How does one easily determine if an algorithm has an exponential time complexity? The Word Break naive solution is known to have a time complexity of O(2n) but most people think its O(n!) because of the decreasing subsets as you go down the stack depth.

Word Break Problem: Given an input string and a dictionary of words, find out if the input string can be segmented into a space-separated sequence of dictionary words.

The brute-force/naive solution to this is:

def wordBreak(self, s, wordDict):
    queue = [0]
    dictionary_set = set(wordDict)
    for left in queue:
        for right in range(left,len(s)):
            if s[left:right+1] in dictionary_set:
                if right == len(s)-1:
                    return True
    return False

With the worst case inputs of:

s = 'aab'
wordDict = ['a','aa','aaa']

Results in a O(2n) time complexity of 23 ≈ 7:


Let me caveat this question with:

  1. I'm not looking for a performant solution so just ignore why the implementation is the way it is. There is a known O(n2) dynamic programming solution.
  2. I know there are similar questions asked on stackexchange, but none of them gave an easy to understand answer to this problem. I would like a clear answer with examples using preferably the Word Break example.
  3. I can definitely figure the complexity by counting every computation in the code, or using recurrence relation substitution/tree methods but what I'm aiming for in this question is how do you know when you look at this algorithm quickly that it has a O(2n) time complexity? An approximation heuristic similar to looking at nested for loops and knowing that it has a O(n2) time complexity. Is there some mathematical theory/pattern here with decreasing subsets that easily answers this?
  • 2
    $\begingroup$ "how do you know when you look at this algorithm quickly?" - Often you don't; it may take an involved calculation. Sometimes with practice you can get quicker at it. I'm not sure you're going to get a simple answer. Have you tried applying the standard techniques for analyzing running time to see what happens? $\endgroup$ – D.W. Sep 27 '17 at 1:47
  • $\begingroup$ "easily", "heuristically" -- what do these words mean to you, exactly? $\endgroup$ – Raphael Sep 28 '17 at 16:51
  • 1
    $\begingroup$ There are two basic ways: formal analysis and experiments. Automated methods are doomed to be incomplete. $\endgroup$ – Raphael Sep 28 '17 at 16:54
  • $\begingroup$ @Raphael, in that way experiments are also doomed, since you can (hypothetically) program a TM that will do it for you. $\endgroup$ – rus9384 Sep 28 '17 at 18:33
  • 1
    $\begingroup$ @perseverance That's called (accurate) intuition. The only way I know to gain it is through working rigorously analyses for as long as it takes. (Some never get there.) $\endgroup$ – Raphael Sep 29 '17 at 20:05

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