ENH: Added a functionality to make it possible to reconstruct an optimal subset for the dynamic programming problem (#1139)

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

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

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

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

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

* some pep8 cleanup too

dynamic_programming/knapsack.py

@@ -1,7 +1,13 @@
 """
-Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack.
+Given weights and values of n items, put these items in a knapsack of
+ capacity W to get the maximum total value in the knapsack.
+
+Note that only the integer weights 0-1 knapsack problem is solvable
+ using dynamic programming.
 """
-def MF_knapsack(i,wt,val,j):
+
+
+def MF_knapsack(i, wt, val, j):
     '''
     This code involves the concept of memory functions. Here we solve the subproblems which are needed
     unlike the below example
@@ -9,34 +15,129 @@ def MF_knapsack(i,wt,val,j):
     '''
     global F  # a global dp table for knapsack
     if F[i][j] < 0:
-        if j < wt[i - 1]:
+        if j < wt[i-1]:
-            val = MF_knapsack(i - 1,wt,val,j)
+            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])
+            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]
 
+
 def knapsack(W, wt, val, n):
     dp = [[0 for i in range(W+1)]for j in range(n+1)]
 
     for i in range(1,n+1):
-        for w in range(1,W+1):
+        for w in range(1, W+1):
-            if(wt[i-1]<=w):
+            if wt[i-1] <= w:
-                dp[i][w] = max(val[i-1]+dp[i-1][w-wt[i-1]],dp[i-1][w])
+                dp[i][w] = max(val[i-1] + dp[i-1][w-wt[i-1]], dp[i-1][w])
             else:
                 dp[i][w] = dp[i-1][w]
 
-    return dp[n][w]
+    return dp[n][W], dp
+
+
+def knapsack_with_example_solution(W: int, wt: list, val:list):
+    """
+    Solves the integer weights knapsack problem returns one of
+    the several possible optimal subsets.
+
+    Parameters
+    ---------
+
+    W: int, the total maximum weight for the given knapsack problem.
+    wt: list, the vector of weights for all items where wt[i] is the weight
+    of the ith item.
+    val: list, the vector of values for all items where val[i] is the value
+    of te ith item
+
+    Returns
+    -------
+    optimal_val: float, the optimal value for the given knapsack problem
+    example_optional_set: set, the indices of one of the optimal subsets
+    which gave rise to the optimal value.
+
+    Examples
+    -------
+    >>> knapsack_with_example_solution(10, [1, 3, 5, 2], [10, 20, 100, 22])
+    (142, {2, 3, 4})
+    >>> knapsack_with_example_solution(6, [4, 3, 2, 3], [3, 2, 4, 4])
+    (8, {3, 4})
+    >>> knapsack_with_example_solution(6, [4, 3, 2, 3], [3, 2, 4])
+    Traceback (most recent call last):
+        ...
+    ValueError: The number of weights must be the same as the number of values.
+    But got 4 weights and 3 values
+    """
+    if not (isinstance(wt, (list, tuple)) and isinstance(val, (list, tuple))):
+        raise ValueError("Both the weights and values vectors must be either lists or tuples")
+
+    num_items = len(wt)
+    if num_items != len(val):
+        raise ValueError("The number of weights must be the "
+                         "same as the number of values.\nBut "
+                         "got {} weights and {} values".format(num_items, len(val)))
+    for i in range(num_items):
+        if not isinstance(wt[i], int):
+            raise TypeError("All weights must be integers but "
+                            "got weight of type {} at index {}".format(type(wt[i]), i))
+
+    optimal_val, dp_table = knapsack(W, wt, val, num_items)
+    example_optional_set = set()
+    _construct_solution(dp_table, wt, num_items, W, example_optional_set)
+
+    return optimal_val, example_optional_set
+
+
+def _construct_solution(dp:list, wt:list, i:int, j:int, optimal_set: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:
+            optimal_set.add(i)
+            _construct_solution(dp, wt, i - 1, j - wt[i-1], optimal_set)
+
 
 if __name__ == '__main__':
     '''
     Adding test case for knapsack
     '''
-    val = [3,2,4,4]
+    val = [3, 2, 4, 4]
-    wt = [4,3,2,3]
+    wt = [4, 3, 2, 3]
     n = 4
     w = 6
-    F = [[0]*(w + 1)] + [[0] + [-1 for i in range(w + 1)] for j in range(n + 1)]
+    F = [[0] * (w + 1)] + [[0] + [-1 for i in range(w + 1)] for j in range(n + 1)]
-    print(knapsack(w,wt,val,n))
+    optimal_solution, _ = knapsack(w,wt,val, n)
-    print(MF_knapsack(n,wt,val,w))  # switched the n and w 
+    print(optimal_solution)
-    
+    print(MF_knapsack(n,wt,val,w))  # switched the n and w
+
+    # testing the dynamic programming problem with example
+    # the optimal subset for the above example are items 3 and 4
+    optimal_solution, optimal_subset = knapsack_with_example_solution(w, wt, val)
+    assert optimal_solution == 8
+    assert optimal_subset == {3, 4}
+    print("optimal_value = ", optimal_solution)
+    print("An optimal subset corresponding to the optimal value", optimal_subset)
+