  We present an average case analysis of two variants of dual-pivot quicksort,
one with a non-algorithmic comparison-optimal partitioning strategy, the other
with a closely related algorithmic strategy. For both we calculate the expected
number of comparisons exactly as well as asymptotically, in particular, we
provide exact expressions for the linear, logarithmic, and constant terms. An
essential step is the analysis of zeros of lattice paths in a certain
probability model. Along the way a combinatorial identity is proven.
