
1.20285701752
=> agf (2)
=> 4500 lists, 15 equally distributed classes, average size 95, max size 997
=> 0.714507102966
alexis
4500 lists, 15 equally distributed classes, average size 95, max size 997
1.1286110878

I tried to summurize everything that's been said and done about this topic in this question and in the duplicate one.
I tried to test and time every solution (all the code here).
Testing
This is the TestCase from the testing module:
class MergeTestCase(unittest.TestCase):
def setUp(self):
with open('./lists/test_list.txt') as f:
self.lsts = json.loads(f.read())
self.merged = self.merge_func(deepcopy(self.lsts))
def test_disjoint(self):
"""Check disjoint-ness of merged results"""
from itertools import combinations
for a,b in combinations(self.merged, 2):
self.assertTrue(a.isdisjoint(b))
def test_coverage(self): # Credit to katrielalex
"""Check coverage original data"""
merged_flat = set()
for s in self.merged:
merged_flat |= s
original_flat = set()
for lst in self.lsts:
original_flat |= set(lst)
self.assertTrue(merged_flat == original_flat)
def test_subset(self): # Credit to WolframH
"""Check that every original data is a subset"""
for lst in self.lsts:
self.assertTrue(any(set(lst) <= e for e in self.merged))
This test is supposing a list of sets as result, so I couldn't test a couple of sulutions that worked with lists.
I couldn't test the following:
katrielalex
steabert
Among the ones I could test, two failed:
-- Going to test: agf (optimized) --
Check disjoint-ness of merged results ... FAIL
-- Going to test: robert king --
Check disjoint-ness of merged results ... FAIL
Timing
The performances are strongly related with the data test employed.
So far three answers tried to time theirs and others solution. Since they used different testing data they had different results.
Niklas benchmark is very twakable. With his banchmark one could do different tests changing some parameters.
I've used the same three sets of parameters he used in his own answer, and I put them in three different files:
filename = './lists/timing_1.txt'
class_count = 50,
class_size = 1000,
list_count_per_class = 100,
large_list_sizes = (100, 1000),
small_list_sizes = (0, 100),
large_list_probability = 0.5,
filename = './lists/timing_2.txt'
class_count = 15,
class_size = 1000,
list_count_per_class = 300,
large_list_sizes = (100, 1000),
small_list_sizes = (0, 100),
large_list_probability = 0.5,
filename = './lists/timing_3.txt'
class_count = 15,
class_size = 1000,
list_count_per_class = 300,
large_list_sizes = (100, 1000),
small_list_sizes = (0, 100),
large_list_probability = 0.1,
This are the results that I got:
From file: timing_1.txt
Timing with: >> Niklas << Benchmark
Info: 5000 lists, average size 305, max size 999
Timing Results:
10.434 -- alexis
11.476 -- agf
11.555 -- Niklas B.
13.622 -- Rik. Poggi
14.016 -- agf (optimized)
14.057 -- ChessMaster
20.208 -- katrielalex
21.697 -- steabert
25.101 -- robert king
76.870 -- Sven Marnach
133.399 -- hochl
From file: timing_2.txt
Timing with: >> Niklas << Benchmark
Info: 4500 lists, average size 305, max size 999
Timing Results:
8.247 -- Niklas B.
8.286 -- agf
8.637 -- Rik. Poggi
8.967 -- alexis
9.090 -- ChessMaster
9.091 -- agf (optimized)
18.186 -- katrielalex
19.543 -- steabert
22.852 -- robert king
70.486 -- Sven Marnach
104.405 -- hochl
From file: timing_3.txt
Timing with: >> Niklas << Benchmark
Info: 4500 lists, average size 98, max size 999
Timing Results:
2.746 -- agf
2.850 -- Niklas B.
2.887 -- Rik. Poggi
2.972 -- alexis
3.077 -- ChessMaster
3.174 -- agf (optimized)
5.811 -- katrielalex
7.208 -- robert king
9.193 -- steabert
23.536 -- Sven Marnach
37.436 -- hochl
With Sven's testing data I got the following results:
Timing with: >> Sven << Benchmark
Info: 200 lists, average size 10, max size 10
Timing Results:
2.053 -- alexis
2.199 -- ChessMaster
2.410 -- agf (optimized)
3.394 -- agf
3.398 -- Rik. Poggi
3.640 -- robert king
3.719 -- steabert
3.776 -- Niklas B.
3.888 -- hochl
4.610 -- Sven Marnach
5.018 -- katrielalex
And finally with Agf's benchmark I got:
Timing with: >> Agf << Benchmark
Info: 2000 lists, average size 246, max size 500
Timing Results:
3.446 -- Rik. Poggi
3.500 -- ChessMaster
3.520 -- agf (optimized)
3.527 -- Niklas B.
3.527 -- agf
3.902 -- hochl
5.080 -- alexis
15.997 -- steabert
16.422 -- katrielalex
18.317 -- robert king
1257.152 -- Sven Marnach
As I said at the beginning all the code is available at this git repository. All the merging functions are in a file called core.py, every function there with its name ending with _merge will be auto loaded during the tests, so it shouldn't be hard to add/test/improve your own solution.
Let me also know if there's something wrong, it's been a lot of coding and I could use a couple of fresh eyes :)

