Let's use the example given, for words say A, B, C, in the problem but not get too tied to the specific values. It helps to think about sorting the search string words by their indexes in the document:

A  C  B  A  A  C  *  *  B  B  *  *  *  *  C
1  2  3  4  5  6  7  8  9  10 11 12 13 14 15


In the code example this is just a call to sort but since each word index list is sorted we could use MergeSort which means this portion of the code is $\mathcal{O}(n)$ for $n$ indexes.

Once we have the list we add push each element into a hash with a separate key for each word and update our estimate of the snippet span that includes all the words.

Here's python code for one simple implementation.

import sys

def smallest_snippet(*args):
'''
args -- K lists of word positions for K words

>>> smallest_snippet([1,10], [2,20], [3,30])
[1, 3]
>>> smallest_snippet([1,9,27], [6,10,19], [8,12,14])
[8, 10]
>>> smallest_snippet([1,4,11,27], [3,6,10,19], [5,8,12,14])
[3, 5]
>>> smallest_snippet([1,4,5], [3,9,10], [2,6,15])
[1, 3]
'''

master = []
for i in range(len(args)):
master.extend(map(lambda j: (i,j), args[i])) # ith word, jth index
master.sort(lambda x,y: cmp(x[1], y[1]))         # TODO: mergesort
tops = {}                                        # { word i: index j }
best = [master[0][1], master[-1][1]]
minspan = best[-1] - best[0] + 1

# update span after each new index tuple
for (i,j) in master:
tops[i] = j
if len(tops) == len(args):
curr = [min(tops.values()), max(tops.values())]
if curr[1] - curr[0] < minspan:
minspan = curr[1] - curr[0]
best = curr

return best

if __name__ == "__main__":
import doctest
doctest.testmod()

sys.exit()