Solve the search problem for sorted input

## Description

Semi-formally, binary search can be described by an algorithm as follows:

• Given is a sorted list l and a value v of the element type of l.
• If l is the empty list then return False.
• Let m be the element in the "middle" of l.
• If m equals v, then return True.
• If m < v, then recursively search v in the list containing all elements to the right of m.
• If m > v, then recursively search v in the list containing all elements to the left of m.
Please note that this formulation also expresses that the element type must admit comparison for comparison.

## Illustration

### Binary search

```-- Polymorphic binary search
-- Assume that input list is sorted
search :: Ord a => [a] -> a -> Bool
search [] _ = False
search xs x =
if x < y then search ys1 x
else if x > y then search ys2 x
else True
where
ys1 = take l xs
(y:ys2) = drop l xs
l = length xs `div` 2
```

The implemented search function can be applied as follows:

```-- Illustrate binary search
main = do
let input = [1,2,3,4,4]
print \$ search input 1 -- True
print \$ search input 5 -- False
```

Note that the input list is readily sorted. The less efficient, linear search does not require the input list to be sorted.

We may also represent the input as a sorted tree. In this manner, the decision of going to the left or the right side after "halving" is much more straightforward. Here is a suitable algebraic data type for trees:

```-- Binary trees
data Tree a = Empty | Fork a (Tree a) (Tree a)
```

The corresponding search function looks like this:

```-- Polymorphic binary search
-- Assume that input tree is sorted
search :: Ord a => Tree a -> a -> Bool
search Empty _ = False
search (Fork x1 l r) x2 =
if x2 < x1
then search l x2
else if x2 > x1
then search r x2
else True
```

A demo follows:

```-- Illustrate binary search
main = do
let input = Fork 3 (Fork 1 Empty Empty) (Fork 42 Empty Empty)
print \$ search input 1 -- True
print \$ search input 3 -- True
print \$ search input 42 -- True
print \$ search input 88 -- False
```

## Citation

(http://en.wikipedia.org/wiki/Binary_search_algorithm, 21 April 2013)

In computer science, a binary search or half-interval search algorithm finds the position of a specified value (the input "key") within a sorted array. ... In each step, the algorithm compares the input key value with the key value of the middle element of the array. If the keys match, then a matching element has been found so its index, or position, is returned. Otherwise, if the sought 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 input key is greater, on the sub-array to the right. If the remaining array to be searched is reduced to zero, then the key cannot be found in the array and a special "Not found" indication is returned.

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