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108-403: Perfect hash functions may be used to implement a lookup table with constant worst-case access time. A perfect hash function can, as any hash function , be used to implement hash tables , with the advantage that no collision resolution has to be implemented. In addition, if the keys are not in the data and if it is known that queried keys will be valid, then the keys do not need to be stored in

216-409: A 1 , a 2 , ..., a n and for any keys a j and a k , j < k implies F ( a j ) < F( a k ) . In this case, the function value is just the position of each key in the sorted ordering of all of the keys. A simple implementation of order-preserving minimal perfect hash functions with constant access time is to use an (ordinary) perfect hash function to store

324-513: A LOOKUP function among its original 20 functions. This has been followed by subsequent spreadsheets, such as Microsoft Excel , and complemented by specialized VLOOKUP and HLOOKUP functions to simplify lookup in a vertical or horizontal table. In Microsoft Excel the XLOOKUP function has been rolled out starting 28 August 2019. Although the performance of a LUT is a guaranteed O ( 1 ) {\displaystyle O(1)} for

432-480: A computational hardness assumption to prove the difficulty of several other problems in computational game theory , property testing , and machine learning . The complexity class QP consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of DTIME as follows. In complexity theory, the unsolved P versus NP problem asks if all problems in NP have polynomial-time algorithms. All

540-529: A constant multiplier , and such a multiplier is irrelevant to big O classification, the standard usage for logarithmic-time algorithms is O ( log ⁡ n ) {\displaystyle O(\log n)} regardless of the base of the logarithm appearing in the expression of T . Algorithms taking logarithmic time are commonly found in operations on binary trees or when using binary search . An O ( log ⁡ n ) {\displaystyle O(\log n)} algorithm

648-459: A lookup table ( LUT ) is an array that replaces runtime computation with a simpler array indexing operation, in a process termed as direct addressing . The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched" ) as part of

756-422: A constant. Linear time is the best possible time complexity in situations where the algorithm has to sequentially read its entire input. Therefore, much research has been invested into discovering algorithms exhibiting linear time or, at least, nearly linear time. This research includes both software and hardware methods. There are several hardware technologies which exploit parallelism to provide this. An example

864-792: A deterministic Turing machine form the complexity class known as EXP . Sometimes, exponential time is used to refer to algorithms that have T ( n ) = 2 , where the exponent is at most a linear function of n . This gives rise to the complexity class E . An algorithm is said to be factorial time if T(n) is upper bounded by the factorial function n! . Factorial time is a subset of exponential time (EXP) because n ! ≤ n n = 2 n log ⁡ n = O ( 2 n 1 + ϵ ) {\displaystyle n!\leq n^{n}=2^{n\log n}=O\left(2^{n^{1+\epsilon }}\right)} for all ϵ > 0 {\displaystyle \epsilon >0} . However, it

972-440: A deterministic machine which is robust in terms of machine model changes. (For example, a change from a single-tape Turing machine to a multi-tape machine can lead to a quadratic speedup, but any algorithm that runs in polynomial time under one model also does so on the other.) Any given abstract machine will have a complexity class corresponding to the problems which can be solved in polynomial time on that machine. An algorithm

1080-399: A given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed as a function of the size of the input. Since this function is generally difficult to compute exactly, and the running time for small inputs is usually not consequential, one commonly focuses on the behavior of the complexity when

1188-465: A grayscale picture of the planet Saturn could be transformed into a color image to emphasize the differences in its rings. In image processing, lookup tables are often called LUT s (or 3DLUT), 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. In computed tomography , "windowing" refers to

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1296-414: A hash function of a sequence of independent fully random hash functions (Φ 1 , Φ 2 , Φ 3 , ...) , starting with Φ 1 . If the hash function does not produce any collisions for the bucket, and the resulting values are not yet occupied by other elements from other buckets, the function is chosen for that bucket. If not, the next hash function in the sequence is tested. To evaluate

1404-474: A high degree of precision: 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 through array indexing operation. This will be close to

