Principle of Locality

When a program executes on a computer, most of the memory references are not made uniformly to a small number of locations. Here the Locality of the reference does matter.

Locality of Reference, also known as the Principle of Locality, the phenomenon of the same value or related storage locations being frequently accessed. Locality occurs in time(temporal locality) and in space (spatial 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, eg, traversing the elements in a one-dimensional array.

To be very simple when exhibiting spatial locality, a program accesses consecutive memory locations and during temporal locality of reference a program repeatedly accesses the same memory location during a short time period. Both forms of locality occur in the following Pascal code segment:

  for i := 0 to 10 do
   A [i] := 0;

In the above Pascal code, the variable 'i' is referenced several times in for loop where 'i' is compared against 10 to see if the loop is complete and also incremented by one at the end of the loop. This shows temporal locality of reference in action since the CPU accesses 'i' at different points in a short time period.

This program also exhibits spatial locality of reference. The loop itself zeros out the elements of array A by writing a zero to the first location in A, then to the second location in A, and so on. Assume Pascal stores elements of A into consecutive memory locations then on each loop iteration it accesses adjacent memory locations.

Cache Memory - Set Associative Mapped Cache

Set Associative mapping scheme combines the simplicity of Direct mapping with the flexibility of Fully Associative mapping. It is more practical than Fully Associative mapping because the associative portion is limited to just a few slots that make up a set.

In this mapping mechanism, the cache memory is divided into 'v' sets, each consisting of 'n' cache lines. A block from Main memory is first mapped onto a specific cache set, and then it can be placed anywhere within that set. This type of mapping has very efficient ratio between implementation and efficiency. The set is usually chosen by

Cache set number = (Main memory block number) MOD (Number of sets in the cache memory)

If there are 'n' cache lines in a set, the cache placement is called n-way set associative i.e. if there are two blocks or cache lines per set, then it is a 2-way set associative cache mapping and four blocks or cache lines per set, then it is a 4-way set associative cache mapping.

Let us assume we have a Main Memory of size 4GB (2³²), with each byte directly addressable by a 32-bit address. We will divide Main memory into blocks of each 32 bytes (2⁵). Thus there are 128M (i.e. 2³²/2⁵ = 2²⁷) blocks in Main memory.

We have a Cache memory of 512KB (i.e. 2¹⁹), divided into blocks of each 32 bytes (2⁵). Thus there are 16K (i.e. 2¹⁹/2⁵ = 2¹⁴) blocks also known as Cache slots or Cache lines in cache memory. It is clear from above numbers that there are more Main memory blocks than Cache slots.

NOTE: The Main memory is not physically partitioned in the given way, but this is the view of Main memory that the cache sees.

NOTE: We are dividing both Main Memory and cache memory into blocks of same size i.e. 32 bytes.

Let us try 2-way set associative cache mapping i.e. 2 cache lines per set. We will divide 16K cache lines into sets of 2 and hence there are 8K (2¹⁴/2 = 2¹³) sets in the Cache memory.

Cache Size = (Number of Sets) * (Size of each set) * (Cache line size)

So even using the above formula we can find out number of sets in the Cache memory i.e.

2¹⁹ = (Number of Sets) * 2 * 2⁵

Number of Sets = 2¹⁹ / (2 * 2⁵) = 2¹³.

When an address is mapped to a set, the direct mapping scheme is used, and then associative mapping is used within a set.

The format for an address has 13 bits in the set field, which identifies the set in which the addressed word will be found if it is in the cache. There are five bits for the word field as before and there is 14-bit tag field that together make up the remaining 32 bits of the address as shown below:
As an example of how the set associative cache views a Main memory address, consider again the address (A035F014)₁₆. The leftmost 14 bits form the tag field, followed by 13 bits for the set field, followed by five bits for the word field as shown below:
In the below example we have chosen the block 14 from Main memory and compared it with the different block replacement algorithms. In Direct Mapped cache it can be placed in Frame 6 since 14 mod 8 = 6. In Set associative cache it can be placed in set 2.
Checkout one more solved problem below.

References

1. Computer Architecture Tutorial - By Gurpur M. Prabhu.
2. Computer Architecture And Organization: An Integrated Approach - By Murdocca & Vincent Heuring

Cache Memory - Fully Associative Mapped Cache

If a Main memory block can be placed in any of the Cache slots, then the cache is said to be mapped in fully associative.

Let us assume we have a Main Memory of size 4GB (2³²), with each byte directly addressable by a 32-bit address. We will divide Main memory into blocks of each 32 bytes (2⁵). Thus there are 128M (i.e. 2³²/2⁵ = 2²⁷) blocks in Main memory.

