Locality of reference

In computer science, locality of reference, also known as the principle of locality, is the phenomenon of the same value or related storage locations being frequently accessed. There are two basic types of reference locality. Temporal locality refers to the reuse of specific data and/or resources within relatively small time durations. Spatial locality refers to the use of data elements within relatively close storage locations. Sequential locality, a special case of spatial locality, occurs when data elements are arranged and accessed linearly, e.g., traversing the elements in a one-dimensional array.

Locality is merely one type of predictable behavior that occurs in computer systems. Systems which exhibit strong locality of reference are good candidates for performance optimization through the use of techniques, like the cache and instruction prefetch technology for memory, or like the advanced branch predictor at the pipelining of processors.

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Locality of reference

The locality of reference, also known as the locality principle, is the phenomenon that the collection of the data locations referenced in a short period of time in a running computer often consists of relatively well predictable clusters. Important special cases of locality are temporal, spatial, equidistant and branch locality.

  • Temporal locality: if at one point in time a particular memory location is referenced, then it is likely that the same location will be referenced again in the near future. There is a temporal proximity between the adjacent references to the same memory location. In this case it is common to make efforts to store a copy of the referenced data in special memory storage, which can be accessed faster. Temporal locality is a very special case of the spatial locality, namely when the prospective location is identical to the present location.
  • Spatial locality: if a particular memory location is referenced at a particular time, then it is likely that nearby memory locations will be referenced in the near future. In this case it is common to attempt to guess the size and shape of the area around the current reference for which it is worthwhile to prepare faster access.
  • Equidistant locality: it is halfway between the spatial locality and the branch locality. Consider a loop accessing locations in an equidistant pattern, i.e. the path in the spatial-temporal coordinate space is a dotted line. In this case, a simple linear function can predict which location will be accessed in the near future.
  • Branch locality: if there are only few amount of possible alternatives for the prospective part of the path in the spatial-temporal coordinate space. This is the case when an instruction loop has a simple structure, or the possible outcome of a small system of conditional branching instructions is restricted to a small set of possibilities. Branch locality is typically not a spatial locality since the few possibilities can be located far away from each other.

In order to make benefit from the very frequently occurring temporal and spatial kind of locality, most of the information storage systems are hierarchical; see below. The equidistant locality is usually supported by the diverse nontrivial increment instructions of the processors. For the case of branch locality, the contemporary processors have sophisticated branch predictors, and on the base of this prediction the memory manager of the processor tries to collect and preprocess the data of the plausible alternatives.

Reasons for locality

There are several reasons for locality. These reasons are either goals to achieve or circumstances to accept, depending on the aspect. The reasons below are not disjoint; in fact, the list below goes from the most general case to special cases.

  • Predictability: In fact, locality is merely one type of predictable behavior in computer systems. Luckily, many of the practical problems are decidable and hence the corresponding program can behave predictably, if it is well written.
  • Structure of the program: Locality occurs often because of the way in which computer programs are created, for handling decidable problems. Generally, related data is stored in nearby locations in storage. One common pattern in computing involves the processing of several items, one at a time. This means that if a lot of processing is done, the single item will be accessed more than once, thus leading to temporal locality of reference. Furthermore, moving to the next item implies that the next item will be read, hence spatial locality of reference, since memory locations are typically read in batches.
  • Linear data structures: Locality often occurs because code contains loops that tend to reference arrays or other data structures by indices. Sequential locality, a special case of spatial locality, occurs when relevant data elements are arranged and accessed linearly. For example, the simple traversal of elements in a one-dimensional array, from the base address to the highest element would exploit the sequential locality of the array in memory.[1] The more general equidistant locality occurs when the linear traversal is over a longer area of adjacent data structures having identical structure and size, and in addition to this, not the whole structures are in access, but only the mutually corresponding same elements of the structures. This is the case when a matrix is represented as a sequential matrix of rows and the requirement is to access a single column of the matrix.

Use of locality in general

If most of the time the substantial portion of the references aggregate into clusters, and if the shape of this system of clusters can be well predicted, then it can be used for speed optimization. There are several ways to make benefit from locality. The common techniques for optimization are:

  • to increase the locality of references. This is achieved usually on the software side.
  • to exploit the locality of references. This is achieved usually on the hardware side. The temporal and spatial locality can be capitalized by hierarchical storage hardwares. The equidistant locality can be used by the appropriately specialized instructions of the processors, this possibility is not only the responsibility of hardware, but the software as well, whether its structure is suitable for compiling a binary program which calls the specialized instructions in question. The branch locality is a more elaborate possibility, hence more developing effort is needed, but there is much larger reserve for future exploration in this kind of locality than in all the remaining ones.

