Lookup table
History
Part of a 20th century table of common logarithms in the reference book Abramowitz and Stegun.
Before the advent of computers, printed lookup tables of values were used by people to speed up hand calculations of complex functions, such as in trigonometry, logarithms, and statistical density functions. School children are often taught to memorize “times tables” to avoid calculations of the most commonly used numbers (up to 9 x 9 or 12 x 12). Even as early as 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were “a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144”
Early in the history of computers, input/output operations were particularly slow – even in comparison to processor speeds of the time. It made sense to reduce expensive read operations by a form of manual caching by creating either static lookup tables (embedded in the program) or dynamic prefetched arrays to contain only the most commonly occurring data items. Despite the introduction of systemwide caching that now automates this process, application level lookup tables can still improve performance for data items that rarely, if ever, change.
Examples
Simple lookup in an array, an associative array or a linked list (unsorted list)
This is known as a linear search or Brute-force search, each element being checked for equality in turn and the associated value, if any, used as a result of the search. This is often the slowest search method unless frequently occurring values occur early in the list. For a one dimensional array or linked list, the lookup is usually to determine whether or not there is a match with an ‘input’ data value.
Linked lists vs. arrays
Main article: Linked lists
Linked lists have several advantages over arrays:
Insertion or deletion of an element at a specific point of a list is a constant time operation. (While one can “delete” an element from an array in constant time by somehow marking its slot as “vacant”, an algorithm that iterates over the elements may have to skip a large number of vacant slots).
arbitrarily many elements may be inserted into a linked list, limited only by the total memory available; while an array will eventually fill up, and then have to be resized an expensive operation, that may not even be possible if memory is fragmented. Similarly, an array from which many elements are removed, may have to be resized in order to avoid wasting too much space.
On the other hand:
arrays allow random access, while linked lists allow only sequential access to elements. Singly-linked lists, in fact, can only be traversed in one direction. This makes linked lists unsuitable for applications where it’s useful to quickly look up an element by its index , such as heapsort. See also trivial hash function below.
Sequential access on arrays is also faster than on linked lists on many machines, because they have greater locality of reference and thus benefit more from processor caching.
linked lists require extra storage needed for references, that often makes them impractical for lists of small data items such as characters or boolean values. It can also be slow, and with a nave allocator, wasteful, to allocate memory separately for each new element, a problem generally solved using memory pools.
Some hybrid solutions try to combine the advantages of the two representations. Unrolled linked lists store several elements in each list node, increasing cache performance while decreasing memory overhead for references. CDR coding does both these as well, by replacing references with the actual data referenced, which extends off the end of the referencing record.
Binary search in an array or an associative array (sorted list)
This is known as a binary chop search, sometimes referred to as a “Divide and conquer algorithm”, each element being found by determining which half of the table a match may be found in and repeating until either success or failure. Only possible if the list is sorted but gives good performance even if the list is lengthy.
Trivial hash function
For a trivial hash function lookup, the unsigned raw data value is used directly as an index to a one dimensional table to extract a result. For small ranges, this can be amongst the fastest lookup, even exceeding binary search speed with zero branches and executing in constant time.
Counting ‘1’ bits in a series of bytes
One discrete problem that is expensive to solve on many computers, is that of counting the number of bits which are set to 1 in a (binary) number, sometimes called the population function. For example, the decimal number “37” is “00100101” in binary, so it contains three bits that are set to binary “1”.
A simple example of C code, designed to count the 1 bits in a int, might look like this:
int count_ones(unsigned int x) {
int result = 0;
while (x != 0)
result++, x = x & (x-1);
return result;
}
This apparently simple algorithm can take potentially hundreds of cycles even on a modern architecture, because it makes many branches in the loop – and branching is slow. This can be ameliorated using loop unrolling and some other more compiler optimizations. There is however a simple and much faster algorithmic solution – using a trivial hash function table lookup.
