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 `HashMap[Int, Int]` keyed by `n` stores each completed subproblem. Before recursing, check `memo.contains(n)`: a hit returns immediately, a miss descends.
explicit memo state The memo is threaded through the recursion as `memo: HashMap[Int, Int]` so the lesson stays about caching, not global state.

Basic Implementation

basic.scala
Replay: real traced execution (multi-file project)
import scala.collection.mutable.HashMap

object Main {
	def fib(n: Int, memo: HashMap[Int, Int]): Int = {
		if (memo.contains(n)) {
			return memo(n)
		}
		if (n < 2) {
			memo(n) = n
			return n
		}
		val value = fib(n - 1, memo) + fib(n - 2, memo)
		memo(n) = value
		value
	}

	def main(args: Array[String]): Unit = {
		val memo = HashMap.empty[Int, Int]
		val result = fib(6, memo)
		println(result)
	}
}

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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    values this step{}memomiss -> descend fib(5)action6n

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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    values this step{}memomiss -> descend fib(4)action5n

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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    values this step{}memomiss -> descend fib(3)action4n

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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    values this step{}memomiss -> descend fib(2)action3n

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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    values this step{}memomiss -> descend fib(1)action2n

  6. memo ← {1: 1}, action ← base 1; memo(1) = 1; return

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    11}12val value = fib(n - 1, memo) + fib(n - 2, memo)13memo(n) = value
    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

    19	val result = fib(6, memo)20	println(result)21}
    values this step8stdout8result

Complexity

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

Implementation notes

  • Scala: the recursion takes the memo as a scala.collection.mutable.HashMap[Int, Int] argument rather than a companion-object var, which keeps state explicit without hiding the lesson behind a shared global. The contains + apply indexer pair stays parallel to the lesson spec instead of leaning on getOrElseUpdate.
  • 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.