How to find smallest substring which contains all characters from a given string?

Question

I have recently come across an interesting question on strings. Suppose you are given following:

Input string1: \"this is a test string\"
Input strin         


        

Answer 1

Edit: apparently there's an O(n) algorithm (cf. algorithmist's answer). Obviously this have this will beat the [naive] baseline described below!

Too bad I gotta go... I'm a bit suspicious that we can get O(n). I'll check in tomorrow to see the winner ;-) Have fun!

Tentative algorithm:
The general idea is to sequentially try and use a character from str2 found in str1 as the start of a search (in either/both directions) of all the other letters of str2. By keeping a "length of best match so far" value, we can abort searches when they exceed this. Other heuristics can probably be used to further abort suboptimal (so far) solutions. The choice of the order of the starting letters in str1 matters much; it is suggested to start with the letter(s) of str1 which have the lowest count and to try with the other letters, of an increasing count, in subsequent attempts.

  [loose pseudo-code]
  - get count for each letter/character in str1  (number of As, Bs etc.)
  - get count for each letter in str2
  - minLen = length(str1) + 1  (the +1 indicates you're not sure all chars of 
                                str2 are in str1)
  - Starting with the letter from string2 which is found the least in string1,
    look for other letters of Str2, in either direction of str1, until you've 
    found them all (or not, at which case response = impossible => done!). 
    set x = length(corresponding substring of str1).
 - if (x < minLen), 
         set minlen = x, 
         also memorize the start/len of the str1 substring.
 - continue trying with other letters of str1 (going the up the frequency
   list in str1), but abort search as soon as length(substring of strl) 
   reaches or exceed minLen.  
   We can find a few other heuristics that would allow aborting a 
   particular search, based on [pre-calculated ?] distance between a given
   letter in str1 and some (all?) of the letters in str2.
 - the overall search terminates when minLen = length(str2) or when 
   we've used all letters of str1 (which match one letter of str2)
   as a starting point for the search