Python: simple list merging based on intersections

Question

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.

Answer 1

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

Answer 2

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).

Answer 3

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.

Answer 4

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

Answer 5

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))

Answer 6

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

Answer 7

I needed this function for myself today and I came upon both this SO thread and the duplicate. First off, a huge thank you to all who participated, especially @alexis, whose answer heavily influenced mine, and @Rik Poggi, whose GitHub repository proved extremely useful for testing and benchmarking my version. This is the kind of thread that make SO invaluable!

A few notes:

• I reworked the repo to move the 'deepcopy(lsts)' from the benched execution to the setup. Using timeit.repeat, we have the exact same behavior and safeties, but the deepcopy doesn't impact timings. This explains why the timings reported below are so low.
• I added a profiler script to the repo, so if you add your function, you can profile it easily against any of the lists.
• My fork of @Rik Poggi's repo is here.
• As said above, my version is a rework of @alexis answer. We both derive from the union-find algorithm, but the main difference is that my version updates all the nodes' parent (bin_ref) upon merging, not just the main node. This allows me to avoid the recursive tree-traversal in @alexis' answer (locatebin) and I think it makes the code clearer.
• This is a rather opaque way of doing what we want (unless you're a mathematician, which I'm not), so I tried to make the code as readable as possible. I believe that "readable" -> "maintainable" in this case.
• My version doesn't make too many assumptions about the input data. My requirements was list[tuple[Path]], so I made a version that didn't care too much about the data inside. As long as you pass a Iterable[Iterable[T]] where T is hashable, you'll be fine.

About the results:

• My version seems to perform better with large datasets. It under-performs slightly (2.5x slower than the best, in the worst case scenario) as the datasets become smaller. For mid-size datasets, prefer @Rik Poggi's version. For small-size datasets, prefer @alexis' version.

from itertools import chain
from typing import Iterable, TypeVar

T = TypeVar('T')


def merge_intersections(lists: Iterable[Iterable[T]]) -> list[set[T]]:
    """
    Merges lists based on whether or not they intersect. Returns a list
    of mutually disjoint sets.

    How it works:
    The general idea is to iterate over the sequences and merge them in
    their "intersection-bins". To do so, for each sequence, we look for
    items that were already encountered during previous iteration. If we
    don't find any, this means the current sequence might be disjoint
    from the rest, so we put it in its own bin. If, on the other hand,
    we find previously encountered items, this means one or more bins
    already exist for these items. If more than one bin exists we merge
    them (updating the reference of all contained items). Finally we put
    the current sequence's items in the bin.

    Usage:
    ::
        >>> result = merge_intersections([[1, 2], [2, 4], [7, 8], [3, 2], [4, 5]])
        >>> result = sorted(sorted(s) for s in result)
        [[1, 2, 3, 4, 5], [7, 8]]
    """
    bins: dict[T: set[T]] = dict()
    bin_refs: dict[T: T] = dict()
    for lst in lists:
        if not lst:
            continue

        # Gather the bin refs of all items in the list that we have
        # already seen.
        encountered_items_bin_refs = {
            bin_refs[item]
            for item in lst
            if item in bin_refs
        }
        if len(encountered_items_bin_refs) >= 1:
            # Some of the items in `lst` have already been seen in a
            # previous iteration. They are therefore already attached
            # to a bin. Select any of their corresponding bin ref.
            bin_ref = encountered_items_bin_refs.pop()
            # If the previously-seen items were not all attached to the
            # same bin, their respective bins need to be merged into
            # the selected one.
            if len(encountered_items_bin_refs) > 0:
                to_merge_bins = [bins.pop(ref) for ref in encountered_items_bin_refs]
                bins[bin_ref].update(chain(*to_merge_bins))
                for item in chain(*to_merge_bins):
                    bin_refs[item] = bin_ref
            bins[bin_ref].update(lst)
        else:
            # None of the items in `lst` have already been seen in a
            # previous iteration. Therefore, we can safely pick any
            # item as our new bin ref and create the corresponding bin.
            bin_ref = next(iter(lst))
            bins[bin_ref] = set(lst)
        for item in lst:
            bin_refs[item] = bin_ref
    return list(bins.values())