1512-412: A lookup operation, no two entities or values can have the same key k {\displaystyle k} . When the size of universe ∪ {\displaystyle \cup } —where the keys are drawn—is large, it might be impractical or impossible to be stored in memory . Hence, in this case, a hash table would be a preferable alternative. For a trivial hash function lookup,

1620-470: A lookup table of the positions of each key. This solution uses O ( n log ⁡ n ) {\displaystyle O(n\log n)} bits, which is optimal in the setting where the comparison function for the keys may be arbitrary. However, if the keys a 1 , a 2 , ..., a n are integers drawn from a universe { 1 , 2 , … , U } {\displaystyle \{1,2,\ldots ,U\}} , then it

1728-620: 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 x 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

1836-425: A lookup table which in turn uses trivial hash function for better performance. The bits array, bits_set with 256 entries is constructed by giving the number of one bits set in each possible byte value (e.g. 0x00 = 0, 0x01 = 1, 0x02 = 1, and so on). Although a runtime algorithm can be used to generate the bits_set array, it's an inefficient usage of clock cycles when the size is taken into consideration, hence

1944-448: 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 sub-range of bits composing an external memory (or disk) address (notably the lowest bits of any possible external 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

2052-466: A paper dictionary. As a result, the search space within the dictionary decreases as the algorithm gets closer to the target word. An algorithm is said to run in polylogarithmic time if its time T ( n ) {\displaystyle T(n)} is O ( ( log ⁡ n ) k ) {\displaystyle O{\bigl (}(\log n)^{k}{\bigr )}} for some constant k . Another way to write this

2160-412: A precomputed table is used—although a compile time script could be used to dynamically generate and append the table to the source file . Sum of ones in each byte of the integer can be calculated through trivial hash function lookup on each byte; thus, effectively avoiding branches resulting in considerable improvement in performance. "Lookup tables (LUTs) are an excellent technique for optimizing

2268-534: A program's initialization phase ( memoization ), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input. FPGAs also make extensive use of reconfigurable, hardware-implemented, lookup tables to provide programmable hardware functionality. LUTs differ from hash tables in

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2376-466: 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. Most computers only perform basic arithmetic operations and 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

2484-417: A range of O ( n ) indices, and then map each index to a range of hash values. The first level of their construction chooses a large prime p (larger than the size of the universe from which S is drawn), and a parameter k , and maps each element x of S to the index If k is chosen randomly, this step is likely to have collisions, but the number of elements n i that are simultaneously mapped to

2592-404: A related concept for determining how to display the intensity of measured radiation. 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

2700-415: 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 it. Depending on the values being precomputed, precomputation with interpolation can also be used to shrink

2808-488: A smaller range of length n + o ( n ) . A more recent method for constructing a perfect hash function is described by Belazzougui, Botelho & Dietzfelbinger (2009) as "hash, displace, and compress". Here a first-level hash function g is also used to map elements onto a range of r integers. An element x ∈ S is stored in the Bucket B g(x) . Then, in descending order of size, each bucket's elements are hashed by

2916-488: A term n c {\displaystyle n^{c}} for any c > 1 {\displaystyle c>1} . Algorithms which run in quasilinear time include: In many cases, the O ( n log ⁡ n ) {\displaystyle O(n\log n)} running time is simply the result of performing a Θ ( log ⁡ n ) {\displaystyle \Theta (\log n)} operation n times (for

3024-406: A way that, to retrieve a value v {\displaystyle v} with key k {\displaystyle k} , a hash table would store the value v {\displaystyle v} in the slot h ( k ) {\displaystyle h(k)} where h {\displaystyle h} is a hash function i.e. k {\displaystyle k}

3132-457: Is O ( log k ⁡ n ) {\displaystyle O(\log ^{k}n)} . For example, matrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine , and a graph can be determined to be planar in a fully dynamic way in O ( log 3 ⁡ n ) {\displaystyle O(\log ^{3}n)} time per insert/delete operation. An algorithm

