Dynamic programming algorithms with log in the run-time
One natural example is finding the longest increasing subsequence of a sequence of $n$ numbers. Candidate subsequences can be linked in the input sequence. This is a fairly common exercise, and works for other type of subsequences, too. It is actually the exercise 15.4-6 in the 3rd edition of the Cormen et al. book too. For an algorithm, see Section 2.2 in these notes.