Create combination_sum.py (#7403)

* Create combination_sum.py

* Update DIRECTORY.md

* Adds doctests

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update combination_sum.py

* Update combination_sum.py

Co-authored-by: Christian Clauss <cclauss@me.com>

DIRECTORY.md

@@ -23,6 +23,7 @@
   * [All Permutations](backtracking/all_permutations.py)
   * [All Subsequences](backtracking/all_subsequences.py)
   * [Coloring](backtracking/coloring.py)
+  * [Combination Sum](backtracking/combination_sum.py)
   * [Hamiltonian Cycle](backtracking/hamiltonian_cycle.py)
   * [Knight Tour](backtracking/knight_tour.py)
   * [Minimax](backtracking/minimax.py)

backtracking/combination_sum.py

@@ -0,0 +1,66 @@
+"""
+In the Combination Sum problem, we are given a list consisting of distinct integers.
+We need to find all the combinations whose sum equals to target given.
+We can use an element more than one.
+
+Time complexity(Average Case): O(n!)
+
+Constraints:
+1 <= candidates.length <= 30
+2 <= candidates[i] <= 40
+All elements of candidates are distinct.
+1 <= target <= 40
+"""
+
+
+def backtrack(
+    candidates: list, path: list, answer: list, target: int, previous_index: int
+) -> None:
+    """
+    A recursive function that searches for possible combinations. Backtracks in case
+    of a bigger current combination value than the target value.
+
+    Parameters
+    ----------
+    previous_index: Last index from the previous search
+    target: The value we need to obtain by summing our integers in the path list.
+    answer: A list of possible combinations
+    path: Current combination
+    candidates: A list of integers we can use.
+    """
+    if target == 0:
+        answer.append(path.copy())
+    else:
+        for index in range(previous_index, len(candidates)):
+            if target >= candidates[index]:
+                path.append(candidates[index])
+                backtrack(candidates, path, answer, target - candidates[index], index)
+                path.pop(len(path) - 1)
+
+
+def combination_sum(candidates: list, target: int) -> list:
+    """
+    >>> combination_sum([2, 3, 5], 8)
+    [[2, 2, 2, 2], [2, 3, 3], [3, 5]]
+    >>> combination_sum([2, 3, 6, 7], 7)
+    [[2, 2, 3], [7]]
+    >>> combination_sum([-8, 2.3, 0], 1)
+    Traceback (most recent call last):
+        ...
+    RecursionError: maximum recursion depth exceeded in comparison
+    """
+    path = []  # type: list[int]
+    answer = []  # type: list[int]
+    backtrack(candidates, path, answer, target, 0)
+    return answer
+
+
+def main() -> None:
+    print(combination_sum([-8, 2.3, 0], 1))
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    main()