Case Study: Cache Efficient Algorithms for Matrix Operations

  • Michael Bader
Chapter
First Online:
Part of the Texts in Computational Science and Engineering book series (TCSE, volume 9)

Abstract

In Chaps. 10 and 11, we discussed applications of space-filling curves for parallelisation, which were motivated by their locality properties. In the following two chapters, we will discuss further applications, which again exploit the intrinsic locality properties of space-filling curves. As both applications will focus on inherently cache-efficient algorithms, we will start with a short introduction to cache-memory architectures, and discuss the resulting requirements on cache-efficient algorithms.

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In Chaps. 10 and 11, we discussed applications of space-filling curves for parallelisation, which were motivated by their locality properties. In the following two chapters, we will discuss further applications, which again exploit the intrinsic locality properties of space-filling curves. As both applications will focus on inherently cache-efficient algorithms, we will start with a short introduction to cache-memory architectures, and discuss the resulting requirements on cache-efficient algorithms.

13.1 Cache Efficient Algorithms and Locality Properties

In computer architecture, a so-called cache (or cache memory) denotes a fast memory component that replicates a certain part of the main memory to allow faster access to the cached (replicated) data. Such caches are necessary, because standard memory hardware is nowadays much slower than the CPUs. The access latency, i.e. the time between a data request and the arrival of the first requested piece of data, is currently1 about 60–70 ns. During that time, a fast CPU core can perform more than 100 floating point operations. This so-called “memory-gap” between CPU speed and main memory is constantly getting worse, because CPU speed is improving much faster (esp. due to using multicore processors) than memory latency. For memory bandwidth (i.e. the maximum rate of data transfer from memory to CPU), the situation is a bit better, but there are already many applications in scientific computing whose performance is limited by memory bandwidth instead of CPU speed.

Cache memory can be made much faster, and can even keep up with CPU speed, but only if it is much smaller than typical main memory. Hence, modern workstations use a hierarchy of cache memory: a very small, so-called first-level cache, running at the same speed as the CPU; a second-level cache that is larger (typically a few megabytes), but only running at half speed, or less; and maybe further levels that are again larger, but slower. Figure 13.1 illustrates such a pyramid of cache memory. The situation is further complicated on multi- and manycore CPUs, because cache memory can then be shared between the CPU cores of one processor, and, in addition, we may have non-uniform memory access (NUMA) to main memory. Figure 13.2 shows a schematic diagram of four quadcore CPUs, where in each CPU, all cores share a common level-3 cache, but have individual level-1 and level-2 caches. Each CPU has a separate memory controller (MC), which is connected to a part of main memory – access to non-local memory runs via an interconnect (IC) between the four cores. As a result, CPUs will have different access speeds to different parts of the main memory.
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Fig. 13.1

A typical pyramid of hierarchical cache memory

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Fig. 13.2

Four quadcore processors (with shared L3 cache) with a NUMA interconnect (IC)

Memory-Bound Performance and Cache Memory

The size and speed of the individual cache levels will have an interesting effect on the runtime of software. In the classical analysis of the runtime complexity of algorithms, a typical job is to determine how the runtime depends on the input size, which leads to the well-known \(\mathcal{O}(N)\) considerations. There, we often assume that all operations are executed with the same speed, which is no longer true on cache-based systems.

Instead, we often observe a situation as in Fig. 13.3. For very small problem size, all data resides in the first-level cache, and the algorithm runs at top speed. As soon as the data does no longer fit into level-1 cache, the speed is reduced to a level determined by the level-2 cache. The respective, step-like decrease of execution speed is repeated for every level of the memory hierarchy, and leads to the velocity profile given in Fig. 13.3. While the complexity of an algorithm will not change in the \(\mathcal{O}(N)\)-sense, such large differences in execution speed cannot be ignored in practice. Hence, we need to make implementations cache efficient, in order to fully exploit the available performance.
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Fig. 13.3

Performance of an optimized vector operation (daxpy-operator: \(y := \alpha x + y\), \(\alpha \in \mathbb{R}\)) and of a naive implementation of matrix multiplication for increasing problem size. Note the substantial performance drops and the different performance levels, which result from data fitting into the cache levels

How Cache Memory Works: Cache Lines and Replacement Strategies

Once we use more data than we can fit into the cache, we need a mechanism to decide what data should be in the cache at what time. For such cache strategies, there are a couple of technical and practical restrictions:

  • For technical reasons, caches do not store individual bytes or words, but small contiguous chunks of memory, so-called cache lines, which are always transferred to and from memory as one block. Hence, cache lines are mapped to corresponding lines in main memory. To simplify (and, thus, speed up) this mapping, lines of memory can often be transferred only to a small subset of lines: we speak of an n-associative cache, if a memory line can be kept in n different cache lines. The most simple case, a 1-associative cache, is called a direct-mapped cache; a fully associative cache, where memory lines may be kept in any cache line, are powerful, but much more expensive to build (if the cache access needs to stay fast).

  • If we want to load a specific cache line from memory, but already have a full cache, we naturally have to remove one cache line from the cache. In an ideal case, we would only remove cache lines that are no longer accessed. As the cache hardware can only guess the future access pattern, certain heuristics are used to replace cache lines. Typical strategies are to remove the cache line that was not used for the longest time (least recently used), or which had the fewest accesses in the recent history (least frequently used).

