# Binary search a generic Array

## Algorithm goal

A binary search finds the index of a target item in a sorted array. The goal is to implement the algorithm using pure-functional Scala.

Binary search runs in $$O(\log{n})$$, which is faster than a linear search ($$O(n)$$).

This algorithm works by iteration: we compare the middle element to the target element; if the middle element is higher than the target, the algorithm checks the left side (ie. run the algorithm against the smaller elements), and likewise for the right side. Once equality is found, we return the result; if the range of checking is exhausted, we return a None to indicate that (in other languages, this would also be -1, or in different implementations of the algorithm, also return the element position closest to the target element, for the purpose of finding where to insert the element).

## Explanation

The Scala implementation uses the Range concept in order to achieve a more terse solution, in particular by defining the range for the next iteration in terms of the previous range, rather than dealing with raw indices.

This is a very powerful concept because you notice that in Scala, algorithms can be quite self-explanatory whereas in some C/Python algorithm implementations one would have to refer to documentation and comments for an explanation. Documentability is crucial in sharing knowledge. (this is © from www.scala-algorithms.com)

## Scala Concepts & Hints

Def Inside Def

A great aspect of Scala is being able to declare functions inside functions, making it possible to reduce repetition.

def exampleDef(input: String): String = {
def surroundInputWith(char: Char): String = s"$char$input\$char"
surroundInputWith('-')
}


It is also frequently used in combination with Tail Recursion.

Drop, Take, dropRight, takeRight

Scala's drop and take methods typically remove or select n items from a collection.

assert(List(1, 2, 3).drop(2) == List(3))

assert(List(1, 2, 3).take(2) == List(1, 2))

assert(List(1, 2, 3).dropRight(2) == List(1))

assert(List(1, 2, 3).takeRight(2) == List(2, 3))

assert((1 to 5).take(2) == (1 to 2))


Option Type

The 'Option' type is used to describe a computation that either has a result or does not. In Scala, you can 'chain' Option processing, combine with lists and other data structures. For example, you can also turn a pattern-match into a function that return an Option, and vice-versa!

assert(Option(1).flatMap(x => Option(x + 2)) == Option(3))

assert(Option(1).flatMap(x => None) == None)


Ordering

In Scala, the 'Ordering' type is a 'type class' that contains methods to determine an ordering of specific types.

assert(List(3, 2, 1).sorted == List(1, 2, 3))

assert(List(3, 2, 1).sorted(Ordering[Int].reverse) == List(3, 2, 1))

assert(Ordering[Int].lt(1, 2))

assert(!Ordering[Int].lt(2, 1))


Range

The (1 to n) syntax produces a "Range" which is a representation of a sequence of numbers.

assert((1 to 5).toString == "Range 1 to 5")

assert((1 to 5).reverse.toString() == "Range 5 to 1 by -1")

assert((1 to 5).toList == List(1, 2, 3, 4, 5))


Stack Safety

Stack safety is present where a function cannot crash due to overflowing the limit of number of recursive calls.

This function will work for n = 5, but will not work for n = 2000 (crash with java.lang.StackOverflowError) - however there is a way to fix it :-)

In Scala Algorithms, we try to write the algorithms in a stack-safe way, where possible, so that when you use the algorithms, they will not crash on large inputs. However, stack-safe implementations are often more complex, and in some cases, overly complex, for the task at hand.

def sum(from: Int, until: Int): Int =
if (from == until) until else from + sum(from + 1, until)

def thisWillSucceed: Int = sum(1, 5)

def thisWillFail: Int = sum(1, 300)


Tail Recursion

In Scala, tail recursion enables you to rewrite a mutable structure such as a while-loop, into an immutable algorithm.

def fibonacci(n: Int): Int = {
@scala.annotation.tailrec
def go(i: Int, previous: Int, beforePrevious: Int): Int =
if (i >= n) previous else go(i + 1, previous + beforePrevious, previous)

go(i = 1, previous = 1, beforePrevious = 0)
}

assert(fibonacci(8) == 21)


## Algorithm in Scala

18 lines of Scala (version 2.13), showing how concise Scala can be!

## Test cases in Scala

assert(binarySearch(Array.empty[Int], 2) == None)
assert(binarySearch(Array(1), 2) == None)
assert(binarySearch(Array(1, 2), 2) == Some(1))
assert(binarySearch(Array(1, 3), 2) == None)
assert(binarySearch(Array(1, 2, 3), 2) == Some(1))
assert(binarySearch(Array(1, 3, 3), 2) == None)
assert(binarySearch(Array(1, 1, 2, 3), 2) == Some(2))
assert(binarySearch(Array(1, 3, 4), 2) == None)
assert(binarySearch(Array(1, 3, 4, 5), 6) == None)
assert(binarySearch(Array(1, 2, 3, 4, 5), 2) == Some(1))

def binarySearch[T](a: Array[T], target: T)(implicit
ordering: Ordering[T]
): Option[Int] = ???