Asked in: Unstop
Image of the Question
All Testcases Passed ✔
Solution
def calculateFScore(features, N):
ans = 0
for i in range(N):
left = i+1
// ... rest of solution available after purchaseExplanation
```
To approach this problem, you first need to clearly understand what is being asked: you have an array representing features of cars, and you want to consider all possible non-empty combinations (subsets) of these cars. For each combination, you calculate its F-score, which is defined as the XOR of the feature values in that subset. Then, you want to find the blend score, which is the XOR of all these F-scores from every possible combination.
1. Understanding the Problem and Requirements:
- The problem involves computing XOR values for every possible non-empty subset of the array.
- Then, XOR all these subset XOR results together to get the final blend score.
- The size of the input array can be up to 10^4, and feature values up to 10^4.
- Enumerating all subsets explicitly would mean considering 2^N subsets, which is computationally impossible for large N (up to 10^4) due to exponential growth.
2. Recognizing the Challenge:
- Direct brute-force enumeration of all subsets and their XORs is not feasible.
- You need to find an algebraic or bitwise property that simplifies the computation.
- The XOR operation is associative and commutative, so it might help to explore how XOR behaves when combined over subsets.
3. Key Insights About XOR Over Subsets:
- Consider the contribution of each bit position (from 0 to the max bit position in Fi values).
- Since XOR works bitwise, you can analyze the problem bit-by-bit.
- For each bit position, count how many subsets have that bit set in their XOR result.
- The final blend score is essentially the XOR of the XORs of all subsets, so the presence of a bit in the final result depends on whether that bit appears an odd number of times across all subset XORs.
4. Breaking Down the Problem by Bits:
- Each feature value is represented as a 14-bit number (since Fi <= 10^4 < 2^14).
- For each bit position, analyze how many subsets have XOR with that bit set.
- Since XOR is a parity operation, subsets that have an odd number of elements with that bit set contribute to that bit being set in their XOR.
- However, counting subsets where the XOR has a specific bit set is non-trivial.
5. Exploring Mathematical Patterns:
- The XOR of all elements in the array (let’s call it X) influences the answer.
- Sometimes, problems like this can be simplified by examining properties of XOR over powersets.
- For example, the XOR of all subsets can relate to XOR of the entire array combined with specific combinatorial counts.
6. Using a Theoretical Result:
- There is a known property: XOR of XORs of all subsets of an array equals 0 if the array size is greater than 1.
- But the problem here is the XOR of all subsets’ XOR values combined (which is the XOR of all F-scores), so the XOR of XORs of all subsets.
- The XOR of XORs of all subsets of an array is zero when the size of the array is greater than 1. The XOR of all subsets is zero because each element appears in half of the subsets, and XORing an element even number of times cancels it out.
- But this problem asks for sum of F-scores, where F-score is the XOR of elements in the subset, and sum means XOR again over these F-scores.
- So, the final answer for an array with N > 1 is zero.
7. What If The Problem Is About XOR of All Subsets' XORs?
- If that’s the case, then for any N > 1, the result is zero.
- If N = 1, the answer is just the value itself.
- If the problem expects a sum (XOR sum) of all subset XORs, then this is the answer.
8. Clarifying the Final Output:
- If the problem wants sum as XOR of all subset XORs, then the answer is zero for N > 1.
- If the problem wants sum as addition (not XOR) of all subset XORs, then a different approach is required.
- Since the problem states "Print the sum of F-scores," it’s important to clarify if sum means XOR sum or arithmetic addition.
- Typically, XOR-related problems imply XOR sum, but if addition is required, more detailed approach is needed.
9. If the Problem Requires Addition (Sum of All Subsets' XORs):
- Calculating the sum of XORs of all subsets is a known problem with a neat solution.
- The sum of XOR of all subsets of an array equals the XOR of all elements multiplied by 2^(N-1).
- This is because each bit contributes to half of the subsets.
- So, calculate the XOR of the entire array.
- Multiply that XOR value by 2^(N-1) to get the sum of XORs of all subsets.
10. Steps to Solve in this Case:
- Compute XOR of all elements in the array.
- Calculate 2^(N-1) (be mindful of large exponentiation, use efficient power function or bit shift).
- Multiply XOR value with 2^(N-1) to get the final sum.
- This approach is efficient and runs in O(N) time.
11. Edge Cases:
- When N=1, sum of XORs is just the element itself.
- When array has zeros, they do not affect the XOR calculation.
- Large N and values should be handled efficiently using fast bit operations.
12. Summary:
- Identify whether sum means arithmetic sum or XOR sum.
- If XOR sum of subset XORs, result is zero for N > 1.
- If arithmetic sum of subset XORs, result is XOR of all elements multiplied by 2^(N-1).
- Calculate XOR of all elements.
- Compute 2^(N-1).
- Multiply and print the result.
- This avoids enumeration and leverages XOR properties and subset counting.
This approach leverages deep understanding of XOR and combinatorial mathematics, enabling an efficient solution to what otherwise would be an infeasible exponential problem.
```