  • A programmer typically has almost no influence on what data is kept in the cache. Only for loading data into the cache, so-called prefetching commands are sometimes available.

Associativity, replacement strategy, and other hardware properties may, of course, vary between the different levels of caches. And it is also clear that these restrictions have an influence of the efficiency of using caches.

Cache Memory and Locality

Caches lead to a speedup, if repeatedly accessed data is kept in the cache, and is thus accessed faster. Due to the typical cache design, algorithms and implementations will be cache efficient, if their data access pattern has good temporal or spatial locality properties:

  • Temporal locality means that a single piece of data will be repeatedly accessed during a short period of time. Replacement strategies such as least recently used or least frequently used will then reduce the probability of removing this data from the cache to a minimum.

  • Spatial locality means that after an access to a data item, the next access(es) will be to items that are stored in a neighbouring memory address. If it belongs to the same cache line as the previously accessed item, it has been loaded into the cache as well, and can be accessed efficiently.

Hence, the cache efficiency of an algorithm depends on its temporal and spatial locality properties.

Cache-Aware and Cache-Oblivious Algorithms

Cache efficient implementation or algorithms can be classified into two categories, depending on whether they consider the exact cache architecture of a platform:

  • Cache-aware algorithms or implementations use detailed information about the cache architecture. They try to increase the temporal or spatial locality by adapting the access pattern to exactly fit the number and size of the cache levels, the length of the cache line, etc. Hence, such a cache-aware implementation is specific for a particular platform, and at least certain parameters need to be tuned, if the architecture changes.

  • Cache-oblivious algorithms, in contrast, do not use explicit knowledge on a specific architecture. Instead, the algorithms are designed to be inherently cache efficient, and profit from any presence of caches, regardless of their size and number of cache levels. Hence, their data access pattern need to have excellent temporal and spatial locality properties.

We have seen that space-filling curves are an excellent tool to create data structures with good locality properties. Hence, we will discuss some examples of how these properties can be exploited to obtain cache-oblivious algorithms.

13.2 Cache Oblivious Matrix-Vector Multiplication

For a vector \(x = ({x}_{1},\ldots ,{x}_{n}) \in {\mathbb{R}}^{n}\) and an n ×n-matrix A with elements aij, the elements of the matrix-vector product y = Ax are given as
$${y}_{i} := \sum \limits _{j=1}^{n}{a}_{ ij}{x}_{j}.$$
The matrix-vector product is a standard task in linear algebra, and also not difficult to implement, which makes it a popular programming task in introductory lectures on programming. We can assume that most implementations will look similar to the one given in Algorithm 13.1 (omitting the initialisation of the vector y).
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To determine the cache efficiency of this algorithm, let’s now examine the temporal and spatial locality properties of this implementation. Regarding the matrix A, we notice that each element is accessed exactly once. Hence, temporal locality of the access will not be an issue, as no element will be reused. The spatial locality depends on the memory layout of the matrix. If the elements are stored in the same sequence as they are traversed by the two nested loops, the spatial locality will be optimal. Hence, for Algorithm 13.1, the matrix elements should be stored row-by-row (so-called row-major layout). However, if our programming language uses column-major layout (as in FORTRAN, for example), we should change Algorithm 13.1: the spatial locality would then be optimal, if we exchange the i- and j-loop. In general, spatial locality is perfect as long as we traverse the matrix element in the same order as they are stored in memory.

Changing the traversal scheme of the matrix elements will, however, also change the access to the two vectors x and y. As both vectors are accessed or updated n times throughout the execution of the algorithm, both temporal and spatial locality are important. For the loop-based access given by Algorithm 13.1, the access to both vectors is spatially local, even if we exchange the two loops. However, the temporal locality is different for the two vectors. The access to vector y is optimal: here, all n accesses to a single element are executed, before we move on to the next element. For the access to x, we have exactly the opposite situation: Before an element of x is re-used, we first access all other elements of the vector. Hence, the temporal locality for this pattern is the worst we can find. A first interesting question is, whether exchanging the loops will improve the situation. Then, x and y will change roles, and which option is faster depends on whether temporal locality is more important for the read access to x or for the write access to y. Even more interesting is the question whether we can derive a traversal of the matrix elements that leads to more local access patterns to both vectors.

Matrix Traversals Using Space-Filling Curves

In Algorithm 13.1, we easily see that the two loops may be exchanged. Actually, there is no restriction at all concerning the order in which we execute the element operations. We should therefore denote Algorithm 13.1 in the following form:

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From Algorithm 13.2, we can now consider execution orders without being influenced by loop constructions. We just need to make sure that each matrix element is processed exactly once, i.e. that we perform a traversal, and can then try to optimise the temporal and spatial locality properties. Our previous discussions of the locality properties of space-filling curves therefore provide us with an obvious candidate.

Assuming that the matrix dimension n is a power of two, we can use a Hilbert iteration, for example, to traverse the matrix. Hence, we modify the traversal algorithm of Chap. 3.2 to obtain an algorithm for matrix-vector multiplication. For that purpose, it is sufficient to interpret our direction operations up, down, left, and right as index operations on the matrix and the two vectors. Operators up and down will increase or decrease i, i.e., the current row index in A and vector index in y. Operators left and right will change j, which is the column index of A and the vector index of x, respectively. Algorithm 13.3 outlines the full algorithm for matrix-vector multiplication.

