  We initiate a systematic study of the time complexity of quantum divide and
conquer algorithms for classical problems. We establish generic conditions
under which search and minimization problems with classical divide and conquer
algorithms are amenable to quantum speedup and apply these theorems to an array
of problems involving strings, integers, and geometric objects. They include
LONGEST DISTINCT SUBSTRING, KLEE'S COVERAGE, several optimization problems on
stock transactions, and k-INCREASING SUBSEQUENCE. For most of these results,
our quantum time upper bound matches the quantum query lower bound for the
problem, up to polylogarithmic factors.