Use of spatial and temporal locality: hierarchical memory

Hierarchical memory is a hardware optimization that takes the benefits of spatial and temporal locality and can be used on several levels of the memory hierarchy. Paging obviously benefits from temporal and spatial locality. A cache is a simple example of exploiting temporal locality, because it is a specially designed faster but smaller memory area, generally used to keep recently referenced data and data near recently referenced data, which can lead to potential performance increases. Data in cache does not necessarily correspond to data that is spatially close in main memory; however, data elements are brought into cache one cache line at a time. This means that spatial locality is again important: if one element is referenced, a few neighboring elements will also be brought into cache. Finally, temporal locality plays a role on the lowest level, since results that are referenced very closely together can be kept in the machine registers. Programming languages such as C allow the programmer to suggest that certain variables are kept in registers.

Data locality is a typical memory reference feature of regular programs (though many irregular memory access patterns exist). It makes the hierarchical memory layout profitable. In computers, memory is divided up into a hierarchy in order to speed up data accesses. The lower levels of the memory hierarchy tend to be slower, but larger. Thus, a program will achieve greater performance if it uses memory while it is cached in the upper levels of the memory hierarchy and avoids bringing other data into the upper levels of the hierarchy that will displace data that will be used shortly in the future. This is an ideal, and sometimes cannot be achieved.

Typical memory hierarchy (access times and cache sizes are approximations of typical values used as of 2006 for the purpose of discussion; actual values and actual numbers of levels in the hierarchy vary):

  • CPU registers (8-32 registers) – immediate access
  • L1 CPU caches (32 KiB to 128 KiB) – fast access
  • L2 CPU caches (128 KiB to 12 MiB) – slightly slower access
  • Main physical memory (RAM) (256 MiB to 24 GiB) – slow access
  • Disk (file system) (100 GiB to 2 TiB) – very slow
  • Remote Memory (such as other computers or the Internet) (Practically unlimited) – speed varies

Modern machines tend to read blocks of lower memory into the next level of the memory hierarchy. If this displaces used memory, the operating system tries to predict which data will be accessed least (or latest) and move it down the memory hierarchy. Prediction algorithms tend to be simple to reduce hardware complexity, though they are becoming somewhat more complicated.

Spatial and temporal locality example: matrix multiplication

A common example is matrix multiplication:

for i in 0..n
  for j in 0..m
    for k in 0..p
      C[i][j] = C[i][j] + A[i][k] * B[k][j];

When dealing with large matrices, this algorithm tends to shuffle data around too much. Since memory is pulled up the hierarchy in consecutive address blocks, in the C programming language it would be advantageous to refer to several memory addresses that share the same row (spatial locality). By keeping the row number fixed, the second element changes more rapidly. In C and C++, this means the memory addresses are used more consecutively. One can see that since j affects the column reference of both matrices C and B, it should be iterated in the innermost loop (this will fix the row iterators, i and k, while j moves across each column in the row). This will not change the mathematical result, but it improves efficiency. By switching the looping order for j and k, the speedup in large matrix multiplications becomes dramatic. (In this case, 'large' means, approximately, more than 100,000 elements in each matrix, or enough addressable memory such that the matrices will not fit in L1 and L2 caches.)

Temporal locality can also be improved in the above example by using a technique called blocking. The larger matrix can be divided into evenly-sized sub-matrices, so that the smaller blocks can be referenced (multiplied) several times while in memory.

for (ii = 0; ii < SIZE; ii += BLOCK_SIZE)
  for (kk = 0; kk < SIZE; kk += BLOCK_SIZE)
    for (jj = 0; jj < SIZE; jj += BLOCK_SIZE)
      for (i = ii; i < ii + BLOCK_SIZE && i < SIZE; i++)
        for (k = kk; k < kk + BLOCK_SIZE && k < SIZE; k++)
          for (j = jj; j < jj + BLOCK_SIZE && j < SIZE; j++)
            C[i][j] = C[i][j] + A[i][k] * B[k][j];

The temporal locality of the above solution is provided because a block can be used several times before moving on, so that it is moved in and out of memory less often. Spatial locality is improved because elements with consecutive memory addresses tend to be pulled up the memory hierarchy together.

See also

Bibliography

  • P.J. Denning, The Locality Principle, Communications of the ACM, Volume 48, Issue 7, (2005), Pages 19–24
  • P.J. Denning, S.C. Schwartz, Communications of the ACM, Volume 15, Issue 3 (March 1972), Pages 191-198

References

  1. ^ Aho, Lam, Sethi, and Ullman. "Compilers: Principles, Techniques & Tools" 2nd ed. Pearson Education, Inc. 2007

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