3240-517: Is content-addressable memory . This concept of linear time is used in string matching algorithms such as the Boyer–Moore string-search algorithm and Ukkonen's algorithm . An algorithm is said to run in quasilinear time (also referred to as log-linear time ) if T ( n ) = O ( n log k ⁡ n ) {\displaystyle T(n)=O(n\log ^{k}n)} for some positive constant k ; linearithmic time

3348-440: Is cuckoo hashing . This scheme maps keys to two or more locations within a range (unlike perfect hashing which maps each key to a single location) but does so in such a way that the keys can be assigned one-to-one to locations to which they have been mapped. Lookups with this scheme are slower, because multiple locations must be checked, but nevertheless take constant worst-case time. Lookup table In computer science ,

Perfect hash function - Misplaced Pages Continue

3456-421: Is k -perfect if at most k elements from S are mapped onto the same value in the range. The "hash, displace, and compress" algorithm can be used to construct k -perfect hash functions by allowing up to k collisions. The changes necessary to accomplish this are minimal, and are underlined in the adapted pseudocode below: A minimal perfect hash function F is order preserving if keys are given in some order

3564-422: Is n . Another example was the graph isomorphism problem , which the best known algorithm from 1982 to 2016 solved in 2 O ( n log ⁡ n ) {\displaystyle 2^{O\left({\sqrt {n\log n}}\right)}} . However, at STOC 2016 a quasi-polynomial time algorithm was presented. It makes a difference whether the algorithm is allowed to be sub-exponential in

3672-461: Is a polynomial time algorithm . The following table summarizes some classes of commonly encountered time complexities. In the table, poly( x ) = x , i.e., polynomial in  x . formerly-best algorithm for graph isomorphism An algorithm is said to be constant time (also written as O ( 1 ) {\textstyle O(1)} time) if the value of T ( n ) {\textstyle T(n)} (the complexity of

3780-420: Is a distinct, unique, directly addressable storage location, the value of the starting address where any object is stored in memory can be considered a de facto perfect hash of that object into the entire memory address range. Using a perfect hash function is best in situations where there is a frequently queried large set, S , which is seldom updated. This is because any modification of the set S may cause

3888-470: Is a minimal perfect hash function if and only if h ( j ) = h ( k ) implies j = k ( injectivity ) and there exists an integer a such that the range of h is a .. a + | S | − 1 . It has been proven that a general purpose minimal perfect hash scheme requires at least log 2 ⁡ e ≈ 1.44 {\displaystyle \log _{2}e\approx 1.44} bits/key. Assuming that S {\displaystyle S}

3996-762: Is a set of size n {\displaystyle n} containing integers in the range [ 1 , 2 o ( n ) ] {\displaystyle [1,2^{o(n)}]} , it is known how to efficiently construct an explicit minimal perfect hash function from S {\displaystyle S} to { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} that uses space n log 2 ⁡ e + o ( n ) {\displaystyle n\log _{2}e+o(n)} bits and that supports constant evaluation time. In practice, there are minimal perfect hashing schemes that use roughly 1.56 bits/key if given enough time. A hash function

4104-427: Is a synonym for "tractable", "feasible", "efficient", or "fast". Some examples of polynomial-time algorithms: These two concepts are only relevant if the inputs to the algorithms consist of integers. The concept of polynomial time leads to several complexity classes in computational complexity theory. Some important classes defined using polynomial time are the following. P is the smallest time-complexity class on

4212-515: 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 field-programmable gate arrays (FPGAs) which provide reconfigurable hardware logic capabilities. In data acquisition and control systems , lookup tables are commonly used to undertake the following operations in: In some systems, polynomials may also be defined in place of lookup tables for these calculations. Polynomial time In theoretical computer science ,

4320-430: Is an open problem. Other computational problems with quasi-polynomial time solutions but no known polynomial time solution include the planted clique problem in which the goal is to find a large clique in the union of a clique and a random graph . Although quasi-polynomially solvable, it has been conjectured that the planted clique problem has no polynomial time solution; this planted clique conjecture has been used as

