Contents
Overview
Last time, in part two, we built and tested our set container class, ModSet(), with a variety of methods that will prove useful when we construct the power set. Methods included copying, nesting/unnesting, filtering to keep only unique members as well as set-union and set-subtraction operations. You will see that most of these will be used, either implicitly or explicitly, in both Binary Mask and Recursive methods of generating power sets.
Preliminaries
Our ModSet() class definition is a must (obviously); let’s run its script so it will be ready and waiting for us when it comes time to make an instance. You can download the script from github here: modset.py
%run modset.py
When we are ready to build our power set, we’ll measure the time it takes each of the two routines to complete, so we can compare their performance. As a preliminary, let’s import the time module so it stands at ready.
import time
Of course we will need an example source set to operate on. Let’s define it now and be done with it.
clumsyPrez = ModSet(set([ 6.1, 2021, "fight like hell", "we love you",('impeachment','#2')]))
Binary Mask power set generator
In this iterative-based approach, we will populate the power set by applying binary masks to extract out member subsets, one-by-one, from the source set in ModSet().val. If you recall, a verbal description of power set generation using a binary mask we discussed way back in part one; you can find it here. Below, we first provide an overview of our binary mask power set generator routine, followed by the routine itself, with heavy commenting, concluded with a detailed walk-through after that.
Overview
The method, named .powerSet_bin(), contains 19 lines and calls .pushDown() and .union() sibling methods. Though we could use NumPy here to make mask generation and member extraction more compact, we did not want to confound the comparison (to our recursive approach to follow) by incorporating functionality of an external module. Already, the code has been somewhat compressed by use of generator expressions.
Flow of control
The routine below can be divided into two main blocks: variable initialization, and the run-time loop that builds each binary mask for extracting the corresponding subset on each iteration. The loop populates the power set member-by-member and terminates when all $2^n$ members have been added. Because indexing is crucial in this approach, you’ll notice heavy use of range() and list() data-types. As you’ll see later, our recursive approach requires no such crutch.
class ModSet():
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# Generate powerset via direct, binary mask approach
def powerSet_bin(self):
## Initialize local variables ##
S = list(self.val) # convert to list for indexing
setSize = len(self.val) # count number of members in source set
psetSize = pow(2, setSize) # calculate the number of elements in the power set
lastIndex = setSize - 1 # index value of last member
setIndices = range(0, setSize) # make indices list for source set
psetIndices = range(0, psetSize) # make indices list for power set to be built
bMasks = [[False for s in setIndices] for p in psetIndices] # Initialize binary mask
pSet = ModSet(set()) # initialize power set as empty ModSet() instance
pSet.pushDown(1) # and nest it down one level for later joining
## Populate powerset with each subset, one at a time ##
for i in psetIndices: # loop through each member of power set
## Generate binary mask for subset i of power set ##
val = i # assign current pSet index as current "value" of mask
for j in setIndices: # loop through each bit-digit of mask
if (val >= pow(2, lastIndex - j)): # if mask value >= value of current bit,
bMasks[i][lastIndex - j] = True # then set corresp. mask bit to "true"
val -= pow(2, lastIndex - j) # subtract value of current bit from
# mask value to determine next bit-digit
## Form subset i of power set ##
# Use generator expression for compactness
dummySet = ModSet(set([S[k] for k in setIndices if bMasks[i][k] == True]))
dummySet.pushDown(1) # nest ModSet instance down one level for union join
pSet.union(dummySet) # include new subset in power set
return pSet, bMasks # return complete power set and binary masks as output
Aside: initializing the mask
To have each and every element alterable in our list of binary masks, bMasks, we need to individually assign each and every element within the nested list. If you attempt to initialize bMasks like this, bMasks = [[False]*len(setIndices)]*len(psetIndices)], (as we did on our first attempt!), you will find that you cannot change individual elements of bMasks later on. That is
bMasks =[[False]*3]*5
bMasks[3][1] = True
bMasks
changes to True all row-entries in the second column, rather than just the fourth, since all point to the same instance of True. However, if you initialize bMasks using a list completion, the nested list assembles with a hash to a unique instance for each and every element therein
bMasks = [[False for j in range(0,3)] for i in range(0, 5)]
bMasks[3][1] = True
bMasks
so that we can make the individual bit flips necessary for our routine. Our apologies for the digression; we felt the need to address this “rookie” Python mistake for those who may be unaccustomed to using list objects in this, possibly unusual, manner.
Testing the binary mask routine
With the binary mask routine complete, we’re ready to build our power set. Below we’ve sandwiched the call within two time.time() reads, so we can measure its runtime duration. Let’s examine bMasks output first.
tStart = time.time() # clock start timestamp
pSet_bin, bMasks = clumsyPrez.powerSet_bin() # call our binary mask power set generator
duration = time.time() - tStart # calc run duration by subtracting tStart from current time.
bMasks # show list of binary masks
Above, you can see that masks progress from all False to all True in a logical pattern. This ordering would be a great feature if we cared about how members are ordered in our power set. But sets, strictly defined, are not distinguished by the ordering of their members (Van Dalen, Doets, De Swart; 2014). That is, set $\{a, b, c\}$ is equivalent to set $\{c, a, b\}$ and $\{b, a, c\}$, and so on.