Using Matrix Manipulations
Let me preface this answer with the following comment:
THIS IS THE WRONG WAY TO DO THIS. IT IS PRONE TO NUMERICAL INSTABILITY AND IS MUCH SLOWER THAN THE OTHER METHODS PRESENTED, USE AT YOUR OWN RISK.
That being said, I couldn't resist solving the problem from a dynamical point of view (and I hope you'll get a fresh perspective on the problem). In theory this should work all the time, but eigenvalue calculations can often fail. The idea is to think of your list as a flow from rows to columns. If two rows share a common value there is a connecting flow between them. If we were to think of these flows as water, we would see that the flows cluster into little pools when they there is a connecting path between them. For simplicity, I'm going to use a smaller set, though it works with your data set as well:
from numpy import where, newaxis
from scipy import linalg, array, zeros
X = [[0,1,3],[2],[3,1]]
We need to convert the data into a flow graph. If row i flows into value j we put it in the matrix. Here we have 3 rows and 4 unique values:
A = zeros((4,len(X)), dtype=float)
for i,row in enumerate(X):
for val in row: A[val,i] = 1
In general, you'll need to change the 4 to capture the number of unique values you have. If the set is a list of integers starting from 0 as we have, you can simply make this the largest number. We now perform an eigenvalue decomposition. A SVD to be exact, since our matrix is not square.
S = linalg.svd(A)
We want to keep only the 3x3 portion of this answer, since it will represent the flow of the pools. In fact we only want the absolute values of this matrix; we only care if there is a flow in this cluster space.
M = abs(S[2])
We can think of this matrix M as a Markov matrix and make it explicit by row normalizing. Once we have this we compute the (left) eigenvalue decomp. of this matrix.
M /= M.sum(axis=1)[:,newaxis]
U,V = linalg.eig(M,left=True, right=False)
V = abs(V)
Now a disconnected (non-ergodic) Markov matrix has the nice property that, for each non-connected cluster, there is a eigenvalue of unity. The eigenvectors associated with these unity values are the ones we want:
idx = where(U > .999)[0]
C = V.T[idx] > 0
I have to use .999 due to the aforementioned numerical instability. At this point, we are done! Each independent cluster can now pull the corresponding rows out:
for cluster in C:
print where(A[:,cluster].sum(axis=1))[0]
Which gives, as intended:
[0 1 3]
[2]
Change X to your lst and you'll get: [ 0 1 3 4 5 10 11 16] [2 8].
Addendum
Why might this be useful? I don't know where your underlying data comes from, but what happens when the connections are not absolute? Say row 1 has entry 3 80% of the time - how would you generalize the problem? The flow method above would work just fine, and would be completely parametrized by that .999 value, the further away from unity it is, the looser the association.
Visual Representation
Since a picture is worth 1K words, here are the plots of the matrices A and V for my example and your lst respectively. Notice how in V splits into two clusters (it is a block-diagonal matrix with two blocks after permutation), since for each example there were only two unique lists!
Faster Implementation
In hindsight, I realized that you can skip the SVD step and compute only a single decomp:
M = dot(A.T,A)
M /= M.sum(axis=1)[:,newaxis]
U,V = linalg.eig(M,left=True, right=False)
The advantage with this method (besides speed) is that M is now symmetric, hence the computation can be faster and more accurate (no imaginary values to worry about).

