[mypy] Fix type annotations for dynamic programming (TheAlgorithms#4687)

* Fix mypy error for knapsack.py

* Fix mypy error for longest_increasing_subsequence

* Fix mypy error for fractional_knapsack_2.py

dynamic_programming/fractional_knapsack_2.py

@@ -7,7 +7,7 @@
 
 def fractional_knapsack(
     value: list[int], weight: list[int], capacity: int
-) -> tuple[int, list[int]]:
+) -> tuple[float, list[float]]:
     """
     >>> value = [1, 3, 5, 7, 9]
     >>> weight = [0.9, 0.7, 0.5, 0.3, 0.1]
@@ -32,8 +32,8 @@ def fractional_knapsack(
     ratio = [v / w for v, w in zip(value, weight)]
     index.sort(key=lambda i: ratio[i], reverse=True)
 
-    max_value = 0
-    fractions = [0] * len(value)
+    max_value: float = 0
+    fractions: list[float] = [0] * len(value)
     for i in index:
         if weight[i] <= capacity:
             fractions[i] = 1
@@ -48,13 +48,6 @@ def fractional_knapsack(
 
 
 if __name__ == "__main__":
-    n = int(input("Enter number of items: "))
-    value = input(f"Enter the values of the {n} item(s) in order: ").split()
-    value = [int(v) for v in value]
-    weight = input(f"Enter the positive weights of the {n} item(s) in order: ".split())
-    weight = [int(w) for w in weight]
-    capacity = int(input("Enter maximum weight: "))
-
-    max_value, fractions = fractional_knapsack(value, weight, capacity)
-    print("The maximum value of items that can be carried:", max_value)
-    print("The fractions in which the items should be taken:", fractions)
+    import doctest
+
+    doctest.testmod()

dynamic_programming/knapsack.py

@@ -91,7 +91,7 @@ def knapsack_with_example_solution(W: int, wt: list, val: list):
             )
 
     optimal_val, dp_table = knapsack(W, wt, val, num_items)
-    example_optional_set = set()
+    example_optional_set: set = set()
     _construct_solution(dp_table, wt, num_items, W, example_optional_set)
 
     return optimal_val, example_optional_set

dynamic_programming/longest_increasing_subsequence.py

@@ -36,7 +36,7 @@ def longest_subsequence(array: list[int]) -> list[int]:  # This function is recu
     pivot = array[0]
     isFound = False
     i = 1
-    longest_subseq = []
+    longest_subseq: list[int] = []
     while not isFound and i < array_length:
         if array[i] < pivot:
             isFound = True