2527. Find Xor-Beauty of Array
Problem Description
This LeetCode problem provides us with an integer array called nums
which is 0-indexed. It introduces a concept of effective value for any triplet of indices in the array (i, j, k)
. The effective value for these indices is calculated as ((nums[i] | nums[j]) & nums[k])
where |
represents the bitwise OR operation and &
represents the bitwise AND operation. The xor-beauty of the array is the result of performing a bitwise XOR operation on all the effective values of all possible triplets of indices within the array. The requirement is to calculate and return this xor-beauty value.
Important things to note:
0 <= i, j, k < n
indicates that each of the indices must be within the bounds of the array.- The bitwise OR (
|
) takes two bit patterns and performs the logical inclusive OR operation on each pair of corresponding bits. - The bitwise AND (
&
) takes two bit patterns and performs the logical AND operation on every pair of corresponding bits.
Intuition
When approaching this problem, one might initially consider iterating through all possible triplets of indices to compute each effective value and then apply the bitwise XOR operation across these values. However, the provided solution suggests a different approach, which implies that there is an observation or property in the bitwise operations that simplifies the problem significantly.
The intuition behind the solution given is that the xor-beauty of the array obtained by computing the effective value for all triplets (i, j, k)
can be simplified to just perform the bitwise XOR of all elements in the input array nums
.
This simplified solution suggests that there must be a property of bitwise OR, AND, and XOR that makes every other value except the numbers themselves cancel out when considering all possible combinations of triplets. Essentially, it seems that the problem boils down to xoring each number with itself for all positions i
, j
, and k
, which results in the original value.
Thus, the solution approach is to take advantage of this property and uses Python's reduce()
function combined with the xor
operator from the operator
module to iteratively apply the xor operation on all elements in the array, leading to the xor-beauty directly without explicitly calculating the effective values for all triplets.
Learn more about Math patterns.
Solution Approach
The provided Python code for this problem uses a functional programming style. It employs the use of the reduce
function from Python's functools
module. The reduce
function applies a given function cumulatively to the items of a sequence, from left to right, so as to reduce the sequence to a single value.
The xor
operator used here is a binary function that performs the "exclusive or" operation, which is imported from Python's built-in operator
module. This operation takes two bits and returns 1
only if exactly one of the bits is 1
.
Knowing that the reduce
function will apply the xor
operator to all elements in the sequence, here is what happens in the given solution:
- The
reduce
function initializes the reduction with the first element of thenums
array. - It then iteratively applies the
xor
operator to the current result and the next element in thenums
array. - This operation continues for all elements in the
nums
array until it has been reduced to a single value.
The final result after this process is the xor-beauty of the array, which is the cumulative result of xoring all the numbers together.
The use of reduce
and xor
indicates no specific need for handling each element position or its relationship with other elements; no iteration over triples or pairs is performed, and no auxiliary data structures are required. This points to an underlying pattern or property within the bitwise operations that allows for this simplified approach.
The code snippet that implements this solution is as follows:
from functools import reduce
from operator import xor
class Solution:
def xorBeauty(self, nums: List[int]) -> int:
return reduce(xor, nums)
Here, the xorBeauty
method is defined inside the Solution class and takes the nums
array as input. It simply returns the xor-beauty, computed by reducing the 'nums' array using bitwise XOR operation.
The key to understanding why this works lies in the properties of bitwise operations, specifically the XOR operation, which include the fact that xoring a number with itself results in zero, and xoring any number with zero results in the number itself. These properties seem to eliminate the need to calculate every single triplet's effective value.
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Start EvaluatorExample Walkthrough
Let's illustrate the solution approach with a small example using an integer array nums
. Imagine that nums
is given to us as [3, 5, 7]
.
Firstly, to follow the problem's statement directly without the optimized solution, we would potentially look at all possible triplets and calculate their effective values, which would be:
- For triplet (i=0, j=0, k=0), the effective value is
((nums[0] | nums[0]) & nums[0])
which equals(3 | 3) & 3
so3 & 3
which results in3
. - For triplet (i=0, j=0, k=1), the effective value is
((nums[0] | nums[0]) & nums[1])
which equals3 & 5
resulting in1
. - This process would continue for all possible triplets
(i, j, k)
wherei, j, k
range from0
to2
.