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To retain the optimal locality of the access to the matrix elements, the matrix elements need to be stored according to the Hilbert order, as well. For the accesses to the vectors x and y, we obtain an overall improved temporal locality. During the matrix traversal, all elements of a \({2}^{k} \times {2}^{k}\)-block will be processed before the Hilbert order moves on to the next \({2}^{k} \times {2}^{k}\)-block. During the corresponding \({({2}^{k})}^{2}\) element operations, 2k elements of vector x and 2k elements of vector y will be accessed – each of them 2k times. On average, we will therefore execute m2 operations on a working set of only 2m elements – for any m ≤ n. Hence, even if only a small amount of elements will fit in a given cache, we can guarantee re-use of both elements of x and y. Our “naive” implementation in Algorithm 13.1 will guarantee this only for one of the two vectors.

13.3 Matrix Multiplication Using Peano Curves

For two \(n \times n\)-matrices A and B, the elements of the matrix product \(C = \mathit{AB}\) are given as
$${C}_{ik} := \sum \limits _{j=1}^{n}{A}_{ ij}{B}_{jk}.$$
Similar to Algorithm 13.1 for the matrix-vector product, we could implement the computation of all Cik via three nested loops – one over j to compute an individual element Cik, and two loops over i and k, respectively – compare Algorithm 13.4
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Again, we can exchange the three loops arbitrarily. However, for large matrices, the cache efficiency will be less than optimal for any order. In library implementations, multiple optimisation methods, such as loop unrolling or blocking and tiling, are applied to improve the performance of matrix multiplication. The respective methods carefully change the execution order to match the cache levels – in particular the sizes of the individual caches (see the references section at the end of this chapter). In the following, we will discuss an approach based on Peano curves, which leads to a cache oblivious algorithm, instead.

As for matrix-vector multiplication, we start with Algorithm 13.5, where we stress that we simply have to execute n3 updates \({C}_{ik} = {C}_{ik} + {A}_{ij}{B}_{jk}\) for all triples (i, j, k). Thus, matrix multiplication is again an instance of a traversal problem.

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The n3 element operations correspond to a 3D traversal of the triple space. For the access to the matrix elements, in contrast, only two out of three indices are needed for each of the involved matrices. Hence, the indices are obtained via respective projections. We will therefore use a Peano curve for the traversal, because the projections of the classical 3D Peano curve to 2D will again lead to 2D Peano curves. In Fig. 13.4, this property is illustrated for the vertical direction. It holds for the other two coordinate directions, as well. Figure 13.4 also tells us that if we execute the matrix operations \({C}_{ik} = {C}_{ik} + {A}_{ij}{B}_{jk}\) according to a 3D Peano order, the matrix elements will be accessed according to 2D Peano orders. As a consequence, we should store the elements in that order, if possible. We will first try this for the simple case of a 3 ×3 matrix multiplication:
$${ \left (\begin{array}{ccc} {c}_{0} & {c}_{5} & {c}_{6} \\ {c}_{1} & {c}_{4} & {c}_{7} \\ {c}_{2} & {c}_{3} & {c}_{8} \end{array} \right )}_{=: C}\,+=\,{\left (\begin{array}{ccc} {a}_{0} & {a}_{5} & {a}_{6} \\ {a}_{1} & {a}_{4} & {a}_{7} \\ {a}_{2} & {a}_{3} & {a}_{8} \end{array} \right )}_{=: A}{\left (\begin{array}{ccc} {b}_{0} & {b}_{5} & {b}_{6} \\ {b}_{1} & {b}_{4} & {b}_{7} \\ {b}_{2} & {b}_{3} & {b}_{8} \end{array} \right )}_{=: B} $$
(13.1)
(here, the element indices indicate the 2D Peano element order). Then, the 3D Peano traversal of the element operations will lead to the following sequence of element updates:
$$ \begin{array}{lllll} {c}_{0}\,+=\,{a}_{0}{b}_{0} & {c}_{5}\,+=\,{a}_{6}{b}_{3}\rightarrow &{c}_{5}\,+=\,{a}_{5}{b}_{4} & {c}_{6}\,+=\,{a}_{5}{b}_{7}\rightarrow &{c}_{6}\,+=\,{a}_{6}{b}_{8} \\ \qquad \downarrow &\qquad \uparrow &\qquad \downarrow &\qquad \uparrow &\qquad \downarrow \\ {c}_{1}\,+=\,{a}_{1}{b}_{0} & {c}_{4}\,+=\,{a}_{7}{b}_{3} & {c}_{4}\,+=\,{a}_{4}{b}_{4} & {c}_{7}\,+=\,{a}_{4}{b}_{7} & {c}_{7}\,+=\,{a}_{7}{b}_{8} \\ \qquad \downarrow &\qquad \uparrow &\qquad \downarrow &\qquad \uparrow &\qquad \downarrow \\ {c}_{2}\,+=\,{a}_{2}{b}_{0} & {c}_{3}\,+=\,{a}_{8}{b}_{3} & {c}_{3}\,+=\,{a}_{3}{b}_{4} & {c}_{8}\,+=\,{a}_{3}{b}_{7} & {c}_{8}\,+=\,{a}_{8}{b}_{8} \\ \qquad \downarrow &\qquad \uparrow &\qquad \downarrow &\qquad \uparrow & \\ {c}_{2}\,+=\,{a}_{3}{b}_{1} & {c}_{2}\,+=\,{a}_{8}{b}_{2} & {c}_{3}\,+=\,{a}_{2}{b}_{5} & {c}_{8}\,+=\,{a}_{2}{b}_{6} & \\ \qquad \downarrow &\qquad \uparrow &\qquad \downarrow &\qquad \uparrow & \\ {c}_{1}\,+=\,{a}_{4}{b}_{1} & {c}_{1}\,+=\,{a}_{7}{b}_{2} & {c}_{4}\,+=\,{a}_{1}{b}_{5} & {c}_{7}\,+=\,{a}_{1}{b}_{6} & \\ \qquad \downarrow &\qquad \uparrow &\qquad \downarrow &\qquad \uparrow & \\ {c}_{0}\,+=\,{a}_{5}{b}_{1}\rightarrow &{c}_{0}\,+=\,{a}_{6}{b}_{2} & {c}_{5}\,+=\,{a}_{0}{b}_{5}\rightarrow &{c}_{6}\,+=\,{a}_{0}{b}_{6} & \\ \end{array} $$
(13.2)
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Fig. 13.4

