Program to recover decode XORed array in Python

Suppose we have a hidden array arr with n non-negative integers. This array is encoded into another array enc of length n-1, where enc[i] = arr[i] XOR arr[i+1]. Given the encoded array and the first element of the original array, we need to recover the original array.

So, if the input is like enc = [8,3,2,7], first = 4, then the output will be [4, 12, 15, 13, 10].

Algorithm

To solve this, we will follow these steps ?

• arr := an array with only one element first

• for i in range 0 to size of enc - 1, do

  • insert arr[i] XOR enc[i] at the end of arr

• return arr

How It Works

Since enc[i] = arr[i] XOR arr[i+1], we can use the XOR property that A XOR B = C implies A XOR C = B. Therefore, arr[i] XOR enc[i] = arr[i+1].

Example

Let us see the following implementation to get better understanding ?

def solve(enc, first):
    arr = [first]
    for i in range(0, len(enc)):
        arr.append(arr[i] ^ enc[i])
    return arr

enc = [8, 3, 2, 7]
first = 4
result = solve(enc, first)
print("Original array:", result)

The output of the above code is ?

Original array: [4, 12, 15, 13, 10]

Step-by-Step Execution

Here's how the algorithm works step by step ?

def solve_with_steps(enc, first):
    arr = [first]
    print(f"Step 0: arr = {arr}")
    
    for i in range(len(enc)):
        next_val = arr[i] ^ enc[i]
        arr.append(next_val)
        print(f"Step {i+1}: arr[{i}] ^ enc[{i}] = {arr[i]} ^ {enc[i]} = {next_val}")
        print(f"         arr = {arr}")
    
    return arr

enc = [8, 3, 2, 7]
first = 4
result = solve_with_steps(enc, first)

Step 0: arr = [4]
Step 1: arr[0] ^ enc[0] = 4 ^ 8 = 12
         arr = [4, 12]
Step 2: arr[1] ^ enc[1] = 12 ^ 3 = 15
         arr = [4, 12, 15]
Step 3: arr[2] ^ enc[2] = 15 ^ 2 = 13
         arr = [4, 12, 15, 13]
Step 4: arr[3] ^ enc[3] = 13 ^ 7 = 10
         arr = [4, 12, 15, 13, 10]

Conclusion

The XOR decoding algorithm leverages the property that XOR is its own inverse. Given the first element and encoded array, we can recover each subsequent element by XORing the previous element with the corresponding encoded value.