Considering there are 3
elements, we have 3^3
or 27
such triplet combinations. We would then run the bitwise XOR operation on all these 27
effective values to get the xor-beauty.
But according to our given solution, we don't need to do all that work. Let's apply the simplified solution on our example array [3, 5, 7]
by performing a cumulative XOR operation on all the elements:
- We start with the first element, which is
3
. - Next, we apply the
xor
operator with the second element:3 xor 5
which gives us6
. - Lastly, we
xor
this result with the third element:6 xor 7
which gives us1
.
So, using the simplified solution, the xor-beauty of the array [3, 5, 7]
is calculated to be 1
.
The process used in the simplified solution involves no explicit combinatorial effective value calculation and animates the bitwise operations' commutative and associative properties to achieve the same result more efficiently.
Here is how the aforementioned Python code snippet from the solution approach would execute this example:
from functools import reduce
from operator import xor
nums = [3, 5, 7]
xor_beauty = reduce(xor, nums)
print(f"The xor-beauty of the array {nums} is: {xor_beauty}")
The output of this code snippet will be: "The xor-beauty of the array [3, 5, 7] is: 1".
Solution Implementation
1from functools import reduce # Import the 'reduce' function from 'functools' module
2from operator import xor # Import 'xor' function from 'operator' module
3from typing import List # Import 'List' type from 'typing' module for type hinting
4
5class Solution:
6 def xor_beauty(self, nums: List[int]) -> int:
7 # Perform a pairwise XOR operation on all elements in `nums`
8 # The `reduce` function applies the `xor` operation cumulatively to the items of `nums`,
9 # from left to right, so as to reduce the iterable `nums` to a single value.
10 # Here `xor` is used as the binary operation function.
11 return reduce(xor, nums)
12
13# Example usage:
14# sol = Solution()
15# result = sol.xor_beauty([1, 2, 3, 4])
16# print(result) # This would output the cumulative XOR of the array elements
17
1class Solution {
2 // Method to calculate the XOR beauty of an array
3 public int xorBeauty(int[] nums) {
4 // Initialize result variable to hold the cumulative XOR
5 int result = 0;
6
7 // Iterate through all elements in the nums array
8 for (int number : nums) {
9 // XOR the current number with the result so far
10 result ^= number;
11 }
12
13 // Return the final XOR result which represents the XOR beauty
14 return result;
15 }
16}
17
1#include <vector>
2
3class Solution {
4public:
5 // Function to calculate the "xorBeauty" of given numbers.
6 int xorBeauty(vector<int>& nums) {
7 int result = 0; // Initialize the result to zero.
8
9 // Iterate over each number in the given vector.
10 for (const int& number : nums) {
11 result ^= number; // Compute the cumulative XOR of the numbers, storing in result.
12 }
13
14 // The result is the XOR of all elements in nums.
15 return result;
16 }
17};
18
1// Function that calculates the XOR of all elements in an array.
2// The XOR of a single number with itself is 0, and the XOR with 0 is the number itself.
3// Hence, the XOR of all numbers in the array gives us the "beauty" of the array.
4// @param {number[]} nums - An array of numbers to find the XOR "beauty" of.
5// @returns {number} - The result of XOR operation on all the elements of the array.
6function xorBeauty(nums: number[]): number {
7 // The reduce() method applies a function against an accumulator
8 // and each element in the array (from left to right) to reduce it to a single value.
9 // Here, it starts with an initial value of 0 and applies XOR operation to all elements.
10 return nums.reduce(
11 (accumulator: number, current: number) => accumulator ^ current, // XOR operation
12 0 // Initial value for the accumulator
13 );
14}
15
Time and Space Complexity
Time Complexity
The time complexity of the given code is O(n)
, where n
is the length of the nums
list. This is because reduce
applies the xor
operation sequentially to the elements of the list, processing each element exactly once.
Space Complexity
The space complexity of the given code is O(1)
. The reduce
function uses a constant amount of space to apply the xor
operation since it only needs to maintain the cumulative xor result and does not require any additional space that depends on the input size.
Learn more about how to find time and space complexity quickly using problem constraints.
In a binary min heap, the maximum element can be found in:
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