Illustration of the projection property of the 3D Peano curve

Note that this scheme follows an inherently local access pattern to the elements: after each element operation, the next operation will re-use one element and access two elements that are direct neighbours of the elements accessed in the previous operation. To extend this simple 3 ×3-scheme to a multiplication algorithm for larger matrices, we need

  • A 2D Peano order that defines the data structure for the matrix elements;

  • A recursive extension of the 3 ×3-scheme in Eq. 13.2, which is basically obtained by using matrix blocks instead of elements;

  • A concept for matrices of arbitrary size, as the standard Peano order will only work for matrices of size \({3}^{k} \times {3}^{k}\).

The 2D Peano order for the elements is derived in a straightforward manner from the iterations of the Peano curve. The respective construction is illustrated in Fig. 13.5. The pattern symbols P, Q, R, and S now denote a numbering scheme for the corresponding subblock in the matrix.
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Fig. 13.5

Recursive construction of the Peano element order to store matrices

13.3.1 Block-Recursive Peano Matrix Multiplication

Let’s now formulate the 3 ×3 multiplication scheme of Eq. (13.2) to a scheme for matrices in Peano order. In a first step, we write the matrices as 3 ×3 block matrices:
$$\left (\begin{array}{ccc} {P}_{A0} & {R}_{A5} & {P}_{A6} \\ {Q}_{A1} & {S}_{A4} & {Q}_{A7} \\ {P}_{A2} & {R}_{A3} & {P}_{A8} \end{array} \right )\left (\begin{array}{ccc} {P}_{B0} & {R}_{B5} & {P}_{B6} \\ {Q}_{B1} & {S}_{B4} & {Q}_{B7} \\ {P}_{B2} & {R}_{B3} & {P}_{B8} \end{array} \right ) = \left (\begin{array}{ccc} {P}_{C0} & {R}_{C5} & {P}_{C6} \\ {Q}_{C1} & {S}_{C4} & {Q}_{C7} \\ {P}_{C2} & {R}_{C3} & {P}_{C8} \end{array} \right ).$$
(13.3)
Here, we named each matrix block according to its numbering scheme (see Fig. 13.5), and indicated the name of the global matrix and the relative position of the block in the Peano order as indices. The element operations of Eq. 13.2 now lead to multiplication of matrix blocks, as in \({P}_{C0}\,+=\,{P}_{A0}{P}_{B0}\), \({Q}_{C1}\,+=\,{Q}_{A1}{P}_{B0}\), \({P}_{C2}\,+=\,{P}_{A2}{P}_{B0}\), etc.
If we just examine the numbering patterns of the matrix blocks, we see that there are exactly eight different types of block multiplications:
$$\begin{array}{lclclcl} P\,+=\,\mathit{PP} &\qquad \qquad &Q\,+=\,\mathit{QP}&\qquad \qquad &R\,+=\,\mathit{PR}&\qquad \qquad &S\,+=\,\mathit{QR} \\ P\,+=\,\mathit{RQ}&&Q\,+=\,\mathit{SQ} &&R\,+=\,\mathit{RS} &&S\,+=\,\mathit{SS}.\end{array}$$
(13.4)
For the block operation \(P\,+=\,\mathit{PP}\), we have already derived the necessary block operations and their optimal sequence of execution in Eq. (13.2). For the other seven types of block multiplications, it turns out that we obtain similar execution patterns, and that no further block operations arise besides those already given in Eq. (13.4). Hence, we obtain a closed system of eight recursive multiplication schemes.

