Binary Search

Instructions

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 the range of indices that equal the search key value are returned. This matches Julia's built-in searchsorted functions. However, to simplify your solution you may assume that the target element is not repeated, except for the bonus task testset on multiple matches.

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. For this exercise, you must do as Julia's built-in searchsorted functions do and return an empty range located at the insertion point, see the docs for searchsorted below for examples.

A binary search halves the number of items to search with each iteration, so locating an item (or determining its absence) takes logarithmic time.

Documentation for Julia's searchsorted function:

searchsorted(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the range of indices of a which compare as equal to x (using binary
search) according to the order specified by the by, lt and rev keywords,
assuming that a is already sorted in that order. 
Return an empty range located at the insertion point if a does not contain
values equal to x.

See also: insorted, searchsortedfirst, sort, findall.

Examples

julia> searchsorted([1, 2, 4, 5, 5, 7], 4) # single match
3:3

julia> searchsorted([1, 2, 4, 5, 5, 7], 5) # multiple matches
4:5

julia> searchsorted([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
3:2

julia> searchsorted([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
7:6

julia> searchsorted([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
1:0

Bonus tasks

• Implement keyword arguments by, lt and rev so that by specifies a transformation applied to all elements of the list, lt specifies a comparison and rev specifies if the list is ordered in reverse. When these parameters are used you must assume that the list has already been sorted using these parameters. See the docs for sort for more details.
• Implement a solution that works on lists with non-unique elements.

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Source

Wikipedia