  In Direct Sum problems [KRW], one tries to show that for a given
computational model, the complexity of computing a collection of finite
functions on independent inputs is approximately the sum of their individual
complexities. In this paper, by contrast, we study the diversity of ways in
which the joint computational complexity can behave when all the functions are
evaluated on a common input. We focus on the deterministic decision tree model,
with depth as the complexity measure; in this model we prove a result to the
effect that the 'obvious' constraints on joint computational complexity are
essentially the only ones.
  The proof uses an intriguing new type of cryptographic data structure called
a `mystery bin' which we construct using a small polynomial separation between
deterministic and unambiguous query complexity shown by Savicky. We also pose a
variant of the Direct Sum Conjecture of [KRW] which, if proved for a single
family of functions, could yield an analogous result for models such as the
communication model.
