Enhancement of the knapsack algorithm with memorization and generalisation (#9295)

* enhance knapsack problem

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* wei/refactor code

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update test_knapsack.py

* Update knapsack.py

* Update test_knapsack.py

* Update knapsack.py

* Update knapsack.py

* Update knapsack.py

* Update knapsack.py

* Update knapsack.py

* Update test_knapsack.py

---------

Co-authored-by: weijiang <weijiang@weijiangs-MacBook-Pro.local>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>

knapsack/README.md

@@ -1,4 +1,4 @@
-# A naive recursive implementation of 0-1 Knapsack Problem
+# A recursive implementation of 0-N Knapsack Problem
 
 This overview is taken from:
 

knapsack/knapsack.py

@@ -1,14 +1,23 @@
-"""A naive recursive implementation of 0-1 Knapsack Problem
+"""A recursive implementation of 0-N Knapsack Problem
 https://en.wikipedia.org/wiki/Knapsack_problem
 """
 
 from __future__ import annotations
 
+from functools import lru_cache
 
-def knapsack(capacity: int, weights: list[int], values: list[int], counter: int) -> int:
+
+def knapsack(
+    capacity: int,
+    weights: list[int],
+    values: list[int],
+    counter: int,
+    allow_repetition=False,
+) -> int:
     """
     Returns the maximum value that can be put in a knapsack of a capacity cap,
-    whereby each weight w has a specific value val.
+    whereby each weight w has a specific value val
+    with option to allow repetitive selection of items
 
     >>> cap = 50
     >>> val = [60, 100, 120]
@@ -17,28 +26,40 @@ def knapsack(capacity: int, weights: list[int], values: list[int], counter: int)
     >>> knapsack(cap, w, val, c)
     220
 
-    The result is 220 cause the values of 100 and 120 got the weight of 50
+    Given the repetition is NOT allowed,
+    the result is 220 cause the values of 100 and 120 got the weight of 50
     which is the limit of the capacity.
+    >>> knapsack(cap, w, val, c, True)
+    300
+
+    Given the repetition is allowed,
+    the result is 300 cause the values of 60*5 (pick 5 times)
+    got the weight of 10*5 which is the limit of the capacity.
     """
 
-    # Base Case
+    @lru_cache
-    if counter == 0 or capacity == 0:
+    def knapsack_recur(capacity: int, counter: int) -> int:
-        return 0
+        # Base Case
+        if counter == 0 or capacity == 0:
+            return 0
 
-    # If weight of the nth item is more than Knapsack of capacity,
+        # If weight of the nth item is more than Knapsack of capacity,
-    #   then this item cannot be included in the optimal solution,
+        #   then this item cannot be included in the optimal solution,
-    # else return the maximum of two cases:
+        # else return the maximum of two cases:
-    #   (1) nth item included
+        #   (1) nth item included only once (0-1), if allow_repetition is False
-    #   (2) not included
+        #       nth item included one or more times (0-N), if allow_repetition is True
-    if weights[counter - 1] > capacity:
+        #   (2) not included
-        return knapsack(capacity, weights, values, counter - 1)
+        if weights[counter - 1] > capacity:
-    else:
+            return knapsack_recur(capacity, counter - 1)
-        left_capacity = capacity - weights[counter - 1]
+        else:
-        new_value_included = values[counter - 1] + knapsack(
+            left_capacity = capacity - weights[counter - 1]
-            left_capacity, weights, values, counter - 1
+            new_value_included = values[counter - 1] + knapsack_recur(
-        )
+                left_capacity, counter - 1 if not allow_repetition else counter
-        without_new_value = knapsack(capacity, weights, values, counter - 1)
+            )
-        return max(new_value_included, without_new_value)
+            without_new_value = knapsack_recur(capacity, counter - 1)
+            return max(new_value_included, without_new_value)
+
+    return knapsack_recur(capacity, counter)
 
 
 if __name__ == "__main__":

knapsack/tests/test_knapsack.py

@@ -30,7 +30,7 @@ class Test(unittest.TestCase):
 
     def test_easy_case(self):
         """
-        test for the base case
+        test for the easy case
         """
         cap = 3
         val = [1, 2, 3]
@@ -48,6 +48,16 @@ class Test(unittest.TestCase):
         c = len(val)
         assert k.knapsack(cap, w, val, c) == 220
 
+    def test_knapsack_repetition(self):
+        """
+        test for the knapsack repetition
+        """
+        cap = 50
+        val = [60, 100, 120]
+        w = [10, 20, 30]
+        c = len(val)
+        assert k.knapsack(cap, w, val, c, True) == 300
+
 
 if __name__ == "__main__":
     unittest.main()