What is LU and what are the axis mean?
@bojianh I'm guessing LU stands for LU decomposition (but not sure). I think the horizontal axis means the number of processors $X$ and vertical axis means the probability of having $X$ sharers.
Yes, LU stands for LU decomposition - "factors a matrix as the product of a lower triangular matrix and an upper triangular matrix" (Wikipedia).
The slow increase in expected number of sharers when P increases is good, because we don't have to communicate to a large number of processors even though there are lot of processors running our algorithm. The communication overhead is kept relatively small.
X axis is # processors; Y is the probability of that being the case. It's just saying cache lines normally (see the highest bar) are not shared by a lot of processors. So it would be wasteful to allocate a bit for each processor.
@misaka X is the number of sharers, Y is the probability.
What is important from this slide? It seems like what we want to get from this is that the directory-based cache coherence limits communication, which is a huge optimization.