Blocking/Tiling is a well-known way to make program faster by leveraging the properties of memory hierarchy.

Take matrix multiplication as an example, multiplying two NxN matrices with the following code (assuming row-major):

```
for i := 1 to N do
for k := 1 to N do
r := A[i,k];
for j := 1 to N do
C[i,j] += r * B[k,j];
```

This code is performing good for several reasons:

- The innermost loop accesses consecutive data in
`B`

and`C`

, which utilizes the cache prefetch mechanism. - The same row of
`C`

is reused in each iteration of the middle loop

And if the matrices are small, `B`

can store entirely in the cache, then the
code can be fast. If the matrices are large, so that the cache cannot even hold
a row of data (`C`

), the cache is never reused. In this case (worst case), there
would be \(2N^3+N^2\) elements read in \(N^3\) iterations.

The blocked code is like the following:

```
for kk := 1 to N step b do {* N/b blocks *}
for jj := 1 to N step b do {* N/b blocks *}
for i := 1 to N do {* each col in C *}
for k := kk to min(kk+b-1, N) do {* computation on a block in B *}
r := A[i,k]; {* loop over a vector in A *}
for j := jj to min(jj+b-1,N) do
C[i,j] += r * B[k,j]; {* vector in C = vector in A x block in B *}
```

This code would be efficient if the block of `B`

of size bxb can reside inside the
cache, so that it can be reused throughout the loop on `i`

. Then the memory read
would be \(2N^3/b+N^2\), reduced by a factor of \(b\) at most. The optimal value (by
Hong & Kung, 1981) of blocking factor \(b\) is roughly square root of \(c\), where \(c\) is
the size of the cache.

In practice of running these algorithms, the performance is not as good as expected because of the interference misses in the cache — because the cache is set associative.

The cache misses is higher than expected because there can be interferences:
there could be *cross interference* when two different variables conflicts each
other in cache, or *self interference* when elements of the same array variable
conflicts with itself in cache.

The paper elaborated that, the intrinsic cache miss is \(N^3(2/b+4b/c)\), which is the best performance that can be achieved. So minimizing this yields blocking factor of \(b=\sqrt{c/2}\). In case of a-way set associative cache is used, the optimal block size would be (heuristic) \(b=\sqrt{c(a-1)/2a}\)

## Bibliographic data

```
@inproceedings{
title = "The Cache Performance and Optimization of Blocked Algorithms",
author = "Monica S. Lam and Edward E. Rothberg and Michael E. Wolf",
booktitle = "Proceedings of the 4th International Conference on Architectural Support for Programming Languages and Operating Systems",
pages = "63--74",
howpublished = "ASPLOS",
year = "1991",
}
```