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e 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 judy iv shop manual pdf hewlett packard pp 80 81 archived pdf from the original on 9 october 2022 1 2 fan bin andersen dave g kaminsky michael mitzenmacher michael d 2014 cuckoo filter practically better than bloom proceedings of the 10th acm international on conference on emerging networking experiments and technologies pp 75 88 doi 10 1145 2674005 2674994 bloom burton h 1970 space time trade offs in hash coding with allowable errors communications of the acm 13 7 422 426 doi 10 1145 362686 362692 s2cid 7931252 knuth 1998 6 2 1 searching an ordered table subsection an important variation knuth 1998 6 2 1 searching an ordered table subsection algorithm u moffat turpin 2002 p 33 1 2 3 knuth 1998 6 2 1 searching an ordered table subsection interpolation search knuth 1998 6 2 1 searching an ordered table subsection exercise 22 perl yehoshua itai alon avni haim 1978 interpolation search a log 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321 56384 2 external links edit the wikibook algorithm implementation has a page on the topic of binary search nist dictionary of algorithms and data structures binary search comparisons and benchmarks of a variety of binary search implementations in c archived 25 september 2019 at the wayback machine v t e data structures and algorithms data structures array associative array binary search tree fenwick tree graph hash table heap linked list queue segment tree stack string tree trie algorithms and algorithmic paradigms backtracking binary search breadth first search brute force search depth first search divide and conquer dynamic programming graph traversal fold greedy hash function minimax online randomized recursion root finding sorting streaming sweep line string searching topological sorting list of data structures list of algorithms retrieved from https en wikipedia org w index php title binary_search oldid 1371884285 categories wikipedia articles published in peer reviewed 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