solve Maximum XOR of Two Numbers in an Array

Author: saubhik

problems/maximum_xor_of_two_numbers_in_an_array.py

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+from typing import List
+
+
+class Solution:
+    # XOR between two numbers is high if there is difference in the most significant
+    # bits.
+    # For each number, during iteration, we need to find if there is another number
+    # which has complementary bits starting with the most significant bit.
+    # We can use a prefix tree (AKA trie) to store the binary bits.
+    # A search in a trie is O(length of trie) = O(32) = O(1) in this case.
+    #
+    # During iteration, we search in the trie for the ideal number using the
+    # complementary bit at each step, starting from the most significant bit.
+    # So, time complexity is O(n).
+    #
+    # Space complexity is the space required to store the trie which is 32 level deep
+    # perfect binary tree in worst case. So, 2^32 - 1 units of storage. But it's O(1).
+    #
+    # Further optimisations:
+    # 1. Store bits instead of as strings in the keys of the trie.
+    # 2. Instead of storing the number as in node["END"], we can also compute the number
+    #    from the path as we traverse.
+    def findMaximumXOR(self, nums: List[int]) -> int:
+        trie = {"ROOT": {}}
+
+        node = trie["ROOT"]
+        for num in nums:
+            node = trie["ROOT"]
+            for ch in "{:032b}".format(num):
+                if not ch in node:
+                    node[ch] = {}
+                node = node[ch]
+            node["END"] = num
+
+        ans = 0
+
+        for num in nums:
+            node = trie["ROOT"]
+            for ch in "{:032b}".format(num):
+                complement = "1" if ch == "0" else "0"
+                node = node[complement] if complement in node else node[ch]
+
+            # "END" must be in node
+            ans = max(ans, num ^ node["END"])
+
+        return ans

tests/test_maximum_xor_of_two_numbers_in_an_array.py

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+import unittest
+
+from maximum_xor_of_two_numbers_in_an_array import Solution
+
+
+class TestMaximumXOROfTwoNumbersInAnArray(unittest.TestCase):
+    def test_example_1(self):
+        assert Solution().findMaximumXOR(nums=[3, 10, 5, 25, 2, 8]) == 28
+
+    def test_example_2(self):
+        assert Solution().findMaximumXOR(nums=[3, 10, 5]) == 15