Recursion and Dynamic Programming

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.

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.

Basic Implementation

basic.R
Replay: real traced execution (multi-file project)
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 = "")

  1. memo ← {}, action ← miss -> descend fib(5)

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{}memomiss -> descend fib(5)action6n

  2. memo ← {}, action ← miss -> descend fib(4)

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{}memomiss -> descend fib(4)action5n

  3. memo ← {}, action ← miss -> descend fib(3)

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{}memomiss -> descend fib(3)action4n

  4. memo ← {}, action ← miss -> descend fib(2)

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{}memomiss -> descend fib(2)action3n

  5. memo ← {}, action ← miss -> descend fib(1)

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{}memomiss -> descend fib(1)action2n

  6. memo ← {1: 1}, action ← base 1; memo[[1]] <- 1; return

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{1: 1}memobase 1; memo[[1]] <- 1; returnaction1n

  7. memo ← {0: 0, 1: 1}, action ← base 0; memo[[0]] <- 0; fib(2)=1; memo[[2]] <- 1

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{0: 0, 1: 1}memobase 0; memo[[0]] <- 0; fib(2)=1; memo[[2]] <- 1action0n

  8. memo ← {0: 0, 1: 1, 2: 1, 3: 2}, action ← hit 1; fib(3)=2; memo[[3]] <- 2

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{0: 0, 1: 1, 2: 1, 3: 2}memohit 1; fib(3)=2; memo[[3]] <- 2action1n

  9. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3}, action ← hit 1; fib(4)=3; memo[[4]] <- 3

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3}memohit 1; fib(4)=3; memo[[4]] <- 3action2n

  10. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5}, action ← hit 2; fib(5)=5; memo[[5]] <- 5

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5}memohit 2; fib(5)=5; memo[[5]] <- 5action3n

  11. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5, 6: 8}, action ← hit 3; fib(6)=8; memo[[6]] <- 8

    9}10r1 <- fib(n - 1, memo); memo <- r1$memo11r2 <- fib(n - 2, memo); memo <- r2$memo
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5, 6: 8}memohit 3; fib(6)=8; memo[[6]] <- 8action4n

  12. stdout ← 8

    19result <- r$value20cat(result, "\n", sep = "")
    values this step8stdout8result

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.