Refactor memoized knapsack to remove global state

Author: nickzerjeski

dynamic_programming/knapsack.py

@@ -6,27 +6,53 @@
 using dynamic programming.
 """
 
+from __future__ import annotations
 
-def mf_knapsack(i, wt, val, j):
+from functools import lru_cache
+
+
+def mf_knapsack(i: int, wt: list[int], val: list[int], j: int) -> int:
     """
-    This code involves the concept of memory functions. Here we solve the subproblems
-    which are needed unlike the below example
-    F is a 2D array with ``-1`` s filled up
+    Return the optimal value for the 0/1 knapsack problem using memoization.
+
+    This implementation caches subproblems with ``functools.lru_cache`` and avoids
+    global mutable state.
+
+    >>> mf_knapsack(4, [4, 3, 2, 3], [3, 2, 4, 4], 6)
+    8
+    >>> mf_knapsack(3, [10, 20, 30], [60, 100, 120], 50)
+    220
+    >>> mf_knapsack(0, [1], [10], 50)
+    0
     """
-    global f  # a global dp table for knapsack
-    if f[i][j] < 0:
-        if j < wt[i - 1]:
-            val = mf_knapsack(i - 1, wt, val, j)
-        else:
-            val = max(
-                mf_knapsack(i - 1, wt, val, j),
-                mf_knapsack(i - 1, wt, val, j - wt[i - 1]) + val[i - 1],
-            )
-        f[i][j] = val
-    return f[i][j]
+    if i < 0:
+        raise ValueError("The number of items to consider cannot be negative.")
+    if j < 0:
+        raise ValueError("The knapsack capacity cannot be negative.")
+    if len(wt) != len(val):
+        raise ValueError("The number of weights must match the number of values.")
+    if i > len(wt):
+        raise ValueError("The number of items to consider cannot exceed input length.")
+
+    weights = tuple(wt)
+    values = tuple(val)
+
+    @lru_cache(maxsize=None)
+    def solve(item_count: int, capacity: int) -> int:
+        if item_count == 0 or capacity == 0:
+            return 0
+        if weights[item_count - 1] > capacity:
+            return solve(item_count - 1, capacity)
+        return max(
+            solve(item_count - 1, capacity),
+            solve(item_count - 1, capacity - weights[item_count - 1])
+            + values[item_count - 1],
+        )
+
+    return solve(i, j)
 
 
-def knapsack(w, wt, val, n):
+def knapsack(w: int, wt: list[int], val: list[int], n: int) -> tuple[int, list[list[int]]]:
     dp = [[0] * (w + 1) for _ in range(n + 1)]
 
     for i in range(1, n + 1):
@@ -36,10 +62,10 @@ def knapsack(w, wt, val, n):
             else:
                 dp[i][w_] = dp[i - 1][w_]
 
-    return dp[n][w_], dp
+    return dp[n][w], dp
 
 
-def knapsack_with_example_solution(w: int, wt: list, val: list):
+def knapsack_with_example_solution(w: int, wt: list, val: list) -> tuple[int, set[int]]:
     """
     Solves the integer weights knapsack problem returns one of
     the several possible optimal subsets.
@@ -100,7 +126,9 @@ def knapsack_with_example_solution(w: int, wt: list, val: list):
     return optimal_val, example_optional_set
 
 
-def _construct_solution(dp: list, wt: list, i: int, j: int, optimal_set: set):
+def _construct_solution(
+    dp: list[list[int]], wt: list[int], i: int, j: int, optimal_set: set[int]
+) -> None:
     """
     Recursively reconstructs one of the optimal subsets given
     a filled DP table and the vector of weights
@@ -139,10 +167,9 @@ def _construct_solution(dp: list, wt: list, i: int, j: int, optimal_set: set):
     wt = [4, 3, 2, 3]
     n = 4
     w = 6
-    f = [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
     optimal_solution, _ = knapsack(w, wt, val, n)
     print(optimal_solution)
-    print(mf_knapsack(n, wt, val, w))  # switched the n and w
+    print(mf_knapsack(n, wt, val, w))
 
     # testing the dynamic programming problem with example
     # the optimal subset for the above example are items 3 and 4