4428-489: Is clearly superpolynomial, but some algorithms are only very weakly superpolynomial. For example, the Adleman–Pomerance–Rumely primality test runs for n time on n -bit inputs; this grows faster than any polynomial for large enough n , but the input size must become impractically large before it cannot be dominated by a polynomial with small degree. An algorithm that requires superpolynomial time lies outside

Perfect hash function - Misplaced Pages Continue

4536-591: Is considered highly efficient, as the ratio of the number of operations to the size of the input decreases and tends to zero when n increases. An algorithm that must access all elements of its input cannot take logarithmic time, as the time taken for reading an input of size n is of the order of n . An example of logarithmic time is given by dictionary search. Consider a dictionary D which contains n entries, sorted in alphabetical order . We suppose that, for 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} , one may access

4644-555: Is continuous and has continuous first derivative , one should use the cubic Hermite spline . When using interpolation, the size of the lookup table can be reduced by using nonuniform 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 . Storage caches (including disk caches for files, or processor caches for either code or data) work also like

4752-436: Is defined to take superpolynomial time if T ( n ) is not bounded above by any polynomial. Using little omega notation , it is ω ( n ) time for all constants c , where n is the input parameter, typically the number of bits in the input. For example, an algorithm that runs for 2 steps on an input of size n requires superpolynomial time (more specifically, exponential time). An algorithm that uses exponential resources

4860-423: Is in O ( n ) , and depends on the achieved range. For example, with m = 1.23 n Belazzougui, Botelho & Dietzfelbinger (2009) achieved a representation size between 3.03 bits/key and 1.40 bits/key for their given example set of 10 million entries, with lower values needing a higher computation time. The space lower bound in this scenario is 0.88 bits/key. The use of O ( n ) words of information to store

4968-423: Is known. Such problems arise in approximation algorithms; a famous example is the directed Steiner tree problem , for which there is a quasi-polynomial time approximation algorithm achieving an approximation factor of O ( log 3 ⁡ n ) {\displaystyle O(\log ^{3}n)} ( n being the number of vertices), but showing the existence of such a polynomial time algorithm

5076-438: Is not a subset of E. An example of an algorithm that runs in factorial time is bogosort , a notoriously inefficient sorting algorithm based on trial and error . Bogosort sorts a list of n items by repeatedly shuffling the list until it is found to be sorted. In the average case, each pass through the bogosort algorithm will examine one of the n ! orderings of the n items. If the items are distinct, only one such ordering

5184-401: Is not generally agreed upon, however the two most widely used are below. A problem is said to be sub-exponential time solvable if it can be solved in running times whose logarithms grow smaller than any given polynomial. More precisely, a problem is in sub-exponential time if for every ε > 0 there exists an algorithm which solves the problem in time O (2 ). The set of all such problems

5292-485: Is of great practical importance. An algorithm is said to be of polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm, that is, T ( n ) = O ( n ) for some positive constant k . Problems for which a deterministic polynomial-time algorithm exists belong to the complexity class P , which is central in the field of computational complexity theory . Cobham's thesis states that polynomial time

5400-432: 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

5508-551: Is possible to construct an order-preserving hash function using only O ( n log ⁡ log ⁡ log ⁡ U ) {\displaystyle O(n\log \log \log U)} bits of space. Moreover, this bound is known to be optimal. While well-dimensioned hash tables have amortized average O(1) time (amortized average constant time) for lookups, insertions, and deletion, most hash table algorithms suffer from possible worst-case times that take much longer. A worst-case O(1) time (constant time even in

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5616-425: Is said to be subquadratic time if T ( n ) = o ( n 2 ) {\displaystyle T(n)=o(n^{2})} . For example, simple, comparison-based sorting algorithms are quadratic (e.g. insertion sort ), but more advanced algorithms can be found that are subquadratic (e.g. shell sort ). No general-purpose sorts run in linear time, but the change from quadratic to sub-quadratic

5724-449: Is said to run in sub-linear time (often spelled sublinear time ) if T ( n ) = o ( n ) {\displaystyle T(n)=o(n)} . In particular this includes algorithms with the time complexities defined above. The specific term sublinear time algorithm commonly refers to randomized algorithms that sample a small fraction of their inputs and process them efficiently to approximately infer properties of