13.3.2 Memory Access Patterns During the Peano Matrix Multiplication

Our next task is to derive execution sequences for the other seven multiplication schemes: \(Q\,+=\,\mathit{QP}\), \(R\,+=\,\mathit{PR}\), etc. We will leave the details for Exercise 13.2, and just state that each multiplication scheme leads to an execution order similar to that given in Eq. (13.2). In addition, all eight execution orders follow the same structure as the scheme for \(P\,+=\,\mathit{PP}\), except that for one, two, or even all three of the involved matrices, the access pattern runs backwards. Table 13.1 lists for the eight different schemes which of the access patterns run backwards.
Table 13.1

Execution orders for the eight different block multiplication schemes. ‘ + ’ indicates that the access pattern for the respective matrix A, B, or C is executed in forward direction (from element 0 to 8). ‘ − ’ indicates backward direction (starting with element 8)

Block scheme

\(P\,+=\,\mathit{PP}\)

\(P\,+=\,\mathit{RQ}\)

\(Q\,+=\,\mathit{QP}\)

\(Q\,+=\,\mathit{SQ}\)

\(R\,+=\,\mathit{PR}\)

\(R\,+=\,\mathit{RS}\)

\(S\,+=\,\mathit{QR}\)

\(S\,+=\,\mathit{SS}\)

 

A

 + 

 + 

 + 

 + 

 − 

 − 

 − 

 − n

 

Access to B

 + 

 + n

 − 

 − 

 + n

 + 

 − 

 − 

 

C

 + 

 − 

 + 

 − n

 + 

 − 

 + 

 − 

 
All eight schemes can be implemented by using only increments and decrements by 1 on the Peano-ordered element indices. Hence, jumps in memory are completely avoided throughout the computation of a 3 ×3 block. Our next step is therefore to make sure that jumps are also avoided between consecutive block schemes. As example, we examine the transfer between the first two block operations in a \(P\,+=\,\mathit{PP}\) scheme: we assume that we just finished the operation \({P}_{C0}\,+=\,{P}_{A0}{P}_{B0}\), and now would go on with \({Q}_{C1}\,+=\,{Q}_{A1}{P}_{B0}\).

  • On matrix A, we have traversed the P-ordered block 0, which means that our last access was to the last element in this block. The \(Q\,+=\,\mathit{QP}\) scheme runs in ‘ + ’ direction on A (see Table 13.1), such that the first access to the Q-ordered block 1 in A will be to the first element, which is of course a direct neighbour of the last element of block 0.

  • In matrix C, we have the identical situation: the P-ordered block 0 has been traversed in ‘ + ’ direction, and the next access will be to the first element in the Q-ordered block 1.

  • In matrix B, both execution sequences work on the P-ordered block 0. However, while the \(P\,+=\,\mathit{PP}\) scheme accesses B in ‘ + ’ direction, the \(Q\,+=\,\mathit{QP}\) scheme will run in ‘ − ’ direction on B. Hence, the last access of the \(P\,+=\,\mathit{PP}\) scheme is to the last element of the block, which will also be the start of the \(Q\,+=\,\mathit{QP}\) scheme.

Hence, the increment/decrement property stays valid between the two block operations. A careful analysis (which is also too tedious to be discussed here) reveals that this is true for all recursive block operations occurring in our multiplication scheme.

As a result, the Peano matrix multiplication can be implemented as illustrated in Algorithm 13.6. There, the schemes \(P\,+=\,\mathit{PP}\), \(Q\,+=\,\mathit{QP}\), etc., are coded via the ‘ + ’- or ‘ − ’-directions of the execution orders for the three matrices, as listed in Table 13.1. The change of direction throughout the recursive calls is coded in the parameters phsA, phsB, and phsC.

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The increment/decrement property leads to an inherently local memory access pattern during the Peano matrix multiplication, which is illustrated in Fig. 13.6. There, we can observe certain locality properties of the memory access:

  • A given range of contiguous operations (highlighted in the left image) will access only a certain contiguous subset of matrix elements.

  • Vice versa, a contiguous subset of matrix elements will be accessed by a set of operations that consists of only a few contiguous sequences.

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Fig. 13.6

Memory access pattern for all three matrices during a 9 ×9 Peano matrix multiplication. The highlighted areas are obtained by partitioning the operations (left image) or the accessed elements of C (right image)

Both, the operator subsets and the range of elements, define partitions that can be used to parallelise the Peano matrix multiplication (following a work-oriented or an owner-computes partitioning, respectively). However, the underlying locality properties also make the algorithm inherently cache efficient, which we will examine in the following section.

13.3.3 Locality Properties and Cache Efficiency

Figure 13.6 illustrates the spatial locality properties of the new matrix multiplication. From the respective chart, we can infer a so-called access locality function, LM(n), which we can define as the maximal possible distance (in memory) between two elements of a matrix M that are accessed within n contiguous operations.

