The analysis presented here mentions an approach that would take O(N sqrt(M)) 
for the large case:
https://code.google.com/codejam/contest/9234486/dashboard#s=a&a=2

However, the approach that is described (as far as I understand it) still 
compares the 26 frequency counts for every word in the dictionary at each 
position in the larger string:

"for each such length, we perform the same process as above, checking against 
only the dictionary words with that length"

This would still be O(N*L) - or O(N*L*K) if we decide to consider the size of 
the alphabet K=26 in the analysis.

What am I missing?

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