Algorithm goal

The Fibonacci sequence is \(1, 1, 2, 3, 5, 8, 13, 21, 33, 54, ...\). To achieve 7, we can use \(5 + 2 = 7\) (2 numbers) or \(3 + 2 + 1 + 1 = 7\) (4 numbers) and to achieve 10, we can use \(2 + 8 = 10\).

Test cases in Scala

assert(fibUntil(5) == List(5, 3, 2, 1, 1))
assert(fillFibonacci(7) == 2)
assert(fillFibonacci(10) == 2)
assert(fillFibonacci(19) == 3)

Algorithm in Scala

19 lines of Scala (compatible versions 2.13 & 3.0), showing how concise Scala can be!

Explanation

Due to how the Fibonacci sequence works, any number in the sequence can be built from any combination of numbers preceding it. Because of this, we know that we need to take the largest available number, before we can use the smaller numbers (due to trying to minimize the number of numbers to use).

To minimize the number of numbers to consume, we then need to start from the largest number down until we reach the target sum. To achieve \(19\), we pick the closest Fibonacci number to it, in which case it is \(13\); adding \(8\) would exceed it now, so we instead pick \(5\), and then the only other number that can add us up to \(19\) is \(1\). (this is © from www.scala-algorithms.com)

Scala concepts & Hints

1. 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('-')
   }
   
   assert(exampleDef("test") == "-test-")
   

   ▶ Run code in the IDE

   It is also frequently used in combination with Tail Recursion.

2. foldLeft and foldRight

   A 'fold' allows you to perform the equivalent of a for-loop, but with a lot less code.

   def foldMutable[I, O](initialState: O)(items: List[I])(f: (O, I) => O): O =
     items.foldLeft(initialState)(f)
   

   ▶ Run code in the IDE

3. Pattern Matching

   Pattern matching in Scala lets you quickly identify what you are looking for in a data, and also extract it.

   assert("Hello World".collect {
     case character if Character.isUpperCase(character) => character.toLower
   } == "hw")
   

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4. 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)
   

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5. 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)
   

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