Program to make the XOR of all segments equal to zero in Python

Suppose we have an array called nums and another value k. The XOR of a segment [left, right] (left <= right) is the XOR of all elements whose indices are in between left and right (inclusive).

We have to find the minimum number of elements to change in the array such that the XOR of all segments of size k is same as zero.

So, if the input is like nums = [3,4,5,2,1,7,3,4,7], k = 3, then the output will be 3 because we can modify elements at indices 2, 3, 4 to make the array [3,4,7,3,4,7,3,4,7].

To solve this, we will follow these steps −

• LIMIT := 1024

• temp := make an array whose size is LIMIT x k, and fill with 0

• for each index i and value x in nums, do

  • temp[i mod k, x] := temp[i mod k, x] + 1

• dp := an array of size LIMIT and fill with -2000

• dp[0] := 0

• for each row in temp, do

  • maxprev := maximum of dp

  • new_dp := an array of size LIMIT and fill with maxprev

  • for each index i and value cnt row, do

    • if cnt > 0, then

      • for each index j and value prev in dp, do

        • new_dp[i XOR j] := maximum of new_dp[i XOR j] and prev+cnt

  • dp := new_dp

• return size of nums - new_dp[0]

Example

Let us see the following implementation to get better understanding

def solve(nums, k):
   LIMIT = 2**10
   temp = [[0 for _ in range(LIMIT)] for _ in range(k)]
   for i,x in enumerate(nums):
      temp[i%k][x] += 1

   dp = [-2000 for _ in range(LIMIT)]
   dp[0] = 0
   for row in temp:
      maxprev = max(dp)
      new_dp = [maxprev for _ in range(LIMIT)]
      for i,cnt in enumerate(row):
         if cnt > 0:
            for j,prev in enumerate(dp):
               new_dp[i^j] = max(new_dp[i^j], prev+cnt)
         dp = new_dp
   return len(nums) - new_dp[0]

nums = [3,4,5,2,1,7,3,4,7]
k = 3
print(solve(nums, k))

Input

[3,4,5,2,1,7,3,4,7], 3

Output

-9