Simply construct a static table, bits_set, with 256 entries giving the number of one bits set in each possible byte value (e.g. 0x00 = 0, 0x01 = 1, 0x02 = 1, and so on). Then use this table to find the number of ones in each byte of the integer using a trivial hash function lookup on each byte in turn, and sum them. This requires no branches, and just four indexed memory accesses, considerably faster than the earlier code.
/* (this code assumes that ‘int’ is 32-bits wide) */
int count_ones(unsigned int x) {
return bits_set[ x & 255] + bits_set[(x >> 8) & 255]
+ bits_set[(x >> 16) & 255] + bits_set[(x >> 24) & 255];
}
The above source can be improved easily, (avoiding OR’ing, and shifting) by ‘recasting’ ‘x’ as a 4 byte unsigned char array and, preferably, coded in-line as a single statement instead of being a function. Note that even this simple algorithm can be too slow now, because the original code might run faster from the cache of modern processors, and (large) lookup tables do not fit well in caches and can cause a slower access to memory (in addition, in the above example, it requires computing addresses within a table, to perform the four lookups needed.
L.U.T.s in Image processing
In data analysis applications, such as image processing, a lookup table (LUT) is used to transform the input data into a more desirable output format. For example, a grayscale picture of the planet Saturn will be transformed into a color image to emphasize the differences in its rings.
A classic example of reducing run-time computations using lookup tables is to obtain the result of a trigonometry calculation, such as the sine of a value. Calculating trigonometric functions can substantially slow a computing application. The same application can finish much sooner when it first precalculates the sine of a number of values, for example for each whole number of degrees (The table can be defined as static variables at compile time, reducing repeated run time costs). When the program requires the sine of a value, it can use the lookup table to retrieve the closest sine value from a memory address, and may also take the step of interpolating to the sine of the desired value, instead of calculating by mathematical formula. Lookup tables are thus used by mathematics co-processors in computer systems. An error in a lookup table was responsible for Intel’s infamous floating-point divide bug.
Functions of a single variable (such as sine and cosine) may be implemented by a simple array. Functions involving two or more variables require multidimensional array indexing techniques. The latter case may thus employ a two-dimensional array of power[x][y] to replace a function to calculate xy for a limited range of x and y values. Functions that have more than one result may be implemented with lookup tables that are arrays of structures.
As mentioned, there are intermediate solutions that use tables in combination with a small amount of computation, often using interpolation. Pre-calculation combined with interpolation can produce higher accuracy for values that fall between two precomputed values. This technique requires slightly more time to be performed but can greatly enhance accuracy in applications that require the higher accuracy. Depending on the values being precomputed, pre-computation with interpolation can also be used to shrink the lookup table size while maintaining accuracy.
In image processing, lookup tables are often called LUTs and give an output value for each of a range of index values. One common LUT, called the colormap or palette, is used to determine the colors and intensity values with which a particular image will be displayed. Windowing in computed tomography refers to a related concept.
While often effective, employing a lookup table may nevertheless result in a severe penalty if the computation that the LUT replaces is relatively simple. Memory retrieval time and the complexity of memory requirements can increase application operation time and system complexity relative to what would be required by straight formula computation. The possibility of polluting the cache may also become a problem. Table accesses for large tables will almost certainly cause a cache miss. This phenomenon is increasingly becoming an issue as processors outpace memory. A similar issue appears in rematerialization, a compiler optimization. In some environments, such as the Java programming language, table lookups can be even more expensive due to mandatory bounds-checking involving an additional comparison and branch for each lookup.
There are two fundamental limitations on when it is possible to construct a lookup table for a required operation. One is the amount of memory that is available: one cannot construct a lookup table larger than the space available for the table, although it is possible to construct disk-based lookup tables at the expense of lookup time. The other is the time required to compute the table values in the first instance; although this usually needs to be done only once, if it takes a prohibitively long time, it may make the use of a lookup table an inappropriate solution. As previously stated however, tables can be statically defined in many cases.