EDIT: OK, the other questions has been closed, posting here.
Nice question! It's much simpler if you think of it as a connected-components problem in a graph. The following code uses the excellent networkx graph library and the pairs function from this question.
def pairs(lst):
i = iter(lst)
first = prev = item = i.next()
for item in i:
yield prev, item
prev = item
yield item, first
lists = [[1,2,3],[3,5,6],[8,9,10],[11,12,13]]
import networkx
g = networkx.Graph()
for sub_list in lists:
for edge in pairs(sub_list):
g.add_edge(*edge)
networkx.connected_components(g)
[[1, 2, 3, 5, 6], [8, 9, 10], [11, 12, 13]]
Explanation
We create a new (empty) graph g. For each sub-list in lists, consider its elements as nodes of the graph and add an edge between them. (Since we only care about connectedness, we don't need to add all the edges -- only adjacent ones!) Note that add_edge takes two objects, treats them as nodes (and adds them if they aren't already there), and adds an edge between them.
Then, we just find the connected components of the graph -- a solved problem! -- and output them as our intersecting sets.

Here's my answer. I haven't checked it against today's batch of answers.
The intersection-based algorithms are O(N^2) since they check each new set against all the existing ones, so I used an approach that indexes each number and runs on close to O(N) (if we accept that dictionary lookups are O(1)). Then I ran the benchmarks and felt like a complete idiot because it ran slower, but on closer inspection it turned out that the test data ends up with only a handful of distinct result sets, so the quadratic algorithms don't have a lot work to do. Test it with more than 10-15 distinct bins and my algorithm is much faster. Try test data with more than 50 distinct bins, and it is enormously faster.
(Edit: There was also a problem with the way the benchmark is run, but I was wrong in my diagnosis. I altered my code to work with the way the repeated tests are run).
def mergelists5(data):
"""Check each number in our arrays only once, merging when we find
a number we have seen before.
"""
bins = range(len(data)) # Initialize each bin[n] == n
nums = dict()
data = [set(m) for m in data ] # Convert to sets
for r, row in enumerate(data):
for num in row:
if num not in nums:
# New number: tag it with a pointer to this row's bin
nums[num] = r
continue
else:
dest = locatebin(bins, nums[num])
if dest == r:
continue # already in the same bin
if dest > r:
dest, r = r, dest # always merge into the smallest bin
data[dest].update(data[r])
data[r] = None
# Update our indices to reflect the move
bins[r] = dest
r = dest
# Filter out the empty bins
have = [ m for m in data if m ]
print len(have), "groups in result"
return have
def locatebin(bins, n):
"""
Find the bin where list n has ended up: Follow bin references until
we find a bin that has not moved.
"""
while bins[n] != n:
n = bins[n]
return n

