add solution to pe169

Author: KerimovEmil

solutions/PE169.py

@@ -0,0 +1,76 @@
+"""
+PROBLEM
+
+Define f(0) = 1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using
+each power no more than twice.
+
+For example, f(10) = 5 since there are five different ways to express 10:
+1+1+8
+1+1+4+4
+1+1+2+2+4
+2+4+4
+2+8
+
+What is f(10^25)?
+
+ANSWER: 178653872807
+Solve time ~0.001 seconds
+"""
+
+# 10 = 8 + 2 = 2^3 + 2^1 = 1010 in binary
+# 1010 = 0210 = 1002 = 0202 = 0122
+# Therefore 5 ways to represent 10 using powers of 2 with no more than 2 of each power
+# idea: 1000 -> 0200 -> 0120 -> 0111,  etc
+# key idea is the (# zeros after 1) when in binary is what matters to count the number of options
+# however we can't simply take the product of all the (# zeros after 1) + 1 since that undercounts the following case:
+# 1010 can become 0210, 1002, 0202, but also 0122, so to count this extra case when we remove the leading one,
+# this is as if we added some cases in which the number of consecutive zeros grows
+
+# e.g. 18
+# - 10010
+# - 10002
+# - 02010
+# - 01210
+# - 02002
+# - 01202
+# - 01122
+from util.utils import timeit
+import unittest
+
+
+class Problem169:
+    def __init__(self):
+        pass
+
+    @timeit
+    def solve(self, n):
+        """f_i = f_{i-1} + zeros[i] * sum_{k=1 to i-1} f_k"""
+        n_binary_str = bin(n)[2:]
+        ls_len_consecutive_zeros = [len(x) for x in n_binary_str.split('1')][1:]  # first one is always 0
+        result, rolling_sum = 1, 1
+        for len_consecutive_zeros in ls_len_consecutive_zeros:
+            result += len_consecutive_zeros * rolling_sum
+            rolling_sum += result
+        return result
+
+
+class Solution169(unittest.TestCase):
+    def setUp(self):
+        self.problem = Problem169()
+
+    def test_sample_solution_1(self):
+        self.assertEqual(5, self.problem.solve(n=10))
+
+    def test_sample_solution_2(self):
+        self.assertEqual(7, self.problem.solve(n=18))
+
+    def test_sample_solution_3(self):
+        self.assertEqual(13, self.problem.solve(n=42))
+
+    def test_solution(self):
+        self.assertEqual(178653872807, self.problem.solve(n=pow(10,25)))
+
+
+if __name__ == '__main__':
+    unittest.main()
+