Add josephus_improved_v3.py

josephus_improved_v3.py

@@ -0,0 +1,40 @@
+# The effective time complexity of this improved version is O(1).
+# For the problem statement, refer `josephus.py`
+
+def josephus_v3(soldiers):
+    # Convert to binary.
+    binary = bin(soldiers)
+    # Get the first digit and put it as last.
+    shift = '0b' + binary[3::] + binary[2:3:]
+    # Convert to decimal.
+    return int(shift, 2)
+
+winning = josephus_v3(soldiers=41)
+print(winning) #Winning Soldier: 19
+
+winning = josephus_v3(soldiers=100)
+print(winning) #Winning Soldier: 73
+
+winning = josephus_v3(soldiers=1000)
+print(winning) #Winning Soldier: 977
+
+# Testing:
+test_josephus = {
+    1: 1,
+    2: 1,
+    3: 3,
+    4: 1,
+    5: 3,
+    6: 5,
+    7: 7,
+    8: 1,
+    9: 3,
+    10: 5,
+    11: 7,
+    12: 9,
+    13: 11,
+    16: 1,
+    41: 19
+}
+for soldiers, expected_winner in test_josephus.items():
+    assert josephus_v3(soldiers=soldiers) == expected_winner