Algorithm goal

Calculate \((b^e) \% m\) without using exponentiation. This could be useful where the exponent is very big, potentially overflowing.

Test cases in Scala

assert(exponentModulo(base = 2, exponent = 3, modulo = 3) == 2)
assert(exponentModulo(base = 4, exponent = 3, modulo = 3) == 1)
assert(
  exponentModulo(base = 5, exponent = 10, modulo = 7) ==
    exponentModuloRaw(base = 5, exponent = 10, modulo = 7)
)

Algorithm in Scala

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

Explanation

Note that \((a*b) \% m = ((a \% m) * (b \% m)) \% m\). Proof: if \(a = xm + y\) and \(b = wm + z\), then , \(a * b = xwm^2 + (w + x)m + yz\) and \((a * b) \% m = yz \% m\); \(((a \% m) * (b \% m)) \% m = (y * z) \% m = yz \% m\).

Then, \((b^e) \% m = (((b ^ {e - 1}) \% m) * (b \% m)) \% m\). (this is © from www.scala-algorithms.com)

Scala concepts & Hints

1. 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

2. 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")
   

   ▶ Run code in the IDE

3. 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))
   

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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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