Recursion and Dynamic Programming

Factorial (Recursive)

factorial(0) = 1, otherwise factorial(n) = n * factorial(n - 1). The smallest example of recursion with a single base case.

Algorithm

Basic Implementation

basic.R 
factorial <- function(n) {
	if (n == 0) {
		return(1)
	}
	return(n * factorial(n - 1))
}

result <- factorial(5)
cat(result, "\n", sep = "")

The pinned run is factorial(5). The diagrams separate the descent, the base case, and the return values so the stack does not feel invisible.

Step 1 - Descend to the base case

Each call waits for one smaller call until f(0) returns 1.

Call tree for factorial(5): f(5) waits on f(4), down to f(0).f(5)waitsf(4)waitsf(3)waitsf(2)waitsf(1)waitsf(0)base = 1

Step 2 - Base value starts the unwind

The first finished frame is f(0) = 1; f(1) can now compute 1 * 1.

Call stack just before unwind begins.top -> bottomknown returnf(0)1f(1)waitingf(2)waitingf(3)waitingf(4)waitingf(5)waiting

Step 3 - Unwind returns 120

Each frame multiplies its n by the completed smaller result.

Return chain for factorial(5).framecalculationreturnsf(0)base1f(1)1 * 11f(2)2 * 12f(3)3 * 26f(4)4 * 624f(5)5 * 24120

Complexity

  • Time: O(n)
  • Space: O(n) call stack

Implementation notes

  • R: same recursive shape as the other languages, with factorial <- function(n) documenting the integer contract. R ships its own factorial() in base R as a vectorised wrapper around gamma(n + 1), which would hide the recursion the lesson teaches.
  • The replay treats the call stack as a vertical list of frames; descent pushes, unwind pops with the computed multiplication shown.
base case `if (n == 0) { return(1) }`
recursive call `return(n * factorial(n - 1))`