5832-620: Is significantly faster than exponential time . The worst case running time of a quasi-polynomial time algorithm is 2 O ( log c ⁡ n ) {\displaystyle 2^{O(\log ^{c}n)}} for some fixed c > 0 {\displaystyle c>0} . When c = 1 {\displaystyle c=1} this gives polynomial time, and for c < 1 {\displaystyle c<1} it gives sub-linear time. There are some problems for which we know quasi-polynomial time algorithms, but no polynomial time algorithm

5940-410: Is that 3SAT , the satisfiability problem of Boolean formulas in conjunctive normal form with at most three literals per clause and with n variables, cannot be solved in time 2 . More precisely, the hypothesis is that there is some absolute constant c > 0 such that 3SAT cannot be decided in time 2 by any deterministic Turing machine. With m denoting the number of clauses, ETH is equivalent to

6048-549: Is the case k = 1 {\displaystyle k=1} . Using soft O notation these algorithms are O ~ ( n ) {\displaystyle {\tilde {O}}(n)} . Quasilinear time algorithms are also O ( n 1 + ε ) {\displaystyle O(n^{1+\varepsilon })} for every constant ε > 0 {\displaystyle \varepsilon >0} and thus run faster than any polynomial time algorithm whose time bound includes

6156-588: Is the class of all parameterized problems ( L , k ) {\displaystyle (L,k)} for which there is a computable function f : N → N {\displaystyle f:\mathbb {N} \to \mathbb {N} } with f ∈ o ( k ) {\displaystyle f\in o(k)} and an algorithm that decides L in time 2 f ( k ) ⋅ poly ( n ) {\displaystyle 2^{f(k)}\cdot {\text{poly}}(n)} . The exponential time hypothesis ( ETH )

6264-452: Is the complexity class SUBEXP which can be defined in terms of DTIME as follows. This notion of sub-exponential is non-uniform in terms of ε in the sense that ε is not part of the input and each ε may have its own algorithm for the problem. Some authors define sub-exponential time as running times in 2 o ( n ) {\displaystyle 2^{o(n)}} . This definition allows larger running times than

6372-409: Is the number of keys in the structure. The important performance parameters for perfect hash functions are the evaluation time, which should be constant, the construction time, and the representation size. A perfect hash function with values in a limited range can be used for efficient lookup operations, by placing keys from S (or other associated values) in a lookup table indexed by the output of

6480-531: Is the size in units of bits needed to represent the input. Algorithmic complexities are classified according to the type of function appearing in the big O notation. For example, an algorithm with time complexity O ( n ) {\displaystyle O(n)} is a linear time algorithm and an algorithm with time complexity O ( n α ) {\displaystyle O(n^{\alpha })} for some constant α > 0 {\displaystyle \alpha >0}

6588-457: Is used to compute the slot, while in the case of LUT, the value v {\displaystyle v} is stored in slot k {\displaystyle k} , thus directly addressable. Before the advent of computers, lookup tables of values were used to speed up hand calculations of complex functions, such as in trigonometry , logarithms , and statistical density functions. In ancient (499 AD) India, Aryabhata created one of

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6696-428: 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: The above implementation requires 32 operations for an evaluation of a 32-bit value, which can potentially take several clock cycles due to branching . It can be " unrolled " into

6804-417: The complexity class P . Cobham's thesis posits that these algorithms are impractical, and in many cases they are. Since the P versus NP problem is unresolved, it is unknown whether NP-complete problems require superpolynomial time. Quasi-polynomial time algorithms are algorithms whose running time exhibits quasi-polynomial growth , a type of behavior that may be slower than polynomial time but yet

6912-485: The floor function . If w = D ( ⌊ n 2 ⌋ ) {\displaystyle w=D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} --that is to say, the word w is exactly in the middle of the dictionary--then we are done. Else, if w < D ( ⌊ n 2 ⌋ ) {\displaystyle w<D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} --i.e., if

