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Cooking with Python, Part 1
Pages: 1, 2

Recipe 5.10: Selecting the nth Smallest Element of a Sequence

Credit: Raymond Hettinger, David Eppstein, Shane Holloway, Chris Perkins


You need to get from a sequence the nth item in rank order (e.g., the middle item, known as the median). If the sequence was sorted, you would just use seq[n]. But the sequence isn't sorted, and you wonder if you can do better than just sorting it first.


Perhaps you can do better, if the sequence is big, has been shuffled enough, and comparisons between its items are costly. Sort is very fast, but in the end (when applied to a thoroughly shuffled sequence of length n) it always takes O(nlogn) time, while there exist algorithms that can be used to get the nth smallest element in time O(n). Here is a function with a solid implementation of such an algorithm:

import random
def select(data, n):
    " Find the nth rank ordered element (the least value has rank 0). "
    # make a new list, deal with <0 indices, check for valid index
    data = list(data)
    if n<0:
        n += len(data)
    if not 0 <= n < len(data):
        raise ValueError, "can't get rank %d out of %d" % (n, len(data))
    # main loop, quicksort-like but with no need for recursion
    while True:
        pivot = random.choice(data)
        pcount = 0
        under, over = [  ], [  ]
        uappend, oappend = under.append, over.append
        for elem in data:
            if elem < pivot:
            elif elem > pivot:
                pcount += 1
        numunder = len(under)
        if n < numunder:
            data = under
        elif n < numunder + pcount:
            return pivot
            data = over
            n -= numunder + pcount

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This recipe is meant for cases in which repetitions count. For example, the median of the list [1, 1, 1, 2, 3] is 1 because that is the third one of the five items in rank order. If, for some strange reason, you want to discount duplications, you need to reduce the list to its unique items first (e.g., by applying the Recipe 18.1), after which you may want to come back to this recipe.

Input argument data can be any bounded iterable; the recipe starts by calling list on it to ensure that. The algorithm then loops, implementing at each leg a few key ideas: randomly choosing a pivot element; slicing up the list into two parts, made up of the items that are "under" and "over" the pivot respectively; continuing work for the next leg on just one of the two parts, since we can tell which one of them the nth element will be in, and the other part can safely be ignored. The ideas are very close to that in the classic algorithm known as quicksort (except that quicksort cannot ignore either part, and thus must use recursion, or recursion-removal techniques such as keeping an explicit stack, to make sure it deals with both parts).

The random choice of pivot makes the algorithm robust against unfavorable data orderings (the kind that wreak havoc with naive quicksort); this implementation decision costs about log2N calls to random.choice. Another implementation issue worth pointing out is that the recipe counts the number of occurrences of the pivot: this precaution ensures good performance even in the anomalous case where data contains a high number of repetitions of identical values.

Extracting the bound methods .append of lists under and over as local variables uappend and oappend may look like a pointless, if tiny, complication, but it is, in fact, a very important optimization technique in Python. To keep the compiler simple, straightforward, unsurprising, and robust, Python does not hoist constant computations out of loops, nor does it "cache" the results of method lookup. If you call under.append and over.append in the inner loop, you pay the cost of lookup each and every time. If you want something hoisted, hoist it yourself. When you're considering an optimization, you should always measure the code's performance with and without that optimization, to check that the optimization does indeed make an important difference. According to my measurements, removing this single optimization slows performance down by about 50% for the typical task of picking the 5000th item of range(10000). Considering the tiny amount of complication involved, a difference of 50% is well worth it.

A natural idea for optimization, which just didn't make the grade once carefully measured, is to call cmp(elem, pivot) in the loop body, rather than making separate tests for elem < pivot and elem > pivot. Unfortunately, measurement shows that cmp doesn't speed things up; in fact, it slows them down, at least when the items of data are of elementary types such as numbers and strings.

So, how does select's performance compare with the simpler alternative of:

def selsor(data, n):
    data = list(data)
    data.sort( )
    return data[n]

On thoroughly shuffled lists of 3,001 integers on my machine, this recipe's select takes about 16 milliseconds to find the median, while selsor takes about 13 milliseconds; considering that sort could take advantage of any partial sortedness in the data, for this kind of length, and on elementary data whose comparisons are fast, it's not to your advantage to use select. For a length of 30,001, performance becomes very close between the two approaches—around 170 milliseconds either way. When you push the length all the way to 300,001, select provides an advantage, finding the median in about 2.2 seconds, while selsor takes about 2.5.

The break-even point will be smaller if the items in the sequence have costly comparison methods, since the key difference between the two approaches is in the number of comparisons performed--select takes O(n), selsor takes O(n log n). For example, say we need to compare instances of a class designed for somewhat costly comparisons (simulating four-dimensional points that will often coincide on the first few dimensions):

class X(object):
    def _ _init_ _(self):
        self.a = self.b = self.c = 23.51
        self.d = random.random( )
    def _dats(self):
        return self.a, self.b, self.c, self.d
    def _ _cmp_ _(self, oth):
        return cmp(self._dats, oth._dats)

Here, select already becomes faster than selsor when what we're computing is the median of vectors of 201 such instances.

In other words, although select has more general overhead, when compared to the wondrously efficient coding of lists' sort method, nevertheless, if n is large enough and each comparison is costly enough, select is still well worth considering.

See Also

Library Reference and Python in a Nutshell docs about method sort of type list, and about module random.

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