This new function only does the minimum necessary number of disjointness tests, something the other similar solutions fail to do. It also uses a deque to avoid as many linear time operations as possible, like list slicing and deletion from early in the list.
from collections import deque
def merge(lists):
sets = deque(set(lst) for lst in lists if lst)
results = []
disjoint = 0
current = sets.pop()
while True:
merged = False
newsets = deque()
for _ in xrange(disjoint, len(sets)):
this = sets.pop()
if not current.isdisjoint(this):
current.update(this)
merged = True
disjoint = 0
else:
newsets.append(this)
disjoint += 1
if sets:
newsets.extendleft(sets)
if not merged:
results.append(current)
try:
current = newsets.pop()
except IndexError:
break
disjoint = 0
sets = newsets
return results
The less overlap between the sets in a given set of data, the better this will do compared to the other functions.
Here is an example case. If you have 4 sets, you need to compare:
1, 2
1, 3
1, 4
2, 3
2, 4
3, 4
If 1 overlaps with 3, then 2 needs to be re-tested to see if it now overlaps with 1, in order to safely skip testing 2 against 3.
There are two ways to deal with this. The first is to restart the testing of set 1 against the other sets after each overlap and merge. The second is to continue with the testing by comparing 1 with 4, then going back and re-testing. The latter results in fewer disjointness tests, as more merges happen in a single pass, so on the re-test pass, there are fewer sets left to test against.
The problem is to track which sets have to be re-tested. In the above example, 1 needs to be re-tested against 2 but not against 4, since 1 was already in its current state before 4 was tested the first time.
The disjoint counter allows this to be tracked.
My answer doesn't help with the main problem of finding an improved algorithm for recoding into FORTRAN; it is just what appears to me to be the simplest and most elegant way to implement the algorithm in Python.
According to my testing (or the test in the accepted answer), it's slightly (up to 10%) faster than the next fastest solution.
def merge0(lists):
newsets, sets = [set(lst) for lst in lists if lst], []
while len(sets) != len(newsets):
sets, newsets = newsets, []
for aset in sets:
for eachset in newsets:
if not aset.isdisjoint(eachset):
eachset.update(aset)
break
else:
newsets.append(aset)
return newsets
No need for the un-Pythonic counters (i, range) or complicated mutation (del, pop, insert) used in the other implementations. It uses only simple iteration, merges overlapping sets in the simplest manner, and builds a single new list on each pass through the data.
My (faster and simpler) version of the test code:
import random
tenk = range(10000)
lsts = [random.sample(tenk, random.randint(0, 500)) for _ in range(2000)]
setup = """
def merge0(lists):
newsets, sets = [set(lst) for lst in lists if lst], []
while len(sets) != len(newsets):
sets, newsets = newsets, []
for aset in sets:
for eachset in newsets:
if not aset.isdisjoint(eachset):
eachset.update(aset)
break
else:
newsets.append(aset)
return newsets
def merge1(lsts):
sets = [set(lst) for lst in lsts if lst]
merged = 1
while merged:
merged = 0
results = []
while sets:
common, rest = sets[0], sets[1:]
sets = []
for x in rest:
if x.isdisjoint(common):
sets.append(x)
else:
merged = 1
common |= x
results.append(common)
sets = results
return sets
lsts = """ + repr(lsts)
import timeit
print timeit.timeit("merge0(lsts)", setup=setup, number=10)
print timeit.timeit("merge1(lsts)", setup=setup, number=10)

Here's an implementation using a disjoint-set data structure (specifically a disjoint forest), thanks to comingstorm's hint at merging sets which have even one element in common. I'm using path compression for a slight (~5%) speed improvement; it's not entirely necessary (and it prevents find being tail recursive, which could slow things down). Note that I'm using a dict to represent the disjoint forest; given that the data are ints, an array would also work although it might not be much faster.
def merge(data):
parents = {}
def find(i):
j = parents.get(i, i)
if j == i:
return i
k = find(j)
if k != j:
parents[i] = k
return k
for l in filter(None, data):
parents.update(dict.fromkeys(map(find, l), find(l[0])))
merged = {}
for k, v in parents.items():
merged.setdefault(find(v), []).append(k)
return merged.values()
This approach is comparable to the other best algorithms on Rik's benchmarks.

