Question

I’ve been going through Skiena’s excellent “The Algorithm Design Manual” and got hung up on one of the exercises.

The question is:
“Given a search string of three words, find the smallest snippet of the document that contains all three of the search words—i.e. , the snippet with smallest number of words in it. You are given the index positions where these words in occur search strings, such as word1: (1, 4, 5), word2: (4, 9, 10), and word3: (5, 6, 15). Each of the lists are in sorted order, as above.”

Anything I come up with is O(n^2)… This question is in the “Sorting and Searching” chapter, so I assume there is a simple and clever way to do it. I’m trying something with graphs right now, but that seems like overkill.

Ideas?
Thanks

Answer 1

I already posted a rather straightforward algorithm that solves exactly that problem in this answer

Google search results: How to find the minimum window that contains all the search keywords?

However, in that question we assumed that the input is represented by a text stream and the words are stored in an easily searchable set.

In your case the input is represented slightly differently: as a bunch of vectors with sorted positions for each word. This representation is easily transformable to what is needed for the above algorithm by simply merging all these vectors into a single vector of (position, word) pairs ordered by position. It can be done literally, or it can be done “virtually”, by placing the original vectors into the priority queue (ordered in accordance with their first elements). Popping an element from the queue in this case means popping the first element from the first vector in the queue and possibly sinking the first vector into the queue in accordance with its new first element.

Of course, since your statement of the problem explicitly fixes the number of words as three, you can simply check the first elements of all three arrays and pop the smallest one at each iteration. That gives you a O(N) algorithm, where N is the total length of all arrays.

Also, your statement of the problem seems to suggest that target words can overlap in the text, which is rather strange (given that you use the term “word”). Is it intentional? In any case, it doesn’t present any problem for the above linked algorithm.