7020-524: The k th entry of the dictionary in a constant time. Let D ( k ) {\displaystyle D(k)} denote this k th entry. Under these hypotheses, the test to see if a word w is in the dictionary may be done in logarithmic time: consider D ( ⌊ n 2 ⌋ ) {\displaystyle D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} , where ⌊ ⌋ {\displaystyle \lfloor \;\rfloor } denotes

7128-410: The time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm . Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by

7236-457: The address signal and whose inputs are the values of the elements contained in the array. These values can either be hard-wired, as in an ASIC whose purpose is specific to a function, or provided by D latches which allow for configurable values. ( ROM , EPROM , EEPROM , or RAM .) An n -bit LUT can encode any n -input Boolean function by storing the truth table of the function in the LUT. This

7344-410: The algorithm are taken to be related by a constant factor . Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity , which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity , which is the average of the time taken on inputs of

7452-454: The algorithm) is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an array takes constant time as only one operation has to be performed to locate it. In a similar manner, finding the minimal value in an array sorted in ascending order; it is the first element. However, finding the minimal value in an unordered array is not a constant time operation as scanning over each element in

7560-437: The array is needed in order to determine the minimal value. Hence it is a linear time operation, taking O ( n ) {\textstyle O(n)} time. If the number of elements is known in advance and does not change, however, such an algorithm can still be said to run in constant time. Despite the name "constant time", the running time does not have to be independent of the problem size, but an upper bound for

7668-437: The best-known algorithms for NP-complete problems like 3SAT etc. take exponential time. Indeed, it is conjectured for many natural NP-complete problems that they do not have sub-exponential time algorithms. Here "sub-exponential time" is taken to mean the second definition presented below. (On the other hand, many graph problems represented in the natural way by adjacency matrices are solvable in subexponential time simply because

7776-565: 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

7884-406: The correct value because sine is a continuous function with a bounded rate of change. For example: 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

7992-425: The entire instance. This type of sublinear time algorithm is closely related to property testing and statistics . Other settings where algorithms can run in sublinear time include: An algorithm is said to take linear time , or O ( n ) {\displaystyle O(n)} time, if its time complexity is O ( n ) {\displaystyle O(n)} . Informally, this means that

8100-413: The evaluation of functions that are expensive to compute and inexpensive to cache. ... For data requests that fall between the table's samples, an interpolation algorithm can generate reasonable approximations by averaging nearby samples." In data analysis applications, such as image processing , a lookup table (LUT) can be used to transform the input data into a more desirable output format. For example,

8208-483: The evaluation time, the construction time, and additionally the range requirement m n {\displaystyle {\frac {m}{n}}} (average number of buckets per key in the hash table). The evaluation time can be as fast as O ( 1 ) , which is optimal. The construction time needs to be at least O ( n ) , because each element in S needs to be considered, and S contains n elements. This lower bound can be achieved in practice. The lower bound for

8316-451: The first sine tables , which he encoded in a Sanskrit-letter-based number system. In 493 AD, 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

8424-403: The first definition of sub-exponential time. An example of such a sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the general number field sieve , which runs in time about 2 O ~ ( n 1 / 3 ) {\displaystyle 2^{{\tilde {O}}(n^{1/3})}} , where the length of the input

8532-439: The formula given by Belazzougui, Botelho & Dietzfelbinger (2009) and for a universe U ⊇ S {\displaystyle U\supseteq S} whose size | U | = u tends towards infinity, the space lower bounds is bits/key, minus log( n ) bits overall. A trivial but pervasive example of perfect hashing is implicit in the (virtual) memory address space of a computer. Since each byte of virtual memory

8640-453: The fractions down to 1/144" Modern 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). 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

8748-433: The function of Fredman, Komlós & Szemerédi (1984) is near-optimal: any perfect hash function that can be calculated in constant time requires at least a number of bits that is proportional to the size of S . For minimal perfect hash functions the information theoretic space lower bound is bits/key. For perfect hash functions, it is first assumed that the range of h is bounded by n as m = (1+ε) n . With