This would be my updated approach:
def merge(data):
sets = (set(e) for e in data if e)
results = [next(sets)]
for e_set in sets:
to_update = []
for i,res in enumerate(results):
if not e_set.isdisjoint(res):
to_update.insert(0,i)
if not to_update:
results.append(e_set)
else:
last = results[to_update.pop(-1)]
for i in to_update:
last |= results[i]
del results[i]
last |= e_set
return results
Note: During the merging empty lists will be removed.
Update: Reliability.
You need two tests for a 100% reliabilty of success:
Check that all the resulting sets are mutually disjointed:
merged = [{0, 1, 3, 4, 5, 10, 11, 16}, {8, 2}, {8}]
from itertools import combinations
for a,b in combinations(merged,2):
if not a.isdisjoint(b):
raise Exception(a,b) # just an example
Check that the merged set cover the original data. (as suggested by katrielalex)
I think this will take some time, but maybe it'll be worth it if you want to be 100% sure.

Just for fun...
def merge(mylists):
results, sets = [], [set(lst) for lst in mylists if lst]
upd, isd, pop = set.update, set.isdisjoint, sets.pop
while sets:
if not [upd(sets[0],pop(i)) for i in xrange(len(sets)-1,0,-1) if not isd(sets[0],sets[i])]:
results.append(pop(0))
return results
and my rewrite of the best answer
def merge(lsts):
sets = map(set,lsts)
results = []
while sets:
first, rest = sets[0], sets[1:]
merged = False
sets = []
for s in rest:
if s and s.isdisjoint(first):
sets.append(s)
else:
first |= s
merged = True
if merged: sets.append(first)
else: results.append(first)
return results

Here's a function (Python 3.1) to check if the result of a merge function is OK. It checks:
Are the result sets disjoint? (number of elements of union == sum of numbers of elements)
Are the elements of the result sets the same as of the input lists?
Is every input list a subset of a result set?
Is every result set minimal, i.e. is it impossible to split it into two smaller sets?
It does not check if there are empty result sets - I don't know if you want them or not...
.
from itertools import chain
def check(lsts, result):
lsts = [set(s) for s in lsts]
all_items = set(chain(*lsts))
all_result_items = set(chain(*result))
num_result_items = sum(len(s) for s in result)
if num_result_items != len(all_result_items):
print("Error: result sets overlap!")
print(num_result_items, len(all_result_items))
print(sorted(map(len, result)), sorted(map(len, lsts)))
if all_items != all_result_items:
print("Error: result doesn't match input lists!")
if not all(any(set(s).issubset(t) for t in result) for s in lst):
print("Error: not all input lists are contained in a result set!")
seen = set()
todo = list(filter(bool, lsts))
done = False
while not done:
deletes = []
for i, s in enumerate(todo): # intersection with seen, or with unseen result set, is OK
if not s.isdisjoint(seen) or any(t.isdisjoint(seen) for t in result if not s.isdisjoint(t)):
seen.update(s)
deletes.append(i)
for i in reversed(deletes):
del todo[i]
done = not deletes
if todo:
print("Error: A result set should be split into two or more parts!")
print(todo)

lists = [[1,2,3],[3,5,6],[8,9,10],[11,12,13]]
import networkx as nx
g = nx.Graph()
for sub_list in lists:
for i in range(1,len(sub_list)):
g.add_edge(sub_list[0],sub_list[i])
print nx.connected_components(g)
#[[1, 2, 3, 5, 6], [8, 9, 10], [11, 12, 13]]
Performance:
5000 lists, 5 classes, average size 74, max size 1000
15.2264976415
Performance of merge1:
print timeit.timeit("merge1(lsts)", setup=setup, number=10)
5000 lists, 5 classes, average size 74, max size 1000
1.26998780571
So it is 11x slower than the fastest.. but the code is much more simple and readable!

