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n h p frac 10 h p comparisons on average where h p p log 2 p 1 p log 2 1 p displaystyle h p p log _ 2 p 1 p log _ 2 1 p is the binary entropy function and τ displaystyle tau is the probability that the procedure yields the wrong position 53 54 55 the noisy binary search problem can be considered as a case of the rényi ulam game 56 a variant of twenty questions where the answers may be wrong 57 quantum binary search edit classical computers are bounded to the worst case of exactly log 2 n 1 textstyle lfloor log _ 2 n 1 rfloor iterations when performing binary search quantum algorithms for binary search are still bounded to a proportion of log 2 n textstyle log _ 2 n queries representing iterations of the classical procedure but the constant factor is less than one providing for a lower time complexity on quantum computers any exact quantum binary search procedure that is a procedure that always yields the correct result requires at least 1 π ln n 1 0 22 log 2 n textstyle frac 1 pi ln n 1 approx 0 22 log _ 2 n queries in the worst case where ln textstyle ln is the natural logarithm 58 there is an exact quantum binary search procedure that runs in 4 log 605 n 0 433 log 2 n textstyle 4 log _ 605 n approx 0 433 log _ 2 n queries in the worst case 59 in comparison grover s algorithm is the optimal quantum algorithm for searching an unordered list of elements and it requires o n displaystyle o sqrt n queries 60 history edit the idea of sorting a list of items to allow for faster searching dates back to antiquity the earliest known example was the inakibit anu tablet from babylon dating back to c 200 bce the tablet contained about 500 sexagesimal numbers and their reciprocals sorted in lexicographical order which made searching for a specific entry easier in addition several lists of names that were sorted by their first letter were discovered on the aegean islands catholicon a latin dictionary finished in 1286 ce was the first work to describe rules for sorting words into alphabetical order as opposed to just the first few letters 9 in 1946 john mauchly made the first mention of binary search as part of the moore school lectures a seminal and foundational college course in computing 9 in 1957 william wesley peterson published the first method for interpolation search 9 61 every published binary search algorithm worked only for arrays whose length is one less than a power of two i until 1960 when derrick henry lehmer published a binary search algorithm that worked on all arrays 63 in 1962 hermann bottenbruch presented an algol 60 implementation of binary search that placed the comparison for equality at the end increasing the average number of iterations by one but reducing to one the number of comparisons per iteration 8 the uniform binary search was developed by a k chandra of stanford university in 1971 9 in 1986 bernard chazelle and leonidas j guibas introduced fractional cascading as a method to solve numerous search problems in computational geometry 50 64 65 implementation issues edit although the basic idea of binary search is comparatively straightforward the details can be surprisingly tricky donald knuth 2 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 mainly because the incorrect implementations failed to run or returned a wrong answer in rare edge cases 66 a study published in 1988 shows that accurate code for it is only found in five out of twenty textbooks 67 furthermore bentley s own implementation of binary search published in his 1986 book programming pearls contained an overflow error that remained undetected for over twenty years the java programming language library implementation of binary search had the same overflow bug for more than nine years 68 in a practical implementation the variables used to represent the indices will often be of fixed size integers and this can result in an arithmetic overflow for very large arrays if the midpoint of the span is calculated as l r 2 displaystyle frac l r 2 then the value of l r displaystyle l r may exceed the range of integers of the data type used to store the midpoint even if l displaystyle l and r displaystyle r are within the range if l displaystyle l and r displaystyle r are nonnegative this can be avoided by calculating the midpoint as l r l 2 displaystyle l frac r l 2 69 an infinite loop may occur if the exit conditions for the loop are not defined correctly once l displaystyle l exceeds r displaystyle 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 most of the programmers who incorrectly implemented binary search made an error in defining the exit conditions 8 70 library support edit many languages standard libraries include binary search routines c provides the function bsearch in its standard library which is typically implemented via binary search although the official standard