What does Binary Search Algorithm do? Given a sorted array of n elements, write a function to search for the index of a given element (target).

Table of Contents

## Process of working of Binary Search Algorithm?

- Search for the array by dividing the array in half repeatedly.
- Initially consider the actual array and pick the element at the middle index
- Keep a lower index i.e. 0 and higher index i.e. length of array
- If it is equal to the target element then return the index
- Else if it is greater than the target element then consider only the left half of array. (lower index = 0, higher = middle – 1)
- Else if it is less than the target element then consider only the right half of array. (lower index = middle + 1, higher = length of array)
- Return -(insertion index + 1) if the target element is not found in the array (If the lower index is greater than or equal to higher index). Some simpler implementations just return -1 if the element is not found. The offset of 1 must be added as the insertion index might be 0 (the searched value might be smaller than all elements in the array). As indexing starts at 0, this must be distinguishable from the case where the target element has the index 0.

## Time Complexity of Binary Search Algorithm

- O(log n) Worst Case
- O(1) Best Case (If middle element of initial array is the target element)

## Space Complexity of Binary Search Algorithm

- O(1) For iterative approach
- O(1) For recursive approach if tail call optimization is used, O(log n) due to recursion call stack, otherwise

## Haskell Code for Binary Search Algorithm

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```
module Misc.BinarySearch where
bsWithIndex :: (Ord a) => [a] -> a -> Int -> Maybe Int
bsWithIndex list n i
| n == head list = Just i
| len == 1 = Nothing -- only candidate in list is not the right elem
| n < head ys = bsWithIndex xs n i
| otherwise = bsWithIndex ys n (i + half)
where
len = length list
half = len `div` 2
(xs, ys) = splitAt half list
bs :: (Ord a) => [a] -> a -> Int
bs list n = case bsWithIndex list n 0 of
Just x -> x
Nothing -> -1
main :: IO ()
main = do
let intList = [1,4,7,10,25,30]
print $ bs intList 29 -- 29 -> -1 as not in list
print $ bs intList 7 -- 7 -> 2 as in list
```

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