Add Fibonacci Sequence (Dynamic Programming)

Author: AhmedLSayed9

dynamic_programming/fibonacci.dart

@@ -0,0 +1,52 @@
+import '../utils/time_measure.dart';
+
+/// Calculate fibonacci number at specific position "n" using Dynamic Programming approach.
+/// Fibonacci Sequence:
+/// 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
+void main() {
+  syncExecTimeMeasure(() => print(fibTabulated(45)), name: 'O(n)');
+  syncExecTimeMeasure(() => print(fibMemoized(45)), name: 'O(n)');
+  syncExecTimeMeasure(() => print(fibRecursive(45)), name: 'O(n^2)');
+}
+
+// Tabulation (Bottom-up approach in Dynamic Programming)
+// Time Complexity: O(n), and better Auxiliary Space.
+fibTabulated(int n) {
+  if (n <= 0) throw ArgumentError('Input must be a positive integer');
+
+  final table = [1, 1];
+  int currentFib = 1;
+
+  for (int i = 3; i <= n; i++) {
+    currentFib = table[0] + table[1];
+    table[0] = table[1];
+    table[1] = currentFib;
+  }
+  return currentFib;
+}
+
+// Memoization (Top-down approach in Dynamic Programming) (involves recursion).
+// Time Complexity: O(n)
+fibMemoized(int n) {
+  if (n <= 0) throw ArgumentError('Input must be a positive integer');
+  if (n <= 2) return 1;
+  if (_memo.containsKey(n)) return _memo[n];
+
+  final result = fibMemoized(n - 1) + fibMemoized(n - 2);
+  _memo[n] = result;
+  return result;
+}
+
+final Map<int, int> _memo = {};
+
+// Naive solution using Recursion.
+// Time Complexity: O(2^n) "Exponential Big O"
+// https://i.stack.imgur.com/kgXDS.png
+// Note: it's exactly (1.6180339887^n), which is known as the golden ratio:
+// https://stackoverflow.com/questions/360748
+fibRecursive(int n) {
+  if (n <= 0) throw ArgumentError('Input must be a positive integer');
+  if (n <= 2) return 1;
+
+  return fibRecursive(n - 1) + fibRecursive(n - 2);
+}