This is slower than the solution offered by Niklas (I got 3.9s on the test.txt instead of 0.5s for his solution), but yields the same result and might be easier to implement in e.g. Fortran, since it doesn't use sets, only sorting of the total amount of elements and then a single run through all of them.
It returns a list with the ids of the merged lists, so also keeps track of empty lists, they stay unmerged.
def merge(lsts):
# this is an index list that stores the joined id for each list
joined = range(len(lsts))
# create an ordered list with indices
indexed_list = sorted((el,index) for index,lst in enumerate(lsts) for el in lst)
# loop throught the ordered list, and if two elements are the same and
# the lists are not yet joined, alter the list with joined id
el_0,idx_0 = None,None
for el,idx in indexed_list:
if el == el_0 and joined[idx] != joined[idx_0]:
old = joined[idx]
rep = joined[idx_0]
joined = [rep if id == old else id for id in joined]
el_0, idx_0 = el, idx
return joined

Firstly I'm not exactly sure if the benchmarks are fair:
Adding the following code to the start of my function:
c = Counter(chain(*lists))
print c[1]
"88"
This means that out of all the values in all the lists, there are only 88 distinct values. Usually in the real world duplicates are rare, and you would expect a lot more distinct values. (of course i don't know where your data from so can't make assumptions).
Because Duplicates are more common, it means sets are less likely to be disjoint. This means the set.isdisjoint() method will be much faster because only after a few tests it will find that the sets aren't disjoint.
Having said all that, I do believe the methods presented that use disjoint are the fastest anyway, but i'm just saying, instead of being 20x faster maybe they should only be 10x faster than the other methods with different benchmark testing.
Anyway, i Thought I would try a slightly different technique to solve this, however the merge sorting was too slow, this method is about 20x slower than the two fastest methods using the benchmarking:
I thought I would order everything
import heapq
from itertools import chain
def merge6(lists):
for l in lists:
l.sort()
one_list = heapq.merge(*[zip(l,[i]*len(l)) for i,l in enumerate(lists)]) #iterating through one_list takes 25 seconds!!
previous = one_list.next()
d = {i:i for i in range(len(lists))}
for current in one_list:
if current[0]==previous[0]:
d[current[1]] = d[previous[1]]
previous=current
groups=[[] for i in range(len(lists))]
for k in d:
groups[d[k]].append(lists[k]) #add a each list to its group
return [set(chain(*g)) for g in groups if g] #since each subroup in each g is sorted, it would be faster to merge these subgroups removing duplicates along the way.
lists = [[1,2,3],[3,5,6],[8,9,10],[11,12,13]]
print merge6(lists)
"[set([1, 2, 3, 5, 6]), set([8, 9, 10]), set([11, 12, 13])]""
import timeit
print timeit.timeit("merge1(lsts)", setup=setup, number=10)
print timeit.timeit("merge4(lsts)", setup=setup, number=10)
print timeit.timeit("merge6(lsts)", setup=setup, number=10)
5000 lists, 5 classes, average size 74, max size 1000
1.26732238315
5000 lists, 5 classes, average size 74, max size 1000
1.16062907437
5000 lists, 5 classes, average size 74, max size 1000
30.7257182826

