Fibonacci with Memoization

Compute fib(n) recursively. Cache each fib(k) in a memo map so each subproblem is solved at most once.

Algorithm

Canonical input n <- 6 produces fib(6) = 8. Replay highlights every memo write and every cache hit.

Basic Implementation

basic.R

fib <- function(n, memo) {
	key <- as.character(n)
	if (!is.null(memo[[key]])) {
		return(list(value = memo[[key]], memo = memo))
	}
	if (n < 2) {
		memo[[key]] <- n
		return(list(value = n, memo = memo))
	}
	r1 <- fib(n - 1, memo); memo <- r1$memo
	r2 <- fib(n - 2, memo); memo <- r2$memo
	value <- r1$value + r2$value
	memo[[key]] <- value
	return(list(value = value, memo = memo))
}

memo <- list()
r <- fib(6, memo)
result <- r$value
cat(result, "\n", sep = "")

Complexity

Time: O(n) with memoization (vs. O(2^n) without)
Space: O(n) memo + O(n) call stack

Implementation notes

R: the recursion takes the memo as a parameter and returns the updated memo alongside the value rather than mutating an enclosing environment, which keeps state explicit without leaning on R's reference-semantic environments (new.env(parent = emptyenv())). Every recursive call threads r1$memo forward so the cache shape is visible in the source.
!is.null(memo[[key]]) is the explicit cache-check predicate; R's named-list default behaviour returns NULL for absent keys, so the predicate stays parallel to the lesson spec instead of leaning on tryCatch defaults or exists()-on-environment checks.
The replay shows the call stack on one side and the memo map on the other so memo writes and cache hits are visually distinct.

memoization A named list `memo` keyed by `as.character(n)` stores each completed subproblem. Before recursing, check `!is.null(memo[[key]])`: a hit returns immediately, a miss descends.

explicit memo state The memo is threaded through the recursion as the second parameter and returned alongside the value so the lesson stays about caching, not a closure upvalue or a shared environment.