Implement a binary search algorithm.
Searching a sorted collection is a common task. A dictionary is a sorted list of word definitions. Given a word, one can find its definition. A telephone book is a sorted list of people's names, addresses, and telephone numbers. Knowing someone's name allows one to quickly find their telephone number and address.
If the list to be searched contains more than a few items (a dozen, say) a binary search will require far fewer comparisons than a linear search, but it imposes the requirement that the list be sorted.
In computer science, a binary search or half-interval search algorithm finds the position of a specified input value (the search "key") within an array sorted by key value.
In each step, the algorithm compares the search key value with the key value of the middle element of the array.
If the keys match, then a matching element has been found and its index, or position, is returned.
Otherwise, if the search key is less than the middle element's key, then the algorithm repeats its action on the sub-array to the left of the middle element or, if the search key is greater, on the sub-array to the right.
If the remaining array to be searched is empty, then the key cannot be found in the array and a special "not found" indication is returned.
A binary search halves the number of items to check with each iteration, so locating an item (or determining its absence) takes logarithmic time. A binary search is a dichotomic divide and conquer search algorithm.
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(* binary-search - 1.3.0 *) open OUnit2 open Binary_search let result_to_string f = function | Error m -> Printf.sprintf "Error \"%s\"" m | Ok x -> f x |> Printf.sprintf "Some %s" let ae exp got _test_ctxt = assert_equal ~printer:(result_to_string string_of_int) exp got let tests = [ "finds a value in an array with one element" >:: ae (Ok 0) (find [|6|] 6); "finds a value in the middle of an array" >:: ae (Ok 3) (find [|1; 3; 4; 6; 8; 9; 11|] 6); "finds a value at the beginning of an array" >:: ae (Ok 0) (find [|1; 3; 4; 6; 8; 9; 11|] 1); "finds a value at the end of an array" >:: ae (Ok 6) (find [|1; 3; 4; 6; 8; 9; 11|] 11); "finds a value in an array of odd length" >:: ae (Ok 9) (find [|1; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 634|] 144); "finds a value in an array of even length" >:: ae (Ok 5) (find [|1; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377|] 21); "identifies that a value is not included in the array" >:: ae (Error "value not in array") (find [|1; 3; 4; 6; 8; 9; 11|] 7); "a value smaller than the array's smallest value is not found" >:: ae (Error "value not in array") (find [|1; 3; 4; 6; 8; 9; 11|] 0); "a value larger than the array's largest value is not found" >:: ae (Error "value not in array") (find [|1; 3; 4; 6; 8; 9; 11|] 13); "nothing is found in an empty array" >:: ae (Error "value not in array") (find [||] 1); "nothing is found when the left and right bounds cross" >:: ae (Error "value not in array") (find [|1; 2|] 0); ] let () = run_test_tt_main ("binary-search tests" >::: tests)
let find haystack needle = let rec search l r = if l > r then Error "value not in array" else let i = (l+r)/2 in match haystack.(i) with | x when x = needle -> Ok i | x when x > needle -> search l (i-1) | x when x < needle -> search (i+1) r in search 0 @@ Array.length haystack - 1
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