Create Enhancement of the knapsack algorithm #9266

Enhancement of the knapsack algorithm #9266

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+The following is a Python implementation of the 0-1 knapsack problem with memorization:
+
+def knapsack_memorization(weights, values, capacity):
+  """Solves the 0-1 knapsack problem using memorization.
+
+  Args:
+    weights: A list of weights of the items.
+    values: A list of values of the items.
+    capacity: The capacity of the knapsack.
+
+  Returns:
+    The maximum value of the items that can be placed in the knapsack.
+  """
+
+  memo = {}
+
+  def knapsack_helper(i, capacity):
+    if i == 0 or capacity == 0:
+      return 0
+
+    key = (i, capacity)
+    if key in memo:
+      return memo[key]
+
+    if weights[i - 1] > capacity:
+      max_value = knapsack_helper(i - 1, capacity)
+    else:
+      max_value = max(knapsack_helper(i - 1, capacity),
+                    values[i - 1] + knapsack_helper(i - 1, capacity - weights[i - 1]))
+
+    memo[key] = max_value
+    return max_value
+
+  return knapsack_helper(len(weights), capacity)
+
+
+
+The following is a Python implementation of the 0-N knapsack problem:
+
+def knapsack_0_n(weights, values, capacity):
+  """Solves the 0-N knapsack problem.
+
+  Args:
+    weights: A list of weights of the items.
+    values: A list of values of the items.
+    capacity: The capacity of the knapsack.
+
+  Returns:
+    The maximum value of the items that can be placed in the knapsack.
+  """
+
+  memo = {}
+
+  def knapsack_helper(i, capacity):
+    if i == 0 or capacity == 0:
+      return 0
+
+    key = (i, capacity)
+    if key in memo:
+      return memo[key]
+
+    max_value = 0
+    for j in range(1, capacity // weights[i - 1] + 1):
+      max_value = max(max_value, values[i - 1] * j + knapsack_helper(i - 1, capacity - weights[i - 1] * j))
+
+    memo[key] = max_value
+    return max_value
+
+  return knapsack_helper(len(weights), capacity)
+