The "Maximum XOR After Operations" coding problem presents you with an array of non-negative integers. You are allowed to perform a specific operation any number of times: choose an index i and replace nums[i] with nums[i] & (nums[i] ^ nums[j]) for any chosen j. The ultimate goal is to maximize the bitwise XOR sum of all elements in the array after performing these operations. This problem delves deep into the properties of bitwise operations, particularly XOR and AND, and challenges you to understand how these operations can be leveraged to achieve the largest possible XOR sum. It's a common array and bit manipulation interview question that often has a surprising and elegant solution.
This Maximum XOR After Operations interview question is frequently asked by companies like Amazon because it thoroughly tests a candidate's understanding of bitwise operations and their properties. It's not just about knowing what XOR and AND do, but how they interact and what invariants or characteristics can be maintained or achieved through the given operation. Interviewers want to see if you can deduce the implications of the operation and arrive at a solution that doesn't involve complex simulations but rather a clever observation about bit manipulation. It assesses your ability to think abstractly and find patterns in low-level data representations, a valuable skill for performance-critical systems.
The "Maximum XOR After Operations" problem has an elegant solution that primarily relies on understanding the properties of Bit Manipulation. The key insight is to observe the effect of the given operation: nums[i] = nums[i] & (nums[i] ^ nums[j]).
Let A = nums[i] and B = nums[j]. The operation is A = A & (A ^ B).
A but not in B, it remains set in A.B but not in A, it remains unset in A.A and B, it becomes unset in A.A and B, it remains unset in A.
Essentially, this operation allows you to unset any bit in nums[i] that is also set in nums[j]. This means you can "transfer" (or effectively remove) bits from nums[i] if nums[j] has those bits. The goal is to maximize the final XOR sum. The maximum possible XOR sum for a set of numbers is achieved if, for every bit position, we can make that bit appear an odd number of times if possible. A crucial property of XOR is that X ^ X = 0 and X ^ 0 = X. If we can make any set bit from any number in the array appear in our final XOR sum, the optimal solution is simply the bitwise OR of all numbers in the array. This is because the operation only allows bits to be turned OFF. If a bit is ON in any number initially, we can choose to keep it ON in that number by performing no operations on it or operations that don't affect that specific bit negatively, or transfer it to another number. The most significant bit (MSB) that is present in any number of the original array will be present in their bitwise OR sum. Therefore, the maximum XOR sum is the bitwise OR of all numbers. This makes it an O(N) array and bit manipulation interview pattern problem.Consider nums = [3, 5, 2].
Binary representations:
3 = 011_25 = 101_22 = 010_2Let's look at the properties of XOR sum of these numbers.
If the maximum XOR sum is the bitwise OR of all numbers:
3 | 5 | 2
011_2 | 101_2 | 010_2
111_2 = 7
The maximum XOR sum after operations is 7. This is achieved by taking the bitwise OR of all elements. The reason this works is that the given operation nums[i] = nums[i] & (nums[i] ^ nums[j]) can only ever turn bits OFF in nums[i]. It cannot turn a bit ON. If a bit is ON in any number in the original array, it is possible to construct a final array such that the bitwise XOR sum has that bit ON. This is a subtle yet powerful observation in this Maximum XOR After Operations coding problem.
A common mistake in the "Maximum XOR After Operations" coding problem is overthinking the operation and trying to simulate complex bit manipulations. Many candidates fail to grasp the crucial property that the operation can only unset bits, never set them. This leads to attempts at dynamic programming or complex greedy strategies that are often incorrect or too slow. Misinterpreting the bitwise AND and XOR properties is another frequent error. Forgetting that the goal is to maximize the XOR sum of the entire array, not just individual elements, can also lead to wrong conclusions about optimal strategies.
To excel in the "Maximum XOR After Operations" interview question, focus on developing a strong intuition for bitwise operations. Practice problems that involve XOR, AND, OR, and NOT, and try to deduce their properties and implications. When presented with an operation, meticulously analyze its effect on individual bits. Ask yourself: Can this operation turn a bit on? Can it turn a bit off? Does it preserve certain properties? For problems like this, the solution often lies in a profound understanding of bit manipulation and how it affects the final array state or sum. This array and bit manipulation interview pattern requires careful observation rather than complex algorithms.
| Title | Difficulty | Topics | LeetCode |
|---|---|---|---|
| Find Xor-Beauty of Array | Medium | Solve | |
| Number of Unique XOR Triplets I | Medium | Solve | |
| Total Hamming Distance | Medium | Solve | |
| Find XOR Sum of All Pairs Bitwise AND | Hard | Solve | |
| Minimum Operations to Make Array Elements Zero | Hard | Solve |