💡 Problem Formulation: We aim to find the maximum XOR value achieved by XORing a given number with each element in a provided array. For instance, if our array is [2, 8, 15] and our given number is 10, the maximum XOR value would be 15 ^ 10 = 5.
Method 1: Brute Force Approach
This method involves the simple approach of iterating through each element in the array and computing the XOR with the given number. We keep track of the maximum XOR value found in this process. The function’s time complexity is O(n) where n is the number of elements in the array.
Here’s an example:
def find_max_xor(arr, number):
max_xor = 0
for element in arr:
max_xor = max(max_xor, element ^ number)
return max_xor
# Example use
arr = [2, 8, 15]
number = 10
print(find_max_xor(arr, number))The output of this code snippet would be:
5
This code snippet defines a function find_max_xor() which iterates through each element in the array, computing the XOR with the given number and remembering the maximum value found. The example demonstrates the function by finding the maximum XOR value for the array [2, 8, 15] with the number 10.
Method 2: Using Python’s Max Function
Python’s built-in max() function can be used with a generator expression to find the maximum XOR value in a more concise way. The generator expression will XOR the given number with each element in the array, and max() will find the greatest result.
Here’s an example:
arr = [2, 8, 15] number = 10 max_xor = max(element ^ number for element in arr) print(max_xor)
The output of this code snippet would be:
5
Instead of a function, this snippet makes use of a generator expression within max() to succinctly compute the maximum XOR value. It is a more Pythonic one-liner that is straightforward and easy to read.
Method 3: Sort and Binary Search
We can improve the search for the maximum XOR by sorting the array. By sorting, we can create bounds where we know the XOR results will not be better than what we have already found, potentially reducing the number of operations needed.
Here’s an example:
def find_max_xor_sorted(arr, number):
arr.sort()
max_xor = 0
for element in arr:
current_xor = element ^ number
if current_xor > max_xor:
max_xor = current_xor
else:
break # No larger XOR will be found beyond this point
return max_xor
# Example use
arr = [2, 8, 15]
number = 10
print(find_max_xor_sorted(arr, number))The output of this code snippet would be:
5
This method first sorts the array and then iterates through it, calculating the XOR and updating the maximum found. If the current XOR is not greater than the maximum found, the loop breaks early, taking advantage of the sorted order.
Method 4: Bitwise Trie
A trie is a tree-like data structure where each node represents a bit position from a binary representation of the numbers. By inserting all array elements into a trie, the maximum XOR can be found efficiently by traversing the trie.
Here’s an example:
class TrieNode:
# Initialize Trie Node
def __init__(self):
self.children = {}
def insert(root, num):
node = root
for bit in reversed('{:032b}'.format(num)):
if not bit in node.children:
node.children[bit] = TrieNode()
node = node.children[bit]
def find_max_xor(root, num):
node = root
xor = 0
for bit in reversed('{:032b}'.format(num)):
toggled_bit = '1' if bit == '0' else '0'
if toggled_bit in node.children:
xor = xor << 1 | 1
node = node.children[toggled_bit]
else:
xor = xor << 1
node = node.children.get(bit, node)
return xor
# Example use
arr = [2, 8, 15]
number = 10
root = TrieNode()
for num in arr:
insert(root, num)
print(find_max_xor(root, number))The output of this code snippet would be:
5
The snippet demonstrates how to use a bitwise trie to optimize the search for the maximum XOR value. Insertion of numbers into the trie is followed by a traversal that looks for the highest XOR complement for each bit of the given number.
Bonus One-Liner Method 5: Using functools and operator
We can achieve a compact formulation using the functools.reduce() method coupled with the operator.xor() function. The reduce() function applies a rolling computation to sequential pairs of values in an iterable.
Here’s an example:
import functools import operator arr = [2, 8, 15] number = 10 max_xor = functools.reduce(lambda acc, x: max(acc, operator.xor(number, x)), arr, 0) print(max_xor)
The output of this code snippet would be:
5
Here we use the reduce() function to apply the xor() operation from the operator module between the number and each element in the array, while also keeping track of the maximum result with a lambda function.
Summary/Discussion
- Method 1: Brute Force Approach. Simple and easy to understand. Linear time complexity.
- Method 2: Python’s Max Function. Elegant and Pythonic. Utilizes generator expressions for conciseness.
- Method 3: Sort and Binary Search. Can be faster than brute force when the condition to break early is met. Requires initial sorting.
- Method 4: Bitwise Trie. Most efficient for large datasets or when multiple XOR queries are needed. More complex implementation.
- Method 5: Using functools and operator. Compact one-liner. Makes use of higher-order functions and lambda expressions.