Recursion and Dynamic Programming

DP

Fibonacci with Memoization

Computes fib(n) with top-down memoization. Each subproblem is solved once and cached; later calls hit the cache instead of recomputing.

Algorithm

Canonical input is fib 6 with an empty memo. The descent fills the cache {0, 1, 2, 3, 4, 5, 6}; the final value is 8.

Basic Implementation

basic.sh 
#!/usr/bin/env bash
set -euo pipefail
declare -A memo
fib_result=0
fib() {
	local n=$1
	local key=$n
	if [ -n "${memo[$key]+_}" ]; then
		fib_result=${memo[$key]}
		return
	fi
	if [ "$n" -lt 2 ]; then
		memo[$key]=$n
		fib_result=$n
		return
	fi
	fib $((n - 1))
	local r1=$fib_result
	fib $((n - 2))
	local r2=$fib_result
	local value=$((r1 + r2))
	memo[$key]=$value
	fib_result=$value
}

fib 6
echo "$fib_result"

Complexity

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

Implementation notes

  • Bash: command substitution $(fib ...) would spawn a subshell and throw away the memo updates, so the function mutates a global memo (declare -A) directly and returns its value through a global fib_result. Callers read fib_result immediately after the call returns.
  • The "${memo[$key]+_}" parameter expansion distinguishes a missing key from a key whose value is 0, which is what makes memo[0]=0 safe to cache.
  • The replay describes the descent / cache miss frames first, then the cache fills as the recursion unwinds, so the viewer sees the cache build out from the base cases.
top-down memoization `fib(n)` first consults a `memo` associative array. On hit, return the stored value; on miss, recursively compute `fib(n - 1) + fib(n - 2)`, store, and return.
base cases `fib(0) = 0` and `fib(1) = 1` populate the cache and stop the recursion.