Binary search algorithm

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In statisticsa categorical variable is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property. Commonly though not in this articleeach of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random categorical variable binary fractional count and limited outcomes called a categorical distribution.

Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cross tabulationsor from observations of quantitative data grouped within given intervals. Often, purely categorical data are summarised in the form of a contingency table.

However, particularly when considering data analysis, it is common to use the term "categorical data" to apply to data binary fractional count and limited outcomes that, while containing some categorical variables, may also contain non-categorical variables.

A categorical variable that can take on exactly two values is termed a binary variable or dichotomous variable ; an important special case is the Bernoulli variable. Categorical variables with more than two possible values are called polytomous variables ; categorical variables are often assumed to be polytomous unless otherwise specified.

Discretization is treating continuous data as if it were categorical. Dichotomization is treating continuous data or polytomous variables as if they were binary variables. Regression analysis often treats category membership with one or more quantitative dummy variables.

For ease in statistical processing, categorical variables may be assigned numeric indices, e. In general, however, the numbers are arbitrary, and have no binary fractional count and limited outcomes beyond simply providing a convenient label for a particular value. In other words, the values in a categorical variable exist on a nominal scale: Instead, valid operations are equivalenceset membershipand other set-related operations.

As a result, the central tendency of a set of categorical variables is given by its mode ; neither the mean nor the median can be defined. As an example, given a set of people, we can consider the set of categorical variables corresponding to their last names. We can consider operations such as equivalence whether two people have the same last nameset membership whether a person has a name in a given listcounting how many people have a given last nameor finding the mode which name occurs most often.

Binary fractional count and limited outcomes a result, we cannot meaningfully ask what the "average name" the mean or binary fractional count and limited outcomes "middle-most name" the median is in a set of names. Note that this ignores the concept of alphabetical orderwhich is a property that is not inherent in the names themselves, but in the way we construct the labels.

However, if we do consider the names as written, e. Categorical random variables are normally described statistically by a categorical distributionwhich allows an arbitrary K -way categorical variable to be expressed with separate probabilities specified for each of the K possible outcomes.

Such multiple-category categorical variables are often analyzed using a multinomial distributionwhich counts the frequency of each possible combination of numbers of occurrences of the various categories. Regression analysis on categorical outcomes is accomplished through multinomial logistic regressionmultinomial probit or a related type of discrete choice model.

Categorical variables that have only two possible outcomes e. Because of their importance, these variables are often considered binary fractional count and limited outcomes separate category, with a separate distribution the Bernoulli distribution and separate regression models logistic regressionprobit regression binary fractional count and limited outcomes, etc. As a result, the term "categorical variable" is often reserved for cases with 3 or more outcomes, sometimes termed a multi-way variable in opposition to a binary variable.

It is also possible to consider categorical variables where the number of categories is not fixed in advance. As an example, for a categorical variable describing a particular word, we might not know in advance the size of the vocabulary, and we would like to allow for the possibility of encountering words that we haven't already seen. Standard statistical models, such as those involving the categorical distribution and multinomial logistic regressionassume that the number of categories is known in advance, and changing the number of categories on the fly binary fractional count and limited outcomes tricky.

In such cases, more advanced techniques must be used. An example is the Dirichlet processwhich falls in the realm of nonparametric statistics. In such a case, it is logically assumed that an infinite number of categories exist, but at any one time most of them in fact, all but a finite number have never been seen. All formulas are phrased in terms of the binary fractional count and limited outcomes of categories actually seen so far rather than the infinite total number of potential categories in existence, and methods are created for incremental updating of statistical distributions, binary fractional count and limited outcomes adding "new" categories.

Categorical variables represent a qualitative method of scoring data i. These can be included as independent variables in a regression analysis or as dependent variables in logistic regression or probit regressionbut must be converted to quantitative binary fractional count and limited outcomes in order to be able to analyze the data. One does so through the use of coding systems. Analyses are conducted such that only g -1 g being binary fractional count and limited outcomes number of groups are coded.

This minimizes redundancy while still representing the complete data set as no additional information would be gained from coding the total g groups: In general, the group that one does not code for is the group of least interest. There are three main coding systems typically used in the analysis of categorical variables in regression: The choice of coding system does not affect the F or R 2 statistics. However, one chooses a coding system based on the comparison of interest since the interpretation of b values will vary.

