add more details to complexity analysis, add output

dynamic_programming/fast_fibonacci.rb

@@ -7,18 +7,46 @@
 # Given n, calculate F(n).
 
 #
-# Approach: Bottom-Up Approach using Memoization
+# Approach: Top-Down Approach using Memoization
 #
 
 # Complexity Analysis:
 #
-# Time complexity: O(n)
-# Space complexity: O(n)
+# Time complexity: O(n). Each number, starting at 2 up to and
+# including N, is visited, computed and then stored for O(1) access
+# later on.
+# Space complexity: O(n). The size of the stack in memory is
+# proportionate to N.
 #
+def fib(n)
+  return n if n <= 1
 
-def fibonacci(number, memo_hash = {})
-    if number == 0 || number == 1
-        return number
-    end
-    memo_hash[number] ||= fibonacci(number - 1, memo_hash) + fibonacci(number - 2, memo_hash)
+  cache = {}
+  cache[0] = 0
+  cache[1] = 1
+
+  memoize(n, cache)
 end
+
+def memoize(n, cache)
+  return cache[n] if cache.keys.include? n
+
+  cache[n] = memoize(n - 1, cache) + memoize(n - 2, cache)
+
+  memoize(n, cache)
+end
+
+n = 2
+# Output: 1
+# Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.
+puts(fib(n))
+
+n = 3
+# Output: 2
+# Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.
+puts(fib(n))
+
+n = 4
+# Output: 3
+# Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.
+puts(fib(n))