For a loop-based implementation of matrix multiplication, we will typically access a range of k successive elements during k operations – which already requires some optimisations in order to avoid stride-n accesses. For the naive implementation given in Algorithm 13.4, we thus have \({L}_{M}(n) \geq n\) for all involved matrices. For an improved algorithm that operates on matrix blocks of size k ×k, we will typically obtain loops over k contiguous elements, so the worst case will reduce to \({L}_{M}(k) \geq k\) with k ≪ n. However, as long as we stay with a k ×k block, we will perform k3 operations on only k2 elements of a matrix. Hence, if our storage scheme for the matrices uses k ×k-blocks, as well, and stores such elements contiguously, we perform k3 operations on k2 contiguous elements. The best case that could be achieved for the access locality function LM should therefore be \({L}_{A}(k) \approx {k}^{2/3}\). Thus, even a blocked approach has the following limitations on LM:

  • We have only one block size, k0, that can lead to the optimal locality – this could be cured, if we change to a recursive blocking scheme (a first step towards space-filling curves).

  • The locality will still be LM(k) ≥ k, if we are within the smallest block.

  • Between two blocks, the locality will only be obtained, if two successively accessed blocks are stored contiguously in memory.

The recursive blocking and the last property (contiguity) is exactly what is achieved by the Peano multiplication scheme. We can therefore obtain \(L(k) \in \mathcal{O}({k}^{2/3})\) as an upper bound of the extent of the index range for any value of k.

While we have \(L(k) = {k}^{2/3}\), if we exactly hit the block boundaries, i.e., compute the distances for the first element operations of two consecutive block multiplications, we obtain a slightly worse ratio for arbitrary element operations. For a tight estimate, we need to determine the longest streak of not reusing matrix blocks in the Peano algorithm. For matrix B, which is always reused three times, the longest streak is two such block multiplications. For three block multiplications, either the first two or the last two work on the same B-block. In the scheme for matrix A, up to nine consecutive block multiplications can occur until a block is re-used. For matrix C, three contiguous block operations occur. During recursion, though, two such streaks can occur right after each other. For matrix A, we therefore obtain 18 as the length of the longest streak. In the worst case, we can therefore do 18n3 block operations on matrix A on 18n2 contiguous elements of A. Thus, for matrix A, we get that
$${L}_{A}(n) \approx \frac{18} {1{8}^{2/3}}{n}^{2/3} = \root{3}\of{18}{n}^{2/3}.$$
(13.5)
For matrix C, we get a maximum streak length of 6, and therefore
$${L}_{B}(n) \approx \root{3}\of{2}{n}^{2/3},\qquad {L}_{ C}(n) \approx \root{3}\of{6}{n}^{2/3}.$$
(13.6)
If we only consider \(\mathcal{O}({n}^{3})\)-algorithms for matrix multiplication, i.e., if we disregard Strassen’s algorithm and similar approaches, then a locality of \({L}_{B}(n) \in \mathcal{O}({n}^{2/3})\) is the optimum we can achieve. The locality functions \({L}_{A}(n) \leq 3{n}^{2/3}\), \({L}_{B}(n) \leq 2{n}^{2/3}\), and \({L}_{C}(n) \leq 2{n}^{2/3}\) are therefore asymptotically optimal. LB and LC, in addition, involve very low constants.

13.3.4 Cache Misses on an Ideal Cache

The access locality functions provide us with an estimate on how many operations will be performed on a given set of matrix elements. If a set of elements is given by a cache line or even the entire cache, we can estimate the number of cache hits and cache misses of the algorithm. To simplify that calculation, we use the model of an ideal cache, which obeys to the following assumptions:

  • The cache consists of M words that are organized as cache lines of L words each. Hence, we have M ∕ L cache lines. The external memory can be of arbitrary size and is structured into memory lines of L words.

  • The cache is fully associative, i.e., can load any memory line into any cache line.

  • If lines need to be evicted from the cache, we assume that the cache can “foresee the future”, and will always evict a line that is no longer needed, or accessed farthest in the future.

Assume that we are about to start a k ×k block multiplication, where k is the largest power of 3, such that three k ×k matrices fit into the cache. Hence, \(3 \cdot {k}^{2} < M\), but \(3 \cdot {(3k)}^{2} > M\), or
$$\frac{1} {3}\sqrt{\frac{M} {3}} < k < \sqrt{\frac{M} {3}}.$$
(13.7)
The cache will usually (except at start) be occupied by blocks required from previous multiplications; however, one of the three involved blocks will be reused and thus already be stored in the cache. To perform the following element operations, we will successively have to fetch all cache lines that contain the elements of the two new matrix blocks. The ideal cache strategy will ensure that the matrix block that is reused from the previous block multiplication will not be evicted from the cache. Instead, cache lines from the other two blocks will be replaced by the elements for the new block operation. We can also be sure that elements of these new blocks will not be evicted during the entire block operation. Hence, there will be only \(2\left \lceil \frac{{k}^{2}} {L} \right \rceil \) cache line transfers for these two k ×k blocks – to simplify the following computation, we assume that \(\left \lceil \frac{{k}^{2}} {L} \right \rceil = \frac{{k}^{2}} {L}\).
As we will perform (n ∕ k)3 such block operations, the total number of cache line transfers throughout an n ×n multiplication will be
$$T(n) ={ \left (\frac{n} {k}\right )}^{3} \cdot 2 \cdot \left (\frac{{k}^{2}} {L} \right ) = \frac{2n} {kL} \leq \left ( \frac{2n} {\frac{1} {3}\sqrt{\frac{M} {3}} \cdot L}\right ) = 6\sqrt{3} \frac{{n}^{3}} {L\sqrt{M}}.$$
(13.8)
For arbitrary size of the cache lines, we still have \(T(N) \in \mathcal{O}\left ( \frac{{n}^{3}} {L\sqrt{M}}\right )\), with a constant close to \(6\sqrt{3}\). Note that the respective calculation also works, if we have multiple levels of cache memory. Then, the estimate of cache line transfers refers to the respective size M of the different caches. For realistic caches, we might obtain more cache transfers because of a bad replacement strategy. However, a cache that always evicts the cache line that was used longest ago (“least recently used” strategy) will expel those cache lines first that contain matrix elements that are farthest away in terms of location in memory, because the access pattern of the multiplication will make sure that all matrix elements that are “closer” in memory have been accessed at a later time. What we cannot consider at this point is effects due to limited associativity of the cache, i.e., if a cache can place memory lines only into a specific set of cache lines.