Dummy coding is used when there is a control or comparison group in mind. One is therefore analyzing the data of one group in relation to the comparison group: It is suggested that three criteria be met for specifying a suitable control group: In dummy coding, the reference group is assigned a value of 0 for each code variable, the group of interest for comparison to the reference group is assigned a value of 1 for its specified code variable, while all other groups are assigned 0 for that particular code variable.

The b values should be interpreted such that the experimental group is being compared against the control group. Therefore, yielding a negative b value would entail the experimental group have scored less than the control group on the dependent binary fractional count and limited outcomes. To illustrate this, suppose that we are measuring optimism among several nationalities and we have decided that French binary fractional count and limited outcomes would serve as a useful control.

If we are comparing them against Italians, and we observe a negative b value, this would suggest Italians obtain lower optimism scores on average.

The following table is an example of dummy coding with French as the control group and C1, C2, and C3 respectively being the codes for ItalianGermanand Other neither French nor Italian nor German:. In the effects coding system, data are analyzed through comparing one group to all other groups.

Unlike dummy coding, there is no control group. Rather, the comparison is being made at the mean of all groups combined a is now the grand mean. Therefore, one is not looking for data in relation to another group but rather, one is seeking data in relation to the grand mean.

Effects coding can either be weighted or unweighted. Weighted effects coding is simply calculating a weighted grand mean, thus taking into account the sample size in each variable. This is most appropriate in situations where the sample is representative of the population in question. Unweighted effects coding is most appropriate in situations where differences in sample size are the result of incidental factors. The interpretation of b is different for each: In effects coding, we code the group of interest with a 1, just as we would for dummy coding.

A code of 0 is assigned to all other groups. The b values should be interpreted such that binary fractional count and limited outcomes experimental group is being compared against the mean of all groups combined or weighted grand mean in the case of weighted effects coding. Therefore, yielding a negative b value would entail the coded group as having scored less than the mean of all groups on the dependent variable.

Using our previous example of optimism scores among nationalities, if the group binary fractional count and limited outcomes interest is Italians, observing a negative b value suggest they obtain a lower optimism score. The following table is an example of effects coding with Other as the group of least interest. The contrast coding system allows a researcher to directly ask specific questions.

Rather than having the coding system dictate the comparison being made i. The hypotheses proposed are generally as follows: Through its a priori focused hypotheses, contrast coding may yield an increase in power of the statistical test when binary fractional count and limited outcomes with the less directed previous coding systems.

Furthermore, in regression, coefficient values must be either in fractional or decimal form. They cannot take on interval values. Violating rule 2 produces accurate R 2 and F values, indicating that we would reach the same conclusions about whether or not there is a significant difference; however, we can no longer interpret the b values as a mean difference. To illustrate the construction of contrast codes consider the following table. Coefficients were chosen to illustrate our a priori hypotheses: This is illustrated through assigning the same coefficient to the French and Italian categories and a different one to the Germans.

The signs assigned indicate the direction of the relationship hence giving Germans a negative sign is indicative of their lower hypothesized optimism scores. Here, assigning a zero value to Germans demonstrates their non-inclusion in the analysis of this hypothesis. Again, the signs assigned are indicative of the proposed relationship. Although it produces correct mean values for the variables, the use of nonsense coding is not recommended as it will lead to uninterpretable statistical results.

An interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive. Interactions may arise with categorical variables in two ways: This type of interaction arises when we have two categorical variables. In order to probe this type of interaction, one would code using the system that addresses the researcher's hypothesis most appropriately.

The product binary fractional count and limited outcomes the codes yields the interaction. One may then calculate the b value and determine whether the interaction is significant.

Simple slopes analysis is a common post hoc test used in regression which is similar to the simple binary fractional count and limited outcomes analysis in ANOVA, used to analyze interactions. In this test, we are examining the simple slopes of one independent variable at specific values of the other independent variable. Such a test is not limited to use with continuous variables, but may also be employed when the independent variable is categorical.

We cannot simply choose values to probe the interaction as we would in the continuous variable case because of the nominal nature of the data i.

In our categorical case we would use a simple regression equation for each group to investigate the simple slopes. It is common practice to standardize or center variables to make the data more interpretable in simple slopes analysis; however, categorical variables should never be standardized or centered. This test can be used with all coding systems.