We have a Cache memory of 512KB (i.e. 2¹⁹), divided into blocks of each 32 bytes (2⁵). Thus there are 16K (i.e. 2¹⁹/2⁵ = 2¹⁴) blocks also known as Cache slots or Cache lines in cache memory. It is clear from above numbers that there are more Main memory blocks than Cache slots.

NOTE: The Main memory is not physically partitioned in the given way, but this is the view of Main memory that the cache sees.

NOTE: We are dividing both Main Memory and cache memory into blocks of same size i.e. 32 bytes.

In fully associative mapping any one of the 128M (i.e. 2²⁷) Main memory blocks can be mapped into any of the single Cache slot. To keep track of which one of the 2²⁷ possible blocks is in each slot, a 27-bit tag field is added to each slot which holds an identifier in the range from 0 to 2²⁷ – 1. The tag field is the most significant 27 bits of the 32-bit memory address presented to the cache.

In an associative mapped cache, each Main memory block can be mapped to any slot. The mapping from main memory blocks to cache slots is performed by partitioning an address into fields for the tag and the word (also known as the “byte” field) as shown below:

When a reference is made to a Main memory address, the cache hardware intercepts the reference and searches the cache tag memory to see if the requested block is in the cache. For each slot, if the valid bit is 1, then the tag field of the referenced address is compared with the tag field of the slot. All of the tags are searched in parallel, using an associative memory. If any tag in the cache tag memory matches the tag field of the memory reference, then the word is taken from the position in the slot specified by the word field. If the referenced word is not found in the cache, then the main memory block that contains the word is brought into the cache and the referenced word is then taken from the cache. The tag, valid, and dirty fields are updated, and the program resumes execution.

Associative mapped cache has the advantage of placing any main memory block into any cache line. This means that regardless of how irregular the data and program references are, if a slot is available for the block, it can be stored in the cache. This results in considerable hardware overhead needed for cache bookkeeping.

Although this mapping scheme is powerful enough to satisfy a wide range of memory access situations, there are two implementation problems that limit performance.

• The process of deciding which slot should be freed when a new block is brought into the cache can be complex. This process requires a significant amount of hardware and introduces delays in memory accesses.
• When the cache is searched, the tag field of the referenced address must be compared with all 2¹⁴ tag fields in the cache.

Cache Memory - Direct Mapped Cache

If each block from main memory has only one place it can appear in the cache, the cache is said to be Direct Mapped. Inorder to determine to which Cache line a main memory block is mapped we can use the formula shown below

Cache Line Number = (Main memory Block number) MOD (Number of Cache lines)

Let us assume we have a Main Memory of size 4GB (2³²), with each byte directly addressable by a 32-bit address. We will divide Main memory into blocks of each 32 bytes (2⁵). Thus there are 128M (i.e. 2³²/2⁵ = 2²⁷) blocks in Main memory.

We have a Cache memory of 512KB (i.e. 2¹⁹), divided into blocks of each 32 bytes (2⁵). Thus there are 16K (i.e. 2¹⁹/2⁵ = 2¹⁴) blocks also known as Cache slots or Cache lines in cache memory. It is clear from above numbers that there are more Main memory blocks than Cache slots.

NOTE: The Main memory is not physically partitioned in the given way, but this is the view of Main memory that the cache sees.

NOTE: We are dividing both Main Memory and cache memory into blocks of same size i.e. 32 bytes.

A set of 8k (i.e. 2²⁷/2¹⁴ = 2¹³) Main memory blocks are mapped onto a single Cache slot. In order to keep track of which of the 2¹³ possible Main memory blocks are in each Cache slot, a 13-bit tag field is added to each Cache slot which holds an identifier in the range from 0 to 2¹³ – 1.

All the tags are stored in a special tag memory where they can be searched in parallel. Whenever a new block is stored in the cache, its tag is stored in the corresponding tag memory location.

When a program is first loaded into Main memory, the Cache is cleared, and so while a program is executing, a valid bit is needed to indicate whether or not the slot holds a block that belongs to the program being executed. There is also a dirty bit that keeps track of whether or not a block has been modified while it is in the cache. A slot that is modified must be written back to the main memory before the slot is reused for another block. When a program is initially loaded into memory, the valid bits are all set to 0. The first instruction that is executed in the program will therefore cause a miss, since none of the program is in the cache at this point. The block that causes the miss is located in the main memory and is loaded into the cache.

This scheme is called "direct mapping" because each cache slot corresponds to an explicit set of main memory blocks. For a direct mapped cache, each main memory block can be mapped to only one slot, but each slot can receive more than one block.

The mapping from main memory blocks to cache slots is performed by partitioning an main memory address into fields for the tag, the slot, and the word as shown below:

The 32-bit main memory address is partitioned into a 13-bit tag field, followed by a 14-bit slot field, followed by a 5-bit word field. When a reference is made to a main memory address, the slot field identifies in which of the 2¹⁴ cache slots the block will be found if it is in the cache.

