# Binary fractional count and limited outcomes

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. Hamilos has received consulting fees from St.

Koo has received institutional research support from St. Jude, Eli Lilly and AstraZeneca. Muller has received research support from the Fondation Vaudoise de Cardiologie , Lausanne, Switzerland; and has received honoraria for presentations from St.

Jude Medical and Medtronic. Van Belle has served as a consultant for St. Jude Medical and received speaker's fees and honoraria from Volcano and St. Fearon has received research support from St. Pijls has served as a consultant to St. De Bruyne has received consulting fees and research support from St.

Jude Medical ; his consulting fees are passed to Cardiovascular Research Center Aalst, a not-for-profit organization. Gould has received internal funding from the Weatherhead PET Center for Preventing and Reversing Atherosclerosis ; signed nonfinancial, nondisclosure agreements with St.

Jude Medical and Volcano Corporation to discuss coronary physiology projects; k applicant for cfrQuant, a software package for quantifying absolute flow using cardiac positron emission tomography, and all royalties will go to a University of Texas scholarship fund, as the University of Texas has a commercial, nonexclusive agreement with Positron Corporation to distribute and market cfrQuant in exchange for royalties, but Dr.

Gould retains the ability to distribute cost-free versions to selected collaborators for research. 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 regression , but must be converted to quantitative data 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 the 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 variable. To illustrate this, suppose that we are measuring optimism among several nationalities and we have decided that French people 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 Italian , German , and 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 the 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 of 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 compared 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.