13.3.5 Multiplying Matrices of Arbitrary Size

For practical implementations, it turns out that regular CPUs will not run at full performance, if we use a fully recursive implementation. For small matrix blocks, loop-based implementations are much more efficient. One of the reasons is that vector-computing extensions of CPUs can then be exploited. We should therefore stop the recursion on a certain block size k ×k, which is chosen to respect such hardware properties. To extend our algorithm for matrices of arbitrary size, we then have three options:
  1. 1.

    We can embed the given matrices into larger matrices of size \({3}^{p} \times {3}^{p}\) (or, with blocks as leaves: \({3}^{p}k \times {3}^{p}k\)). The additional zeros should, of course, not be stored. A respective approach is described in Sect. 13.4, where such a padding approach is given for band matrices or sparse matrices in general. In such an approach, we will typically stop the recursion on matrix blocks, as soon as these fit into the innermost cache of a CPU.

     
  2. 2.

    In Sect. 8.2.3, we introduced Peano iterations on 3D grids of size \(k \times l \times m\), where k, l, and m may be arbitrary odd numbers. The Peano matrix multiplication also works on such Peano iterations. For the leaf-block operations, we have to introduce schemes for matrix multiplication on \({n}_{1} \times {n}_{2}\), where n1 and n2 may be 3, 5, or 7, respectively. Actually, the scheme will work for leaf blocks of any odd size.

     
  3. 3.

    We can stick to the classical 3 ×3 recursion, if we stop the recursion on larger block sizes. If these are stored in regular row- or column-major order (compare the approach in Sect. 13.4), we just need to specify an upper limit for the size of these blocks, and can then use an implementation that is optimised for small (but arbitrary) matrix sizes.

     

What approach performs best will depend on the specific scenario. The first approach is interesting, for example, if we can choose the size of the leaf-level matrix block such that three such blocks fit exactly into the innermost cache. The constant factor for the number of cache line transfers, as estimated in Eq. (13.8) will then be further reduced. The third approach is especially interesting for coarse-grain parallel computations. There, the leaf-level blocks will be implemented by a call to a sequential library, where the size of the leaf blocks is of reduced influence. See the works indicated in the references section for more details.

13.4 Sparse Matrices and Space-Filling Curves

Sparse matrices are matrices that contain so many zero elements that it becomes worthwhile to change to different data structures and algorithms to store and process them.2 Considering our previous experience with space-filling-curve data structures for matrices and arrays, but also for adaptively refined grids, we can try to use a quadtree-type structure to store matrices that contain a lot of zeros. As illustrated in Fig. 13.7, we recursively subdivide a matrix into smaller and smaller blocks. The substructuring scheme can be a quadtree scheme, but due to our previously introduced Peano algorithm, we again use a 3 ×3 refinement, i.e., a 32-spacetree. Once a matrix block consists entirely of zero elements, we can stop refinement, mark the respective tree with a zero-block leaf, and thus not store the individual elements. For blocks that contain elements, we stop the recursion on blocks of a certain minimal sizes. On such blocks, we either store a dense matrix (could be even in row- or column-major order) or a small sparse matrix (using a respective simple storage scheme). The tree structure for such a storage scheme is also illustrated in Fig. 13.7.
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Fig. 13.7

Illustration of a spacetree storage scheme for sparse matrices. The tree representation of the sparsity pattern is sequentialised according to modified depth-first scheme, where information on the child tree is stored within a parent node

The matrix blocks, either sparse blocks or dense blocks, are stored in the sequential order given by the Peano curve. However, in contrast to the Peano scheme for dense blocks, the matrix blocks will now have varying size – because of the different sparsity patterns of the individual blocks, but also because the zero blocks are not stored at all. Hence, to access a specified block, it will usually be necessary to traverse the sparsity tree from its root. To issue recursive calls on one or several child matrix blocks, a parent node needs information on the exact position of all child blocks in the data structure. Thus, in a node, we will store the start address of all nine child blocks of the matrix. Zero blocks are automatically considered by this scheme by just storing two consecutive identical start addresses. As we typically need both start and end addresses for the children, we also need to store the end address of the last block. Hence, for every parent node, we will store a sequence of ten integers, as indicated by the data stream in Fig. 13.7. We thus obtain a data structure that follows the same idea as the modified depth-first traversal that was introduced for the refinement trees of adaptive grids, in Sect. 10.5.