I found #Niklas B.'s answer really helpful... but it took me a while to read through it and understand the logic. This is a re-write of exactly the same code with new variable names and more explanation... to help the other N00bs out there!
def mergeUntilOnlyDisjointSetsRemain(_listsOfLists):
"""Function for merging lists until there are only disjoint sets"""
"""Imagine this algorithm as if it were processing train cars filled with
integers. It takes the first car of the train, separates it from the rest,
and then compares the first car to each subsequent car.
Start by renaming the first car to 'common'
If the two train cars have a common integer, you merge the two cars into
common, and continue down the line until you reach the end of the train.
Once you reach the end of the train, place the common car in results, (which
is essentially a collection of train cars that have already been compared
to all other cars).
You can exit the loop as soon as you get to the end of the train without
merging any of the cars. This is controlled by the continueMerge variable.
This variable is only reset to True after a merge operation.
"""
# Start by creating a trainCar(i.e. a set) from each list in our listOfLists
freightTrain = [set(trainCar) for trainCar in _listsOfLists if trainCar]
# This continueMerge means that we have not yet compared all cars in the train.
continueMerge = True
while continueMerge:
# Reset the while loop trigger.
continueMerge = False
# Create a fresh empty list of cars that have already been cross checked
crossCheckedCars = []
# While there are still elements in freightTrain
while freightTrain:
# Split the freightTrain up, into first car vs all the remaining cars
commonFirstTrainCar = freightTrain[0]
remainingCars = freightTrain[1:]
# The freightTrain is now empty
freightTrain = []
# Iterate over all the remaining traincars
for currentTrainCar in remainingCars:
# If there are not any common integers with the first car...
if currentTrainCar.isdisjoint(commonFirstTrainCar):
# Add each train car back onto the freightTrain
freightTrain.append(currentTrainCar)
# But if they share a common integer...
else:
# Trigger the reset switch to continueMerging cars
continueMerge = True
# and Join he cars together
commonFirstTrainCar |= currentTrainCar
# Once we have checked commonFirstTrainCar, remove it from the
# freightTrain and place it in crossCheckedCars
crossCheckedCars.append(commonFirstTrainCar)
# Now we have compared the first car to all subsequent cars
# (... but we aren't finished because the 5th and 7th cars might have
# had a common integer with each other... but only 1st and 5th cars
# shared an integer the first time around... so the 1st and 5th cars
# were merged, but the 7th car is still alone!)
# Reset the system by creating a new freightTrain
freightTrain = crossCheckedCars
# Post-process the freight train to turn it into lists instead of sets
listsForReturnTripHome = []
for processedTraincar in freightTrain:
listsForReturnTripHome.append(list(processedTraincar))
return listsForReturnTripHome

My solution, works well on small lists and is quite readable without dependencies.
def merge_list(starting_list):
final_list = []
for i,v in enumerate(starting_list[:-1]):
if set(v)&set(starting_list[i+1]):
starting_list[i+1].extend(list(set(v) - set(starting_list[i+1])))
else:
final_list.append(v)
final_list.append(starting_list[-1])
return final_list
Benchmarking it:
lists = [[1,2,3],[3,5,6],[8,9,10],[11,12,13]]
%timeit merge_list(lists)
100000 loops, best of 3: 4.9 µs per loop

This can be solved in O(n) by using the union-find algorithm. Given the first two rows of your data, edges to use in the union-find are the following pairs:
(0,1),(1,3),(1,0),(0,3),(3,4),(4,5),(5,10)

Use flag to ensure you get the final mutual exclusive results
def merge(lists):
while(1):
flag=0
for i in range(0,len(lists)):
for j in range(i+1,len(lists)):
if len(intersection(lists[i],lists[j]))!=0:
lists[i]=union(lists[i],lists[j])
lists.remove(lists[j])
flag+=1
break
if flag==0:
break
return lists

from itertools import combinations
def merge(elements_list):
d = {index: set(elements) for index, elements in enumerate(elements_list)}
while any(not set.isdisjoint(d[i], d[j]) for i, j in combinations(d.keys(), 2)):
merged = set()
for i, j in combinations(d.keys(), 2):
if not set.isdisjoint(d[i], d[j]):
d[i] = set.union(d[i], d[j])
merged.add(j)
for k in merged:
d.pop(k)
return [v for v in d.values() if v]
lst = [[65, 17, 5, 30, 79, 56, 48, 62],
[6, 97, 32, 93, 55, 14, 70, 32],
[75, 37, 83, 34, 9, 19, 14, 64],
[43, 71],
[],
[89, 49, 1, 30, 28, 3, 63],
[35, 21, 68, 94, 57, 94, 9, 3],
[16],
[29, 9, 97, 43],
[17, 63, 24]]
print(merge(lst))