Computing sines
Most computers, which only perform basic arithmetic operations, cannot directly calculate the sine of a given value. Instead, they use the CORDIC algorithm or a complex formula such as the following Taylor series to compute the value of sine to a high degree of precision:
(for x close to 0)
However, this can be expensive to compute, especially on slow processors, and there are many applications, particularly in traditional computer graphics, that need to compute many thousands of sine values every second. A common solution is to initially compute the sine of many evenly distributed values, and then to find the sine of x we choose the sine of the value closest to x. This will be close to the correct value because sine is a continuous function with a bounded rate of change. For example:
real array sine_table[-1000..1000]
for x from -1000 to 1000
sine_table[x] := sine(pi * x / 1000)
function lookup_sine(x)
return sine_table[round(1000 * x / pi)]
Linear interpolation on a portion of the sine function
Unfortunately, the table requires quite a bit of space: if IEEE double-precision floating-point numbers are used, over 16,000 bytes would be required. We can use fewer samples, but then our precision will significantly worsen. One good solution is linear interpolation, which draws a line between the two points in the table on either side of the value and locates the answer on that line. This is still quick to compute, and much more accurate for smooth functions such as the sine function. Here is our example using linear interpolation:
function lookup_sine(x)
x1 := floor(x*1000/pi)
y1 := sine_table[x1]
y2 := sine_table[x1+1]
return y1 + (y2-y1)*(x*1000/pi-x1)
When using interpolation, it is often beneficial to use non-uniform sampling, which means that where the function is close to straight, we use few sample points, while where it changes value quickly we use more sample points to keep the approximation close to the real curve. For more information, see interpolation.
Example in C
// C 8-bit Sine Table
const unsigned char sine_table = {14,36,48,63,105,2,41,37,44,104,60,32,116,17,93,68,16,88,102,2,50,57,44};
Other usage of lookup tables
Caches
Main article: cache
Storage caches (including disk caches for files, or processor caches for either for code or data) work also like a lookup table. The table is built with very fast memory instead of being stored on slower external memory, and maintains two pieces of data for a subrange of bits composing an external memory (or disk) address (notably the lowest bits of any possible external address):
one piece (the tag) contains the value of the remaining bits of the address; if these bits match with those from the memory address to read or write, then the other piece contains the cached value for this address.
the other piece maintains the data associated to that address.
A single (fast) lookup is performed to read the tag in the lookup table at the index specified by the lowest bits of the desired external storage address, and to determine if the memory address is hit by the cache. When a hit is found, no access to external memory is needed (except for write operations, where the cached value may need to be updated asynchronously to the slower memory after some time, or if the position in the cache must be replaced to cache another address).
Hardware LUTs
In digital logic, an n-bit lookup table can be implemented with a multiplexer whose select lines are the inputs of the LUT and whose outputs are constants. An n-bit LUT can encode any n-input Boolean function by modeling such functions as truth tables. This is an efficient way of encoding Boolean logic functions, and LUTs with 4-6 bits of input are in fact the key component of modern FPGAs.
See also
Branch table
Memoization
Memory bound function
Palette and Colour Look-Up Table – for the usage in computer graphics
3D LUT usage in film
External links
Fast table lookup using input character as index for branch table
Art of Assembly: Calculation via Table Lookups
Color Presentation of Astronomical Images
“Bit Twiddling Hacks” (includes lookup tables) By Sean Eron Anderson of Stanford university
Memoization in C++ by Paul McNamee, Johns Hopkins University showing savings
“The Quest for an Accelerated Population Count” by Henry S. Warren, Jr.
References
^ http://apl.jhu.edu/~paulmac/c++-memoization.html
^ http://www.amazon.com/dp/0198508417 “The History of Mathematical Tables: From Sumer to Spreadsheets”
^ Maher & Makowski 2001, p.383
Categories: Arrays | Associative arrays | Computer performance | Software optimization | Articles with example C code
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