Solve InterviewCake's highest-product-of-3

Write a function that returns the highest product of three integers within a
list of integers. This solution uses a greedy algorithm that solves for the
answer in linear time. The space complexity is constant.

scratch/deepmind/part_two/highest-product-of-3.py

@@ -0,0 +1,81 @@
+import unittest
+import sys
+import trace
+
+
+def highest_product_of_3(xs):
+    if len(xs) < 3:
+        raise Exception("List needs to contain at least three elements.")
+    hp3 = xs[0] * xs[1] * xs[2]
+    hp2 = xs[0] * xs[1]
+    lp2 = xs[0] * xs[1]
+    hn = max(xs[0], xs[1])
+    ln = min(xs[0], xs[1])
+    for x in xs[2:]:
+        hp3 = max(hp3, hp2 * x, lp2 * x)
+        hp2 = max(hp2, hn * x, ln * x)
+        lp2 = min(lp2, hn * x, ln * x)
+        hn = max(hn, x)
+        ln = min(ln, x)
+    return hp3
+
+
+# Tests
+class Test(unittest.TestCase):
+    def test_short_list(self):
+        actual = highest_product_of_3([1, 2, 3, 4])
+        expected = 24
+        self.assertEqual(actual, expected)
+
+    def test_longer_list(self):
+        actual = highest_product_of_3([6, 1, 3, 5, 7, 8, 2])
+        expected = 336
+        self.assertEqual(actual, expected)
+
+    def test_list_has_one_negative(self):
+        actual = highest_product_of_3([-5, 4, 8, 2, 3])
+        expected = 96
+        self.assertEqual(actual, expected)
+
+    def test_list_has_two_negatives(self):
+        actual = highest_product_of_3([-10, 1, 3, 2, -10])
+        expected = 300
+        self.assertEqual(actual, expected)
+
+    def test_list_is_all_negatives(self):
+        actual = highest_product_of_3([-5, -1, -3, -2])
+        expected = -6
+        self.assertEqual(actual, expected)
+
+    def test_error_with_empty_list(self):
+        with self.assertRaises(Exception):
+            highest_product_of_3([])
+
+    def test_error_with_one_number(self):
+        with self.assertRaises(Exception):
+            highest_product_of_3([1])
+
+    def test_error_with_two_numbers(self):
+        with self.assertRaises(Exception):
+            highest_product_of_3([1, 1])
+
+    def test_custom(self):
+        actual = highest_product_of_3([9, 5, 2, 1, 7, 3])
+        expected = 9 * 7 * 5
+        self.assertEqual(actual, expected)
+
+
+unittest.main(verbosity=2)
+
+
+def main():
+    highest_product_of_3([-5, -1, -3, -2])
+
+
+tracer = trace.Trace(ignoredirs=[sys.prefix, sys.exec_prefix],
+                     trace=0,
+                     count=1)
+
+tracer.run('main()')
+r = tracer.results()
+r.write_results(show_missing=True, coverdir=".")