  The problem of sorting with priced information was introduced by [Charikar,
Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this
setting, different comparisons have different (potentially infinite) costs. The
goal is to find a sorting algorithm with small competitive ratio, defined as
the (worst-case) ratio of the algorithm's cost to the cost of the cheapest
proof of the sorted order.
  The simple case of bichromatic sorting posed by [CFGKRS] remains open: We are
given two sets $A$ and $B$ of total size $N$, and the cost of an $A-A$
comparison or a $B-B$ comparison is higher than an $A-B$ comparison. The goal
is to sort $A \cup B$. An $\Omega(\log N)$ lower bound on competitive ratio
follows from unit-cost sorting. Note that this is a generalization of the
famous nuts and bolts problem, where $A-A$ and $B-B$ comparisons have infinite
cost, and elements of $A$ and $B$ are guaranteed to alternate in the final
sorted order.
  In this paper we give a randomized algorithm InversionSort with an
almost-optimal w.h.p. competitive ratio of $O(\log^{3} N)$. This is the first
algorithm for bichromatic sorting with a $o(N)$ competitive ratio.