does not require it to do so 71 c s standard library provides the functions binary_search lower_bound upper_bound and equal_range 72 using the c 20 std ranges library it can be applied over a range as std ranges binary_search d s standard library phobos in std range module provides a type sortedrange returned by sort and assumesorted functions with methods contains equalrange lowerbound and trisect that use binary search techniques by default for ranges that offer random access 73 cobol provides the search all verb for performing binary searches on cobol ordered tables 74 go s sort standard library package contains the functions search searchints searchfloat64s and searchstrings which implement general binary search as well as specific implementations for searching slices of integers floating point numbers and strings respectively 75 java offers a set of overloaded binarysearch static methods in the classes arrays and collections in the standard java util package for performing binary searches on java arrays and on list s respectively 76 77 microsoft s net framework 2 0 offers static generic versions of the binary search algorithm in its collection base classes an example would be system array s method binarysearch t t array t value 78 for objective c the cocoa framework provides the nsarray indexofobject insortedrange options usingcomparator method in mac os x 10 6 79 apple s core foundation c framework also contains a cfarraybsearchvalues function 80 python provides the bisect module that keeps a list in sorted order without having to sort the list after each insertion 81 ruby s array class includes a bsearch method with built in approximate matching 82 rust s slice primitive provides binary_search binary_search_by binary_search_by_key and partition_point 83 see also edit bisection method algorithm for finding a zero of a function the same idea used to solve equations in the real numbers multiplicative binary search binary search variation with simplified midpoint calculation notes and references edit this article was submitted to wikijournal of science for external academic peer review in 2018 reviewer reports the updated content was reintegrated into the wikipedia page under a cc by sa 3 0 license 2019 the version of record as reviewed is anthony lin et al 2 july 2019 binary search algorithm pdf wikijournal of science 2 1 5 doi 10 15347 wjs 2019 005 issn 2470 6345 wikidata q81434400 notes edit the o displaystyle o is big o notation and log displaystyle log is the logarithm in big o notation the base of the logarithm does not matter since every logarithm of a given base is a constant factor of another logarithm of another base that is log b n log k n log k b displaystyle log _ b n log _ k n div log _ k b where log k b displaystyle log _ k b is a constant any search algorithm based solely on comparisons can be represented using a binary comparison tree an internal path is any path from the root to an existing node let i displaystyle i be the internal path length the sum of the lengths of all internal paths if each element is equally likely to be searched the average case is 1 i n displaystyle 1 frac i n or simply one plus the average of all the internal path lengths of the tree this is because internal paths represent the elements that the search algorithm compares to the target the lengths of these internal paths represent the number of iterations after the root node adding the average of these lengths to the one iteration at the root yields the average case therefore to minimize the average number of comparisons the internal path length i displaystyle i must be minimized it turns out that the tree for binary search minimizes the internal path length knuth 1998 proved that the external path length the path length over all nodes where both children are present for each already existing node is minimized when the external nodes the nodes with no children lie within two consecutive levels of the tree this also applies to internal paths as internal path length i displaystyle i is linearly related to external path length e displaystyle e for any tree of n displaystyle n nodes i e 2 n displaystyle i e 2n when each subtree has a similar number of nodes or equivalently the array is divided into halves in each iteration the external nodes as well as their interior parent nodes lie within two levels it follows that binary search minimizes the number of average comparisons as its comparison tree has the lowest possible internal path length 14 knuth 1998 showed on his mix computer model which knuth designed as a representation of an ordinary computer that the average running time of this variation for a successful search is 17 5 log 2 n 17 textstyle 17 5 log _ 2 n 17 units of time compared to 18 log 2 n 16 textstyle 18 log _ 2 n 16 units for regular binary search the time complexity for this variation grows slightly more