Question

I’ve been trying to find the optimal solution to the following (interesting?) problem that came up at work: Eventually I settled for a good enough solution but I’d like to know if there’s a better one.

Let a₁…a_n be an array of strings.

Let s₁…s_k be an unordered list of strings, all of them also members of the array.

The task is to find the minimum set of index ranges eleements of s cover in a.

So for example if a = [ “x”, “y”, “a”, “f”, “c” ] and s = { “c”,”y”,”f” }, the answer would be (1;1), (3;4), assuming that the array is indexed from zero.

a is typically fairly large (hundreds of thousands of elements), while s is relatively small, typically length(s) < log(length(a)).

So the question is: can you find a time-efficient algorithm for this problem? (Space efficiency is not a concern within reasonable limits.)

Just a quick but important update: I need to perform this operation with different s values but the same a a lot. So precomputing stuff based on a is allowed, indeed it is the only way.

Answer 1

Build a hash table H(a) to map from element to index: a_x->x in O(n) time and space. Then look up each s_y in H(a) (in O(1) time on average for a total of O(k) for s) and keep track of the ranges. For that you can use an array of pair(min_index, max_index) sorted by min_index and do a binary search to either locate the range or where you should insert the new 1 element range.
So overall, the solution above would take O( n + k + k * log( nb_ranges ) ) time and O( n + nb_ranges ) space.