8856-401: The function. One can then test whether a key is present in S , or look up a value associated with that key, by looking for it at its cell of the table. Each such lookup takes constant time in the worst case . With perfect hashing, the associated data can be read or written with a single access to the table. The important performance parameters for perfect hashing are the representation size,

8964-514: The hash function to no longer be perfect for the modified set. Solutions which update the hash function any time the set is modified are known as dynamic perfect hashing , but these methods are relatively complicated to implement. A minimal perfect hash function is a perfect hash function that maps n keys to n consecutive integers – usually the numbers from 0 to n − 1 or from 1 to n . A more formal way of expressing this is: Let j and k be elements of some finite set S . Then h

9072-418: The hypothesis that k SAT cannot be solved in time 2 for any integer k ≥ 3 . The exponential time hypothesis implies P ≠ NP . An algorithm is said to be exponential time , if T ( n ) is upper bounded by 2 , where poly( n ) is some polynomial in n . More formally, an algorithm is exponential time if T ( n ) is bounded by O (2 ) for some constant k . Problems which admit exponential time algorithms on

9180-523: The input size increases—that is, the asymptotic behavior of the complexity. Therefore, the time complexity is commonly expressed using big O notation , typically O ( n ) {\displaystyle O(n)} , O ( n log ⁡ n ) {\displaystyle O(n\log n)} , O ( n α ) {\displaystyle O(n^{\alpha })} , O ( 2 n ) {\displaystyle O(2^{n})} , etc., where n

9288-439: The known inapproximability results for the set cover problem. The term sub-exponential time is used to express that the running time of some algorithm may grow faster than any polynomial but is still significantly smaller than an exponential. In this sense, problems that have sub-exponential time algorithms are somewhat more tractable than those that only have exponential algorithms. The precise definition of "sub-exponential"

9396-430: The lookup table size while maintaining accuracy. 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

9504-422: The lookup table, saving space. Disadvantages of perfect hash functions are that S needs to be known for the construction of the perfect hash function. Non-dynamic perfect hash functions need to be re-constructed if S changes. For frequently changing S dynamic perfect hash functions may be used at the cost of additional space. The space requirement to store the perfect hash function is in O ( n ) where n

9612-410: The lower bound is log e ≈ 1.44 bits per element. A perfect hash function for a specific set S that can be evaluated in constant time, and with values in a small range, can be found by a randomized algorithm in a number of operations that is proportional to the size of S. The original construction of Fredman, Komlós & Szemerédi (1984) uses a two-level scheme to map a set S of n elements to

9720-414: 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). In digital logic , a lookup table can be implemented with a multiplexer whose select lines are driven by

9828-655: The notation, see Big O notation § Family of Bachmann–Landau notations ). For example, binary tree sort creates a binary tree by inserting each element of the n -sized array one by one. Since the insert operation on a self-balancing binary search tree takes O ( log ⁡ n ) {\displaystyle O(\log n)} time, the entire algorithm takes O ( n log ⁡ n ) {\displaystyle O(n\log n)} time. Comparison sorts require at least Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} comparisons in

9936-454: The perfect hash function h ( x ) one only has to save the mapping σ of the bucket index g ( x ) onto the correct hash function in the sequence, resulting in h(x) = Φ σ(g(x)) . Finally, to reduce the representation size, the ( σ(i)) 0 ≤ i < r are compressed into a form that still allows the evaluation in O ( 1 ) . This approach needs linear time in n for construction, and constant evaluation time. The representation size

10044-437: 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. Lookup tables were one of the earliest functionalities implemented in computer spreadsheets , with the initial version of VisiCalc (1979) including

10152-433: The representation size depends on m and n . Let m = (1+ε) n and h a perfect hash function. A good approximation for the lower bound is log ⁡ e − ε log ⁡ 1 + ε ε {\displaystyle \log e-\varepsilon \log {\frac {1+\varepsilon }{\varepsilon }}} Bits per element. For minimal perfect hashing, ε = 0 ,

