A covering array is an \$N \times k\$ array in which each element is a from a set of \$v\$ symbols, and for every \$t\$ columns, every possible set of \$v^t\$ choices of the symbols appears at least once. The covering array number \$\mathrm{CA}(t,k,v)\$ is the smallest \$N\$ for which a covering array exists, given \$t\$, \$k\$, and \$v\$. A list of known covering array numbers (CANs) is available here.

I want to parse all of the known CANs from these pages and to find how "efficient" they are—what I mean by this is the ratio of \$\mathrm{CA}(t,k,v)\$ compared to \$v^t\$.

I developed Python code that access every covering array page, and parses the tables. I then sort the list of covering arrays by this ratio, and plot it using matplotlib.pyplot (using a log-scale for the y axis).

from urllib.request import urlopen
from bs4 import BeautifulSoup
import matplotlib.pyplot as plt

# covering array object
class CAElement:
    def __init__(self, N, t, k, v):
        self.N = N
        self.t = t
        self.k = k
        self.v = v
    def set_ratio(self, ratio):
        self.ratio = ratio
    def __str__(self):
        return "CA(N=%d; t=%d, k=%d, v=%d) -> %f" % (self.N, self.t, self.k, self.v, self.ratio)

CAArray = []

# iterate over v, t in the known table ranges
for v in range(2, 26):
    for t in range(2, 7):
        # download the webpage and find the elements
        url = "http://www.public.asu.edu/~ccolbou/src/tabby/%d-%d-ca.html" % (t, v)
        response = urlopen(url)
        soup = BeautifulSoup(response)
        tables = soup.findChildren('table')
        table = tables[0]
        rows = table.findChildren('tr')

        # iterate over all rows in the one table
        for row in rows:
            cells = row.findChildren('td') # has all of the table's elements
            elements = []
            for cell in cells:
                value = cell.string
                if value is not None and value != "k" and value != "N" and value != "Source":
            if len(elements) >= 2:
                kParsed = int(elements[0])
                NParsed = int(elements[1])
                element = CAElement(NParsed, t, kParsed, v)
                ratio = element.N / pow(element.v, element.t)

# sort by N/(v^t)
CAArray.sort(key=lambda x: (x.ratio, x.N, x.v, x.t, x.k), reverse=True)

# print each element (in sorted order)
for element in CAArray:

# plotting - using log scale for y axis
# each point is colored according to t (i.e., the "strength" of the CA)
xs = range(0, len(CAArray))
ys = [y.ratio for y in CAArray]
colors = {2:"red", 3:"blue", 4:"green", 5:"yellow", 6:"orange"}
plt.scatter(xs, ys, c=[colors[x.t] for x in CAArray])
plt.axis([min(xs), max(xs), min(ys), max(ys)])

This code does exactly what I want it to do. However, there are some problems:

  • Accessing the webpages is somewhat slow, and could be faster.
  • The code does not seem very Pythonic (and by making it so could make iterating over the array much faster).

Any suggestions are welcome!


2 Answers 2


Here are some of the quick performance wins you may get:

  • switch to using requests and reuse a Session instance which should provide a performance boost:

    if you're making several requests to the same host, the underlying TCP connection will be reused, which can result in a significant performance increase (see HTTP persistent connection).

    import requests
    session = requests.Session()
    # iterate over v, t in the known table ranges
    for v in range(2, 26):
        for t in range(2, 7):
            # download the webpage and find the elements
            url = "http://www.public.asu.edu/~ccolbou/src/tabby/%d-%d-ca.html" % (t, v)
            response = session.get(url)
  • use the fastest available underlying parser - lxml (requires to install lxml):

    soup = BeautifulSoup(response, 'lxml')
  • parse only table elements from a page source via SoupStrainer:

    parse_only = SoupStrainer('table')
    soup = BeautifulSoup(response, 'lxml', parse_only=parse_only)

    Requires importing SoupStrainer from bs4.

And, applying some bs4 related shortcuts, the HTML parsing part would transform into:

table = soup.table
for row in table('tr'):
    elements = []
    for cell in row('td'):
        value = cell.string
        if value is not None and value not in ("k", "N", "Source"):
  1. When you have two or more independent loops:

    # iterate over v, t in the known table ranges
    for v in range(2, 26):
        for t in range(2, 7):

    you can use itertools.product to combine these into one loop:

    for v, t in product(range(2, 26), range(2, 7)):

    This saves a level of indentation.

  2. The CAElement class doesn't have any behaviour, it's just a repository for data. So I would recommend making it a collections.namedtuple:

    CAElement = namedtuple('CAElement', 'ratio N v t k')

    Putting the attributes in this order means that you no longer need a key function in the call to CAArray.sort.

    This requires a small change to the initialization, which becomes:

    ratio = NParsed / pow(v, t)
    element = CAElement(ratio, NParsed, t, kParsed, v)

    Note that pow(v, t) is the same for every row so you could cache this in a local variable, but I doubt this makes a noticeable difference to the runtime.


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