Benchmark results:

Timing with: >> Niklas << Benchmark
Info: 5000 lists, average size 305, max size 999
(from file: ./lists/timing_1.txt)
0.200 (1x) -- takeshi
0.328 (1.6x) -- alexis
0.473 (2.4x) -- agf
0.746 (3.7x) -- Rik. Poggi
0.776 (3.9x) -- Niks rewrite
0.792 (4x) -- Niklas B.
1.419 (7.1x) -- agf (optimized)
1.480 (7.4x) -- ChessMaster
1.579 (7.9x) -- katrielalex
1.627 (8.2x) -- steabert
1.938 (9.7x) -- robert king
9.273 (46x) -- Sven Marnach
33.779 (169x) -- hochl

Timing with: >> Niklas << Benchmark
Info: 4500 lists, average size 305, max size 999
(from file: ./lists/timing_2.txt)
0.158 (1x) -- takeshi
0.166 (1x) -- agf
0.231 (1.5x) -- Niks rewrite
0.232 (1.5x) -- Rik. Poggi
0.233 (1.5x) -- Niklas B.
0.268 (1.7x) -- alexis
0.408 (2.6x) -- ChessMaster
0.422 (2.7x) -- agf (optimized)
1.408 (8.9x) -- katrielalex
1.483 (9.4x) -- steabert
1.924 (12x) -- robert king
8.622 (55x) -- Sven Marnach
25.952 (164x) -- hochl

Timing with: >> Niklas << Benchmark
Info: 4500 lists, average size 98, max size 999
(from file: ./lists/timing_3.txt)
0.059 (1x) -- takeshi
0.068 (1.1x) -- agf
0.094 (1.6x) -- alexis
0.095 (1.6x) -- Rik. Poggi
0.095 (1.6x) -- Niklas B.
0.097 (1.6x) -- Niks rewrite
0.183 (3.1x) -- ChessMaster
0.192 (3.2x) -- agf (optimized)
0.425 (7.2x) -- katrielalex
0.586 (9.9x) -- robert king
0.787 (13x) -- steabert
2.827 (48x) -- Sven Marnach
9.162 (155x) -- hochl

Timing with: >> Sven << Benchmark
Info: 200 lists, average size 10, max size 10
0.000 (1x) -- alexis
0.001 (1.3x) -- ChessMaster
0.001 (1.8x) -- agf (optimized)
0.001 (1.9x) -- takeshi
0.002 (3.7x) -- steabert
0.002 (3.7x) -- robert king
0.002 (4.2x) -- katrielalex
0.002 (4.6x) -- Rik. Poggi
0.002 (4.8x) -- agf
0.002 (5.1x) -- Niklas B.
0.002 (5.5x) -- hochl
0.003 (6.1x) -- Niks rewrite
0.003 (7.7x) -- Sven Marnach

Timing with: >> Agf << Benchmark
Info: 2000 lists, average size 253, max size 500
0.030 (1x) -- Rik. Poggi
0.035 (1.2x) -- ChessMaster
0.036 (1.2x) -- agf
0.036 (1.2x) -- agf (optimized)
0.037 (1.2x) -- Niks rewrite
0.038 (1.3x) -- Niklas B.
0.060 (2x) -- hochl
0.074 (2.5x) -- takeshi
0.118 (3.9x) -- alexis
0.520 (17x) -- katrielalex
0.528 (18x) -- steabert
0.694 (23x) -- robert king
34.746 (1158x) -- Sven Marnach

Test results:

### Going to test: takeshi ###

test_coverage (__main__.MergeTestCase)
Check coverage original data ... ok
test_disjoint (__main__.MergeTestCase)
Check disjoint-ness of merged results ... ok
test_subset (__main__.MergeTestCase)
Check that every original data is a subset ... ok

----------------------------------------------------------------------
Ran 3 tests in 0.008s

OK

Answer 8

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

Answer 9

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)

Answer 10

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.

Answer 11

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

Answer 12

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.

Answer 13

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)

Answer 14

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)

Answer 15

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!

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

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.

1. 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
   

2. 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
   

3. 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 :)