function for the knapsack problem which returns one of the optimal subsets

Author: maxwell-aladago

dynamic_programming/knapsack.py

@@ -85,11 +85,32 @@ def knapsack_with_example_solution(W, wt, val):
 
 
 def _construct_solution(dp, wt, i, j, optimal_set):
+    """
+    Recursively reconstructs one of the optimal subsets given
+    a filled DP table and the vector of weights
+
+    Parameters
+    ---------
+
+    dp: list of list, the table of a solved integer weight dynamic programming problem
+
+    wt: list or tuple, the vector of weights of the items
+    i: int, the index of the  item under consideration
+    j: int, the current possible maximum weight
+    optimal_set: set, the optimal subset so far. This gets modified by the function.
+
+    Returns
+    -------
+    None
+
+    """
+    # for the current item i at a maximum weight j to be part of an optimal subset,
+    # the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
+    # where i - 1 means considering only the previous items at the given maximum weight
     if i > 0 and j > 0:
         if dp[i - 1][j] == dp[i][j]:
             _construct_solution(dp, wt, i - 1, j, optimal_set)
         else:
-            # item i is certainly part of the solution
             optimal_set.add(i)
             _construct_solution(dp, wt, i - 1, j - wt[i-1], optimal_set)