So, after taking a look at our power set in pSet_bin, we see that the nice ordering was all for naught.
# Report duration of binary mask power set generation
print('The Binary Mask approach took %0.6f seconds to complete.'%(duration))
print('The power set contains %i subset elements'%(len(pSet_bin.val)))
pSet_bin.val # show power set output; remember, this is a set!
From the power set output string above, it looks like we have many sub-set instances as members of one big super-set. But, if you recall from all the way back in part one, this is not the case; our power set pSet_bin is actually a ModSet() object, whose .val attribute is a set, each element of which is another ModSet() instance that contains one unique sub-set of the original source set (as its .val attribute). We defined the ModSet.__repr__() method to return the code representation of its .val attribute; that’s the reason why the return string from calling pSet_bin.val appears in this set-resembling format.
Recursive power set generator
Overview
Our recursion-based method for generating power sets contains just 11 lines of code but requires five sibling methods, as well as itself, to run. Though .powerSet_rec() has fewer lines than .powerSet_bin(), it relies more heavily upon behavior of sibling methods in the ModSet() class, i.e. .__copy__(), .pushDown(), .diffFunc(), .union() and .removeDuplicates() to perform the respective obligatory processing tasks of duplication, nesting, extraction, joining and filtering. Unlike the Binary Mask approach above, indexing is not needed here, and hence, not employed. Through recursion, we can generate the identical power set using thoughtfully-placed set-subtraction, -union and -filtering operations.
class ModSet():
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# Generate power set recursively.
def powerSet_rec(self):
pSet = self.__copy__() # Preserve self instance; its copy, pSet
# will be altered
pSet.pushDown(1) # Nest pSet for later joining.
if len(self.val) > 0: # Recursion termination condition
for elSet in self.val: # Iterate through members of set self.val
# Generate subset that remains after removing current
# element, elSet, from set self.val
dummySet = self.diffFunc(ModSet(set([elSet])))
# To current powerset, append the powerset of the
# subset from previous step
pSet.union(dummySet.powerSet_rec()) # Self-call power set method,
# union join power set of
# dummySet with pSet
return pSet # Return power set at current
# level of recursion
else:
dummySet = ModSet(set()) # Generate instance of ModSet of empty set
dummySet.pushDown(1) # Nest empty set down one level so it can
return dummySet # be union-joined in the recursion level
# above (that called this current run).
Flow of control
First a duplicate of the calling instance is made to serve as the (local) power set within our routine. This instance is promptly pushed down one level so it can be joined by union later, if necessary. Next, the number of elements in the calling instance are counted. If empty, an empty ModSet() instance is returned on exit of the routine. This case path is the exit condition for the recursive flow; eventually all calling instances will dwindle down to empty as members are stripped from them in the alternative, non-empty, case that we’ll describe next.
If the calling instance is not empty, .powerSet_rec() iterates over the elements therein, each time subtracting out the current element, elSet, and calling itself again to build the power set from elements that remain. Output from this recursive call is then joined by union with pSet. Remember that every time pSet.union() runs, it calls the filtering method removeDuplicates() to retain only unique members in pSet as it is assembled.
When the loop completes, and all subsets have been included at the present level, the local instance of pSet is returned so that it can be union-joined with the pSet instance inside the calling-function one level above. The method continues in this fashion until pSet at the top-most level of recursion is fully populated, at which point output is returned in response to the initial method call.
[Phew..!]
Let’s give it a whirl and gauge its run-time duration.
Testing the recursive routine
tStart = time.time()
pSet_rec = clumsyPrez.powerSet_rec()
duration = time.time() - tStart
print('The Recursive approach took %0.6f seconds to complete.'%(duration))
print('The power set contains %i subset elements'%(len(pSet_rec.val)))
pSet_rec.val
As promised, we see the same power set as we saw from the Binary Mask approach. But unlike that approach, there is no mask output for us to examine this time.
Quick-and-dirty efficiency comparison
For a given programming objective, recursive algorithms tend to be less efficient than their iterative equivalents when used in imperative-based languages, like Python, where iteration is preferred (Recursion (computer science), Wikipedia). Below we hobbled together a quick script to measure, and statistically compare, processing times of the two methods of power set generation of clumsyPrez. We instruct the two power set generators to each run 500 times in a randomly alternating sequence to destroy potential processing-related biases (e.g. sequential, timing) that could otherwise arise by running the two in separate blocks.
