  In this paper, we put forward the model of zero-error distributed function
compression system of two binary memoryless sources X and Y, where there are
two encoders En1 and En2 and one decoder De, connected by two channels (En1,
De) and (En2, De) with the capacity constraints C1 and C2, respectively. The
encoder En1 can observe X or (X,Y) and the encoder En2 can observe Y or (X,Y)
according to the two switches s1 and s2 open or closed (corresponding to taking
values 0 or 1). The decoder De is required to compress the binary arithmetic
sum f(X,Y)=X+Y with zero error by using the system multiple times. We use
(s1s2;C1,C2;f) to denote the model in which it is assumed that C1 \geq C2 by
symmetry. The compression capacity for the model is defined as the maximum
average number of times that the function f can be compressed with zero error
for one use of the system, which measures the efficiency of using the system.
We fully characterize the compression capacities for all the four cases of the
model (s1s2;C1,C2;f) for s1s2= 00,01,10,11. Here, the characterization of the
compression capacity for the case (01;C1,C2;f) with C1>C2 is highly nontrivial,
where a novel graph coloring approach is developed. Furthermore, we apply the
compression capacity for (01;C1,C2;f) to an open problem in network function
computation that whether the best known upper bound of Guang et al. on
computing capacity is in general tight.