From Wikipedia, the free encyclopedia. The Practice of Statistics 2nd ed. Regression with dummy variables. Mean arithmetic geometric harmonic Median Mode.

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In computer science , binary search , also known as half-interval search , [1] logarithmic search , [2] or binary chop , [3] is a search algorithm that finds the position of a target value within a sorted array. If the search ends with the remaining half being empty, the target is not in the array.

Binary search runs in at worst logarithmic time , making O log n comparisons, where n is the number of elements in the array, the O is Big O notation , and log is the logarithm. Binary search takes constant O 1 space, meaning that the space taken by the algorithm is the same for any number of elements in the array.

Although the idea is simple, implementing binary search correctly requires attention to some subtleties about its exit conditions and midpoint calculation. There are numerous variations of binary search.

In particular, fractional cascading speeds up binary searches for the same value in multiple arrays, efficiently solving a series of search problems in computational geometry and numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree and B-tree data structures are based on binary search. Binary search works on sorted arrays. Binary search begins by comparing the middle element of the array with the target value. If the target value matches the middle element, its position in the array is returned.

If the target value is less than or greater than the middle element, the search continues in the lower or upper half of the array, respectively, eliminating the other half from consideration. Given an array A of n elements with values or records A 0 , A 1 , In the above procedure, the algorithm checks whether the middle element m is equal to the target t in every iteration. Some implementations leave out this check during each iteration. This results in a faster comparison loop, as one comparison is eliminated per iteration.

However, it requires one more iteration on average. The above procedure only performs exact matches, finding the position of a target value. However, due to the ordered nature of sorted arrays, it is trivial to extend binary search to perform approximate matches. For example, binary search can be used to compute, for a given value, its rank the number of smaller elements , predecessor next-smallest element , successor next-largest element , and nearest neighbor.

Range queries seeking the number of elements between two values can be performed with two rank queries. The performance of binary search can be analyzed by reducing the procedure to a binary comparison tree, where the root node is the middle element of the array. The middle element of the lower half is the left child node of the root and the middle element of the upper half is the right child node of the root.

The rest of the tree is built in a similar fashion. This model represents binary search; starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration, representing the successive elimination of elements.

The worst case is reached when the search reaches the deepest level of the tree, equivalent to a binary search that has reduced to one element and, in each iteration, always eliminates the smaller subarray out of the two if they are not of equal size.

The worst case may also be reached when the target element is not in the array. In the best case, where the target value is the middle element of the array, its position is returned after one iteration. In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary search.

This is because the comparison tree representing binary search has the fewest levels possible as each level is filled completely with nodes if there are enough. This is the case for other search algorithms based on comparisons, as while they may work faster on some target values, the average performance over all elements is affected. This problem is solved by binary search, as dividing the array in half ensures that the size of both subarrays are as similar as possible.

Fractional cascading can be used to speed up searches of the same value in multiple arrays. Each iteration of the binary search procedure defined above makes one or two comparisons, checking if the middle element is equal to the target in each iteration. Again assuming that each element is equally likely to be searched, each iteration makes 1. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search, eliminating on average half a comparison from each iteration.

This slightly cuts the time taken per iteration on most computers, while guaranteeing that the search takes the maximum number of iterations, on average adding one iteration to the search. For implementing associative arrays , hash tables , a data structure that maps keys to records using a hash function , are generally faster than binary search on a sorted array of records; [19] most implementations require only amortized constant time on average.

In addition, all operations possible on a sorted array can be performed—such as finding the smallest and largest key and performing range searches. A binary search tree is a binary tree data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time. Insertion and deletion also require on average logarithmic time in binary search trees.

This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a sorted array, including range and approximate queries. However, binary search is usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search.

This applies even to balanced binary search trees , binary search trees that balance their own nodes—as they rarely produce optimally -balanced trees—but to a lesser extent. Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can effectively be structured in filesystems. The B-tree generalizes this method of tree organization; B-trees are frequently used to organize long-term storage such as databases and filesystems.

Linear search is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a linked list, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short. Sorting the array also enables efficient approximate matches and other operations.

A related problem to search is set membership. Any algorithm that does lookup, like binary search, can also be used for set membership. There are other algorithms that are more specifically suited for set membership. For approximate results, Bloom filters , another probabilistic data structure based on hashing, store a set of keys by encoding the keys using a bit array and multiple hash functions. Bloom filters are much more space-efficient than bit arrays in most cases and not much slower: However, Bloom filters suffer from false positives.