import numpy as np
import matplotlib.pyplot as plt
# routine to repeatedly collect run-time durations
# of functions genFunc1 and genFunc2
def genDurationsDist(genFunc1, genFunc2, nReps):
durations = np.zeros(2*nReps) # initialize output array
genFuncs = [genFunc1, genFunc2] # make list of the two functions to run
# initialize an array to contain the run-sequence of the two functions
funcSeq = np.concatenate([np.zeros(nReps), np.ones(nReps)]).astype(int)
np.random.shuffle(funcSeq) # shuffle the ordering of the sequence
for i in np.arange(0, funcSeq.shape[0]): # iterate over random sequence
tStart = time.time() # start timestamp
genFuncs[funcSeq[i]]() # run one of the two functions based on funcSeq
durations[i] = time.time() - tStart # end timestamp, calc duration
# separate and return the two sets of durations
return durations[funcSeq == 0], durations[funcSeq == 1]
durations_bin, durations_rec = genDurationsDist(clumsyPrez.powerSet_bin,
clumsyPrez.powerSet_rec, 500)
Now that we have distributions of processing times from the two power set generators, we can plot histograms and Q-Q plots to get an idea of their locations and shapes.
import nonparametric_stats as nps
plt.rcParams['figure.figsize'] = [7, 5]
durs = (10**3)*np.array([durations_bin, durations_rec])
# Obtain color cycle that matplotlib uses
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
# plot histograms of duration distributions
labels=['Bin. Mask', 'Recursive']
ax = nps.histPlotter(50, *durs, labels=labels, colors=colors)
ax.set_title('Runtime durations of 2 power set generators')
ax.set_xlabel('duration (msec.)')
ax.set_ylabel('counts')
ax.legend()
# Examine Q-Q plots to compare distributions to
# their corresponding theoretical normal dists
fig, axs = plt.subplots(nrows=1, ncols=2)
fig.tight_layout()
for i in range(0,2):
nps.qqPlotter_normal(durs[i], 30, axes=axs[i], color=colors[i])
axs[i].set_title(labels[i] + ' gen. Q-Q')
axs[i].set_xlabel('theoretical normal')
if i == 0:
axs[i].set_ylabel('data')
The two distributions certainly do not appear to be normally distributed; they depart from Normal theoretical very early on. Consequently the typical two-tailed Student t-test cannot be used here. Below, we import and run the Mann-Whitney non-parametric comparison test to assess if the difference between the two reaches statistical significance.
from scipy.stats import mannwhitneyu as mwu
med_bin = (10**3)*np.median(durations_bin) # compute median duration of Bin. Mask ps durations
med_rec = (10**3)*np.median(durations_rec) # compute median duration of Recursive ps durations
print('Median Bin. Mask duration: %0.2f msec., Median Recursive duration: %0.2f msec.'\
%(med_bin, med_rec))
# Run non-parametric test to determine if differences between
# distributions are statistically significant
uStat, pVal = mwu(durations_bin, durations_rec)
print('Mann-Whitney U statistic: %0.2f, p-value: %0.2e'%(uStat, pVal))
Measuring the time elapsed for each of the two approaches, we find that, on average, our recursive algorithm does indeed take about twice the time to complete on our machine as the binary-mask method (4.59 ms versus 2.11 ms, U = 487, p < 0.01). Of course, ours is certainly not a very controlled comparison; so please take the result with “a grain of salt” if you will.
Summary
In part one we discussed power sets and their relation to the binary theorem. In addition, we mapped out a recursive procedure for generating the power set. A brief investigation revealed that Python’s set() class could not accommodate our need to include subsets within a set as in the case of power sets. To solve the problem, we introduced a custom container class with a set-valued attribute, whose instances are “hashable” and thus, eligible for membership in set objects.
Starting with the container strategy from part one, in part two we expanded our custom class, ModSet(), to include methods that perform copying, nesting, uniqueness-filtering, set-union and set-difference operations as well as others. We verified that all ModSet() methods functioned as desired. Many of the methods added would prove necessary for both methods of power set generation to follow.
In part three, we realized our goal of generating power sets. First, we detailed program flow of both the Binary Mask and Recursive approaches of power set generation. The mask routine used list indexing and took more lines of code than recursive method, but called fewer class methods. Our Recursive approach was shorter in terms of lines of code because it incorporated more sibling methods of the ModSet() class. We then explained the control flow of the recursive generator hoping to illustrate how recursive algorithms function. Finally, we compared relative run times of the two approaches. It turned out that, as expected, the Binary Mask routine took less time to run than its recursive counterpart.
Lastly, stay tuned for part foura where we’ll walk-through and test a more efficient algorithm for generating power sets in the ModSet() class.
Sources (part 3)
1. D. Van Dalen; H. C. Doets; H. De Swart (9 May 2014). Sets: Naïve, Axiomatic and Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students. Elsevier Science. ISBN 978-1-4831-5039-0.
2. Recursion (computer science), Recursion versus Iteration, Wikipedia.
The complete ModSet() class
You can download ModSet()‘s complete definition from github: modset.py
The ModSet() class by nullexit.org is licensed under a Creative Commons Attribution 4.0 International License.
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