Problem Description

In the given problem, there is an array named arr that contains n non-negative integers which we can't see because it is hidden. This arr array has been transformed into another array named encoded with a length of n - 1. This transformation uses the bitwise XOR operation: each element of encoded is the result of XORing consecutive elements in arr, i.e., encoded[i] = arr[i] XOR arr[i + 1]. The XOR operation is a bitwise operation that outputs 1 only if the input bits are different, and 0 otherwise.

The challenge is to reconstruct the original arr array given the encoded array and the first element of arr (referred to as first or arr[0]). The description assures that there is a unique solution for arr.

Intuition

The XOR operation plays a crucial role here because it has a unique property: it's reversible if we know one of the operands. Meaning, if we have a XOR b = c, we can find a by doing c XOR b, and similarly, find b by doing c XOR a. This is because performing XOR twice with the same number cancels out the operation (e.g., a XOR b XOR b is equal to a).

This property makes it possible to recover the arr array starting from the provided first element (arr[0]). The intuition behind the solution is simply to iterate through the encoded array and apply the XOR operation between the last known value of arr and the current element in encoded to find the next value of arr. In essence, given arr[i] and encoded[i], we can solve for arr[i + 1] by calculating arr[i] XOR encoded[i].

By starting with arr[0] as first and iteratively applying the XOR operation with the encoded values, we can decode the entire arr array. The solution process unfolds one element at a time, until all values in arr are revealed.

Solution Approach

The implementation utilizes a simple for loop and the append method on lists in Python. For those unfamiliar with Python, appending to a list adds a new element to the end of the list. The algorithm works as follows:

• Initially, the known first element of arr (given as first) is appended to an empty list named ans.
• We then iterate over each element e in the encoded array.
• In each iteration, we XOR the last element of ans with e. In terms of the algorithm, if ans[-1] is the last element of the list ans, then e is XORed with ans[-1], and the result is the next element in arr, which is then appended to ans.
• This method leverages the reversible property of XOR mentioned earlier: encoded[i] = arr[i] XOR arr[i + 1] implies arr[i + 1] = encoded[i] XOR arr[i]. Since arr[i] is the last known element (initially first), we can decode arr[i + 1] using the elements of encoded.
• The loop continues until we have reconstructed the entire arr array.

The Solution class and its decode method, provided in the reference solution, are examples of the use of Python's object-oriented programming paradigm. The decode method encapsulates the aforementioned algorithm. There are no specific data structures besides the list used for the result, and no complex patterns -- it's a straight application of XOR to decode each subsequent number in the sequence.

Here's an explicit breakdown of the steps in the decode function:

• Start by creating a result list ans with the first element first.
• Iterate over the encoded array using a for loop.
• In each iteration, XOR the last element of ans with the current element in encoded and append the result to ans.
• Continue until all elements in encoded have been used.
• The list ans is now the decoded arr array, which we return from the function.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following encoded array and first element:

• encoded = [6, 1, 4]
• first = 5 (which is actually arr[0])

We know that encoded[i] is derived from arr[i] XOR arr[i + 1]. So the original array arr starts with first and has n elements, with n = len(encoded) + 1.

Now we will use the given solution approach to decode the original array arr.

Step 1: Initialize the result list with first

• ans = [5]

Step 2: XOR first with the first element of encoded

• Calculate 5 XOR 6 which equals 3
• Append 3 to ans
• ans becomes [5, 3]

Step 3: XOR the last element of ans (now 3) with the next element of encoded

• Calculate 3 XOR 1 which equals 2
• Append 2 to ans
• ans becomes [5, 3, 2]

Step 4: XOR the last element of ans (now 2) with the next element of encoded

• Calculate 2 XOR 4 which equals 6
• Append 6 to ans
• ans becomes [5, 3, 2, 6]

At this point, we've performed an XOR operation with all elements of the encoded array, and our result list now contains all elements of the original array arr.

Final Result:

The decoded original array arr is [5, 3, 2, 6].

Using the given solution approach, we are able to reconstruct arr from encoded and first. This illustrates how the reversible property of the XOR operation can be utilized to solve this type of decoding problem iteratively.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python function decode consists of a single loop that iterates through the encoded list, which has n elements, where n is the length of the encoded list. Within the loop, there is a constant time operation which performs an XOR operation (^) and appends the result to the ans list. Therefore, since each operation in the loop takes O(1) time and the loop runs for n iterations, the overall time complexity is O(n).

Space Complexity

The space complexity of the function decode is determined by the ans list which the function populates and returns. Since the ans list will contain exactly n + 1 elements after processing an encoded list of length n, the space complexity is O(n). The space required grows linearly with the input size, making the space complexity linear as well.

Learn more about how to find time and space complexity quickly using problem constraints.