Create beautiful-arrangement

dynamic_programming/beautiful_arrangement.py

@@ -0,0 +1,64 @@
+"""
+Suppose you have n integers labeled 1 through n. 
+A permutation of those n integers
+perm (1-indexed) is considered a 
+"beautiful arrangement" if for every i (1 <= i <= n),
+either of the following is true:
+
+-> perm[i] is divisible by i.
+-> i is divisible by perm[i].
+Given an integer n, return the number of the 
+"beautiful arrangements" that you can construct.
+
+"""
+# Solution using Backtracking
+
+
+class BeautifulArrange:
+    # funtion call; n is the size of the permutation (numbers 1..n)
+    def countarrangement(self, n: int) -> int:
+        self.count = 0
+        """
+        We initialize a counter to record how 
+            many valid arrangements we find.
+        Using self.count lets the nested 
+            function modify it without nonlocal.
+        """
+
+        used = [False] * (n + 1)
+        """
+        A boolean list to mark which numbers have
+        already been placed in the permutation.
+        """
+
+        def backtrack(pos):
+            """
+            Define the recursive backtracking function.
+            pos is the current position in the 
+                permutation we are filling (1-indexed).
+            We try to assign a number to position pos.
+            """
+            if pos > n:
+                self.count += 1
+                # We found a complete valid arrangement, so increment the total count.
+                return
+            for num in range(
+                1, n + 1
+            ):  # Try every candidate number num for the current position pos.
+                """
+                Two checks in one:
+                1. not used[num] — the number num has 
+                    not been placed yet (we can use it).
+                2. (num % pos == 0 or pos % num == 0) — 
+                    the beautiful-arrangement condition:
+                    either num divides pos or pos divides num.
+                If both are true, num is a valid choice for pos.
+
+                """
+                if not used[num] and (num % pos == 0 or pos % num == 0):
+                    used[num] = True
+                    backtrack(pos + 1)
+                    used[num] = False
+
+        backtrack(1)
+        return self.count