Ok i found the answer. It doesn't actually reduce the computations. It just reduces the space complexity. This is a stub from wikipedia:
Using a Bayesian network can save considerable amounts of memory, if the dependencies i the joint distribution are sparse. For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for 2^{10} = 1024 values. If the local distributions of no variable depends on more than 3 parent variables, the Bayesian network representation only needs to store at most 10*2^3 = 80 values. One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distribution.