slowly but at the cost of higher initial complexity 18 knuth 1998 performed a formal time performance analysis of both of these search algorithms on knuth s mix computer which knuth designed as a representation of an ordinary computer binary search takes on average 18 log n 16 textstyle 18 log n 16 units of time for a successful search while linear search with a sentinel node at the end of the list takes 1 75 n 8 5 n mod 2 4 n textstyle 1 75n 8 5 frac n text mod 2 4n units linear search has lower initial complexity because it requires minimal computation but it quickly outgrows binary search in complexity on the mix computer binary search only outperforms linear search with a sentinel if n 44 textstyle n 44 14 27 inserting the values in sorted order or in an alternating lowest highest key pattern will result in a binary search tree that maximizes the average and worst case search time 32 it is possible to search some hash table implementations in guaranteed constant time 37 this is because simply setting all of the bits which the hash functions point to for a specific key can affect queries for other keys which have a common hash location for one or more of the functions 42 there exist improvements of the bloom filter which improve on its complexity or support deletion for example the cuckoo filter exploits cuckoo hashing to gain these advantages 42 that is arrays of length 1 3 7 15 31 62 citations edit williams jr louis f 22 april 1976 a modification to the half interval search binary search method proceedings of the 14th acm southeast conference acm pp 95 101 doi 10 1145 503561 503582 archived from the original on 12 march 2017 retrieved 29 june 2018 1 2 knuth 1998 6 2 1 searching an ordered table subsection binary search butterfield ngondi 2016 p 46 cormen et al 2009 p 39 weisstein eric w binary search mathworld 1 2 flores ivan madpis george 1 september 1971 average binary search length for dense ordered lists communications of the acm 14 9 602 603 doi 10 1145 362663 362752 issn 0001 0782 s2cid 43325465 1 2 3 knuth 1998 6 2 1 searching an ordered table subsection algorithm b 1 2 3 4 bottenbruch hermann 1 april 1962 structure and use of algol 60 journal of the acm 9 2 161 221 doi 10 1145 321119 321120 issn 0004 5411 s2cid 13406983 procedure is described at p 214 43 titled program for binary search 1 2 3 4 5 6 knuth 1998 6 2 1 searching an ordered table subsection history and bibliography 1 2 kasahara morishita 2006 pp 8 9 1 2 3 sedgewick wayne 2011 3 1 subsection rank and selection 1 2 3 goldman goldman 2008 pp 461 463 sedgewick wayne 2011 3 1 subsection range queries 1 2 3 4 5 6 7 8 9 10 11 12 knuth 1998 6 2 1 searching an ordered table subsection further analysis of binary search knuth 1998 6 2 1 searching an ordered table theorem b chang 2003 p 169 1 2 3 knuth 1997 2 3 4 5 path length 1 2 knuth 1998 6 2 1 searching an ordered table subsection exercise 23 rolfe timothy j 1997 analytic derivation of comparisons in binary search acm signum newsletter 32 4 15 19 doi 10 1145 289251 289255 s2cid 23752485 herf michael december 2001 radix tricks stereopsis graphics implement total_cmp for f32 f64 by golddranks pull request 72568 rust lang rust github contains relevant quotations from ieee 754 2008 and 2019 contains a type pun implementation and explanation binary search eliminates branch mispredictions paul khuong some lisp pvk ca khuong paul virak morin pat 2017 array layouts for comparison based searching journal of experimental algorithmics 22 article 1 3 arxiv 1509 05053 doi 10 1145 3053370 s2cid 23752485 binary search is a pathological case for caches paul khuong some lisp pvk ca knuth 1997 2 2 2 sequential allocation 1 2 3 4 beame paul fich faith e 2001 optimal bounds for the predecessor problem and related problems journal of computer and system sciences 65 1 38 72 doi 10 1006 jcss 2002 1822 knuth 1998 answers to exercises 6 2 1 for exercise 5 knuth 1998 6 2 1 searching an ordered table knuth 1998 5 3 1 minimum comparison sorting sedgewick wayne 2011 3 2 ordered symbol tables sedgewick wayne 2011 3 2 binary search trees subsection order based methods and deletion knuth 1998 6 2 2 binary tree searching subsection but what about the worst case sedgewick wayne 2011 3 5 applications which symbol table implementation should i use knuth 1998 5 4 9 disks and drums knuth 1998 6 2 4 multiway trees knuth 1998 6 4 hashing knuth 1998 6 4 hashing subsection history dietzfelbinger martin karlin anna mehlhorn kurt meyer auf der heide friedhelm rohnert hans tarjan robert e august 1994 dynamic perfect hashing upper and lower bounds siam journal on computing 23 4 738 761 doi 10 1137 s0097539791194094 morin pat hash tables pdf p 1 archived pdf from the original on 9 october 2022 retrieved 28 march 2016 knuth 2011 7 1 3 bitwise tricks and techniques 1 2 silverstein alan jud...
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