The sparsity structure information and the matrix elements of the leaf blocks can either be stored in a single data stream or in two separate streams. If we choose to store the elements together with the structure, we will only need to change the data structure for the leaf blocks. These blocks will then require information on the block size and type (dense or sparse), as well as on the extent of the block in terms of bytes in memory. If matrix elements are stored in a separate stream, the pre-leaf-level nodes need to store respective pointers to the start of the respective blocks in the elements stream.

As already indicated in Sect. 13.3.5, we can also use the presented sparse-matrix data structure for matrices that are only dense (or comparably dense) in certain parts of the matrix – one simple example would be band matrices. In that case, we can further simplify the data structure by allowing only dense matrix blocks in the leaves.

13.5 References and Further Readings

The Peano-curve algorithm for matrix multiplication was introduced in [24], where we particularly discussed the cache properties of the algorithm. The locality properties, as given in Eqs. 13.5 and 13.8 are asymptotically optimal – respective lower bounds were proven by Hong and Kung [132]. Hardware-oriented, efficient implementations of the algorithms, including the parallelisation on shared-memory multicore platforms, were presented in [21, 127]. In [19], we discussed the parallelisation of the algorithm using message passing on distributed-memory. There, the result for cache line transfers in the ideal-cache model can be used to estimate the number of transferred matrix blocks in a distributed-memory setting. The extension for sparse matrices was presented in [22], which also includes a discussion of LU-decomposition based on the Peano matrix multiplication.

Since the advent of cache-based architectures, improving the cache efficiency of linear algebra operations has been an active area of research. Blocking approaches to improve the cache-efficiency were already applied in the first level-3 BLAS routines, when introduced by Dongarra et al. [78]. Blocking approaches and the block-oriented matrix operations were also a driving factor in the development of LAPACK [13], whose implementation was consequently based on exploiting the higher performance of BLAS 3 routines. Since then, blocking and tiling of matrix operations has become a standard technique for high performance libraries. The ATLAS project [267] uses automatic tuning of blocking sizes to the available memory hierarchy, and GotoBLAS [104] explicitly considers the translation lookaside buffer (TLB) and even virtual memory as further cache levels [103].

Block matrix layouts to improve cache efficiency were introduced by Chatterjee et al. [65], who studied Morton order and 4D tiled arrays (i.e., 2D arrays of matrix blocks), and by Gustavson [116], who demonstrated that recursive blocking automatically leads to cache-efficient algorithms. Frens and Wise [90] used quadtree decompositions of dense matrices and respective recursive implementations of matrix multiplication and of QR decomposition [91]. The term cache-oblivious for such inherently cache-efficient algorithms has been introduced by Frigo et al. [92]. For a review of both cache-oblivious and cache-aware algorithms in linear algebra, see Elmroth et al. [81]. For further, recent work, see [106, 255], e.g. Yotov et al. [112, 281] compared the performance of cache-oblivious algorithms with carefully tuned cache-aware approaches, and identified efficient prefetching of matrix blocks as a crucial question for recursive algorithms. For the Peano algorithm, the increment/decrement access to blocks apparently solves this problem. In any case, block-oriented data structures and algorithms are more and more considered a necessity in the design of linear algebra routines. The designated LAPACK-successor PLASMA [56, 57], for example, aims for a stronger block orientation, and similar considerations drive the FLAME project [113]. As libraries will often depend on row-major or column-major storage, changing the matrix storage to blocked layouts on-the-fly becomes necessary; respective in-place format conversions were studied in [115].

For sparse matrices, quadtree data structures were, for example, examined by Wise et al. [271, 272]. Hybrid “hypermatrices” that mix dense and sparse blocks in recursively structured matrices were discussed by Herrero et al. [130]. Haase [117] used a Hilbert-order data structure for sparse-matrix-vector multiplication to improve cache efficiency.

Chapter 14 will present a cache-oblivious approach to solve partial differential equations on adaptive discretisation grids – hence, we save all further references to work on cache oblivious algorithms, be it grid-based or other simulation approaches, for the references section of Chap. 14.

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13.6 Exercises

Try to determine the number of cache misses caused by Algorithm 13.3, following the same cache model as in Sect. 13.3.4. Focus, in particular, on the misses caused by the access to the vectors x and y.

Use the graph-based illustration of Eq. (13.2) to derive the execution orders for the matrix multiplication schemes \(Q\,+=\,\mathit{QP}\), \(R\,+=\,\mathit{PR}\), \(S\,+=\,\mathit{QR}\), etc.

Consider an algorithm that uses 2D Morton order to store the matrix elements, and 3D Morton order to execute the individual element operations of matrix multiplication. Analyse the number of cache misses causes by this algorithm, using a cache model and computation as in Sect. 13.3.4.

As a data structure for sparse matrices, we could also refine the sparsity tree up to single elements, i.e., not stop the recursion on larger blocks. Give an estimate on how much memory we will require to store a sparse matrix (make suitable assumptions on the sparsity pattern or sparsity tree, if necessary).

Footnotes

  1. 1.

    Currently, here, means in the year 2010.

  2. 2.

    Following a definition given by Wilkinson.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Bader
    • 1
  1. 1.Department of InformaticsTechnische Universität MünchenMunichGermany

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