Editing Binary search (section)

=== Noisy binary search ===
[[File:Noisy binary search.svg|thumb|right|In noisy binary search, there is a certain probability that a comparison is incorrect.]]
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. Every noisy binary search procedure must make at least <math>(1 - \tau)\frac{\log_2 (n)}{H(p)} - \frac{10}{H(p)}</math> comparisons on average, where <math>H(p) = -p \log_2 (p) - (1 - p) \log_2 (1 - p)</math><!-- Attribution of LaTeX code: see history of https://en.wikipedia.org/wiki/Binary_entropy_function --> is the [[binary entropy function]] and <math>\tau</math> is the probability that the procedure yields the wrong position.<ref>{{cite conference |last1=Ben-Or |first1=Michael |last2=Hassidim |first2=Avinatan |title=The Bayesian learner is optimal for noisy binary search (and pretty good for quantum as well) |date=2008 |book-title=49th [[Annual IEEE Symposium on Foundations of Computer Science|Symposium on Foundations of Computer Science]] |pages=221–230 |doi=10.1109/FOCS.2008.58 |url=http://www2.lns.mit.edu/~avinatan/research/search-full.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www2.lns.mit.edu/~avinatan/research/search-full.pdf |archive-date=2022-10-09 |url-status=live |isbn=978-0-7695-3436-7}}</ref><ref name="pelc1989">{{cite journal|last1=Pelc|first1=Andrzej|title=Searching with known error probability|journal=Theoretical Computer Science|date=1989|volume=63|issue=2|pages=185–202|doi=10.1016/0304-3975(89)90077-7|doi-access=free}}</ref><ref>{{cite conference|last1=Rivest|first1=Ronald L.|last2=Meyer|first2=Albert R.|last3=Kleitman|first3=Daniel J.|last4=Winklmann|first4=K.|author-link1=Ronald Rivest|author-link2=Albert R. Meyer|author-link3=Daniel Kleitman|title=Coping with errors in binary search procedures|conference=10th [[Symposium on Theory of Computing|ACM Symposium on Theory of Computing]]|doi=10.1145/800133.804351|doi-access=free}}</ref> The noisy binary search problem can be considered as a case of the [[Ulam's game|Rényi-Ulam game]],<ref>{{cite journal|last1=Pelc|first1=Andrzej|title=Searching games with errors—fifty years of coping with liars|journal=Theoretical Computer Science|date=2002|volume=270|issue=1–2|pages=71–109|doi=10.1016/S0304-3975(01)00303-6|doi-access=free}}</ref> a variant of [[Twenty Questions]] where the answers may be wrong.<ref>{{Cite journal | last1=Rényi | first1=Alfréd | title=On a problem in information theory | language=hu | mr=0143666 | year=1961 | journal=Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei| volume=6 | pages=505–516}}</ref>