There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays.

For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as van Emde Boas trees , fusion trees , tries , and bit arrays.

However, while these operations can always be done at least efficiently on a sorted array regardless of the keys, such data structures are usually only faster because they exploit the properties of keys with a certain attribute usually keys that are small integers , and thus will be time or space consuming for keys that lack that attribute.

Uniform binary search stores, instead of the lower and upper bounds, the index of the middle element and the change in the middle element from the current iteration to the next iteration. Each step reduces the change by about half. For example, if the array to be searched was [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] , the middle element would be 6. Uniform binary search works on the basis that the difference between the index of middle element of the array and the left and right subarrays is the same.

In this case, the middle element of the left subarray [1, 2, 3, 4, 5] is 3 and the middle element of the right subarray [7, 8, 9, 10, 11] is 9. Uniform binary search would store the value of 3 as both indices differ from 6 by this same amount.

The main advantage of uniform binary search is that the procedure can store a table of the differences between indices for each iteration of the procedure, which may improve the algorithm's performance on some systems. It starts by finding the first element with an index that is both a power of two and greater than the target value.

Afterwards, it sets that index as the upper bound, and switches to binary search. Exponential search works on bounded lists, but becomes an improvement over binary search only if the target value lies near the beginning of the array.

Instead of calculating the midpoint, interpolation search estimates the position of the target value, taking into account the lowest and highest elements in the array as well as length of the array. This is only possible if the array elements are numbers. It works on the basis that the midpoint is not the best guess in many cases. For example, if the target value is close to the highest element in the array, it is likely to be located near the end of the array.

In practice, interpolation search is slower than binary search for small arrays, as interpolation search requires extra computation. Although its time complexity grows more slowly than binary search, this only compensates for the extra computation for large arrays.

Fractional cascading is a technique that speeds up binary searches for the same element for both exact and approximate matching in "catalogs" arrays of sorted elements associated with vertices in graphs. Fractional cascading was originally developed to efficiently solve various computational geometry problems, but it also has been applied elsewhere, in domains such as data mining and Internet Protocol routing.

Fibonacci search is a method similar to binary search that successively shortens the interval in which the maximum of a unimodal function lies.

Given a finite interval, a unimodal function, and the maximum length of the resulting interval, Fibonacci search finds a Fibonacci number such that if the interval is divided equally into that many subintervals, the subintervals would be shorter than the maximum length. After dividing the interval, it eliminates the subintervals in which the maximum cannot lie until one or more contiguous subintervals remain.

Noisy binary search algorithms solve the case where the algorithm cannot reliably compare elements of the array. For each pair of elements, there is a certain probability that the algorithm makes the wrong comparison. Noisy binary search can find the correct position of the target with a given probability that controls the reliability of the yielded position. In , John Mauchly made the first mention of binary search as part of the Moore School Lectures , the first ever set of lectures regarding any computer-related topic.

Guibas introduced fractional cascading as a method to solve numerous search problems in computational geometry. Although the basic idea of binary search is comparatively straightforward, the details can be surprisingly tricky When Jon Bentley assigned binary search as a problem in a course for professional programmers, he found that ninety percent failed to provide a correct solution after several hours of working on it, [56] and another study published in shows that accurate code for it is only found in five out of twenty textbooks.

The Java programming language library implementation of binary search had the same overflow bug for more than nine years. In a practical implementation, the variables used to represent the indices will often be of fixed size, and this can result in an arithmetic overflow for very large arrays. If the target value is greater than the greatest value in the array, and the last index of the array is the maximum representable value of L , the value of L will eventually become too large and overflow.

A similar problem will occur if the target value is smaller than the least value in the array and the first index of the array is the smallest representable value of R. In particular, this means that R must not be an unsigned type if the array starts with index 0.

An infinite loop may occur if the exit conditions for the loop are not defined correctly. Once L exceeds R , the search has failed and must convey the failure of the search. In addition, the loop must be exited when the target element is found, or in the case of an implementation where this check is moved to the end, checks for whether the search was successful or failed at the end must be in place.

Bentley found that, in his assignment of binary search, most of the programmers who implemented binary search incorrectly made an error defining the exit conditions. Many languages' standard libraries include binary search routines:.