10260-418: The running time has to be independent of the problem size. For example, the task "exchange the values of a and b if necessary so that a ≤ b {\textstyle a\leq b} " is called constant time even though the time may depend on whether or not it is already true that a ≤ b {\textstyle a\leq b} . However, there is some constant t such that

10368-418: The running time increases at most linearly with the size of the input. More precisely, this means that there is a constant c such that the running time is at most c n {\displaystyle cn} for every input of size n . For example, a procedure that adds up all elements of a list requires time proportional to the length of the list, if the adding time is constant, or, at least, bounded by

10476-403: The same index i is likely to be small. The second level of their construction assigns disjoint ranges of O ( n i ) integers to each index i . It uses a second set of linear modular functions, one for each index i , to map each member x of S into the range associated with g ( x ) . As Fredman, Komlós & Szemerédi (1984) show, there exists a choice of the parameter k such that

10584-405: The second-level linear modular functions. Computing the hash value of a given key x may be performed in constant time by computing g ( x ) , looking up the second-level function associated with g ( x ) , and applying this function to x . A modified version of this two-level scheme with a larger number of values at the top level can be used to construct a perfect hash function that maps S into

10692-505: 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 interpolate to the sine of the desired value, instead of calculating by mathematical formula. Lookup tables can thus used by mathematics coprocessors in computer systems. An error in

10800-430: The size of the input is the square of the number of vertices.) This conjecture (for the k-SAT problem) is known as the exponential time hypothesis . Since it is conjectured that NP-complete problems do not have quasi-polynomial time algorithms, some inapproximability results in the field of approximation algorithms make the assumption that NP-complete problems do not have quasi-polynomial time algorithms. For example, see

10908-427: The size of the instance, the number of vertices, or the number of edges. In parameterized complexity , this difference is made explicit by considering pairs ( L , k ) {\displaystyle (L,k)} of decision problems and parameters k . SUBEPT is the class of all parameterized problems that run in time sub-exponential in k and polynomial in the input size n : More precisely, SUBEPT

11016-516: The sum of the lengths of the ranges for the n different values of g ( x ) is O ( n ) . Additionally, for each value of g ( x ) , there exists a linear modular function that maps the corresponding subset of S into the range associated with that value. Both k , and the second-level functions for each value of g ( x ) , can be found in polynomial time by choosing values randomly until finding one that works. The hash function itself requires storage space O ( n ) to store k , p , and all of

11124-419: The time required is always at most t . An algorithm is said to take logarithmic time when T ( n ) = O ( log ⁡ n ) {\displaystyle T(n)=O(\log n)} . Since log a ⁡ n {\displaystyle \log _{a}n} and log b ⁡ n {\displaystyle \log _{b}n} are related by

11232-431: 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 an example using linear interpolation: Linear interpolation provides for an interpolated function that is continuous, but will not, in general, have continuous derivatives . For smoother interpolation of table lookup that

11340-419: 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 . One discrete problem that is expensive to solve on many computers is that of counting the number of bits that are set to 1 in a (binary) number, sometimes called

11448-418: The word w comes earlier in alphabetical order than the middle word of the whole dictionary--we continue the search in the same way in the left (i.e. earlier) half of the dictionary, and then again repeatedly until the correct word is found. Otherwise, if it comes after the middle word, continue similarly with the right half of the dictionary. This algorithm is similar to the method often used to find an entry in

11556-460: The worst case because log ⁡ ( n ! ) = Θ ( n log ⁡ n ) {\displaystyle \log(n!)=\Theta (n\log n)} , by Stirling's approximation . They also frequently arise from the recurrence relation T ( n ) = 2 T ( n 2 ) + O ( n ) {\textstyle T(n)=2T\left({\frac {n}{2}}\right)+O(n)} . An algorithm

11664-426: The worst case) would be better for many applications (including network router and memory caches ). Few hash table algorithms support worst-case O(1) lookup time (constant lookup time even in the worst case). The few that do include: perfect hashing; dynamic perfect hashing ; cuckoo hashing ; hopscotch hashing ; and extendible hashing . A simple alternative to perfect hashing, which also allows dynamic updates,

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