If the valid bit is 1, then the tag field of the referenced address is compared with the tag field of the cache slot. If the tag fields are the same, then the word is taken from the position in the slot specified by the word field. If the valid bit is 1 but the tag fields are not the same, then the slot is written back to main memory if the dirty bit is set, and the corresponding main memory block is then read into the slot. For a program that has just started execution, the valid bit will be 0, and so the block is simply written to the slot. The valid bit for the block is then set to 1, and the program resumes execution.

Check out one more solved problem below


References

1. Computer Architecture Tutorial - By Gurpur M. Prabhu.

Cache Memory - Part1

Cache memory is a small (in size) and very fast (zero wait state) memory which sits between the CPU and main memory. Unlike normal memory, the bytes appearing within a cache do not have fixed addresses. Instead, cache memory can reassign the address of a data object. This allows the system to keep recently accessed values in the cache.

Cache Memory

Using the Principle of Locality to improve performance while keeping the memory system affordable we can pose 4 questions about any level of memory hierarchy and we will answer those questions considering one level of memory hierarchy for e.g. cache in our case.

• Block Placement - Where should a block be placed in the cache?
• Block Identification -How to confirm if a block is in the cache or not?
• Block Replacement -Which block frame in the cache should be replaced upon a miss?
• Interaction Policies with Main Memory - What happens when reads and writes are done in the cache?

Block Placement
A number of hardware schemes have been developed for translating main memory addresses to cache memory addresses. The user does not need to know much about the address translation circuitry, which has the advantage, that cache memory enhancements can be introduced into a computer without a corresponding need for modifying application software.

Basically number of cache lines are very less than the number of main memory blocks. As a result an algorithm is needed for mapping main memory blocks into cache lines. Also a means is needed for determining which main memory block currently occupies a cache line.

The choice of cache mapping scheme affects cost and performance, and there is no single best method that is appropriate for all situations. There are three methods in block placement namely

• Direct Mapped Cache
• Fully Associative Mapped Cache
• Set Associative Mapped Cache

Block Identification
In general a cache has two important parts; the cache data line and the cache tags. But in granular it can shown as below

• Valid Bit : is set to 1 when a valid data is stored in cache.
• Dirty Bit : is set to 1 when data is changed and is not updated to main memory in the same time.
• Tag : this field tells which address is in that line.
• Data : the data fetched from main memory.

Since a cache is typically smaller than an entire address space, there is a possibility that any particular requested data is not present in the cache. Therefore there must be some mechanism to determine whether any requested data is present in the cache or not. The tags fill this purpose.

Cache has an address tag on each block frame that gives the block address. Therefore tag entry of every cache block is checked to see if it matches the block address from the CPU. As a rule, all possible tags are searched in parallel because speed is critical.

There must be a way to know that a cache block does/doesn’t have valid information. The most common procedure is to add a valid bit to the tag to say whether or not this entry contains a valid address. If the bit is not set, there cannot be a match on this address. Accordingly an address, generated by CPU (or main memory address) is divided as shown below:

Courtesy: Various parts into which main menory address is divided.

The first division is between the block address and the block offset. The block frame address can be further divided into the tag field and the index field. The block-offset field selects the desired data from the block, the index field selects the set, and the tag field is compared against it for a hit. Although the comparison could be made on more of the address than the tag, there is no need because of the following:

If the total cache size is kept the same, increasing associativity increases the number of blocks per set, thereby decreasing the size of the index and increasing the size of the tag.

Block Replacement
When a miss occurs, the cache controller must select a block to be replaced with the desired data. A benefit of direct-mapped placement is that hardware decisions are simplified - in fact, so simple that there is no choice: Only one block frame is checked for a hit, and only that block can be replaced. With fully associative or set-associative placement, there are many blocks to choose from on a miss. There are three primary strategies employed for selecting which block to replace:

Random - To spread allocation uniformly, candidate blocks are randomly selected. Some systems generate pseudorandom block numbers to get reproducible behavior, which is particularly useful when debugging hardware.

Advantage : simple to implement in hardware
Disadvantage : ignores Principle of Locality

Least-recently used (LRU) - To reduce the chance of throwing out information that will be needed soon, accesses to blocks are recorded. Relying on the past to predict the future, the block replaced is the one that has been unused for the longest time. LRU relies on a corollary of locality: If recently used blocks are likely to be used again, then a good candidate for disposal is the least-recently used block.

Advantage : takes locality into account
Disadvantage : as the number of blocks to keep track of increases, LRU becomes more expensive (harder to implement, slower and often just approximated).

First In First Out (FIFO) - Because LRU can be complicated to calculate, this approximates LRU by determining the oldest block rather than the LRU.

Checkout part2.

References
1. Computer Architecture - A Quantitative Approach, Third Edition.