Three-Hundred-Sixteen Commit: Amend Bitwise_XOR_Tutorial.txt in Bitwise XOR section

AAdewunmi · AAdewunmi · commit 70f695aa0d07 · 2024-07-26T11:08:04.000+01:00
diff --git a/src/Bitwise_XOR/Bitwise_XOR_Tutorial.txt b/src/Bitwise_XOR/Bitwise_XOR_Tutorial.txt
@@ -1,175 +1,218 @@
 Bitwise XOR
-===========
+============
 
 - Introduction:
+===============
 
-Bitwise XOR (exclusive OR) is a binary operation that operates on the individual bits of its operands. It is a fundamental
-operation in digital logic and computer science, commonly used in tasks like encryption, error detection, and data manipulation.
+    Bitwise XOR (exclusive OR) is a binary operation that operates on the individual bits of its operands. It is a fundamental
+    operation in digital logic and computer science, commonly used in tasks like encryption, error detection, and data manipulation.
 
-- How Bitwise XOR Works
+    - How Bitwise XOR Works:
+    ------------------------
 
-Bitwise XOR compares corresponding bits of two numbers and returns a new number where each bit is set to 1 if the corresponding
-bits of the operands are different, and 0 if they are the same.
+    Bitwise XOR compares corresponding bits of two numbers and returns a new number where each bit is set to 1 if the corresponding
+    bits of the operands are different, and 0 if they are the same.
 
-Here's a bit-by-bit comparison for clarification:
+    Here's a bit-by-bit comparison for clarification:
 
-- 0 XOR 0 = 0
-- 0 XOR 1 = 1
-- 1 XOR 0 = 1
-- 1 XOR 1 = 0
+        - 0 XOR 0 = 0
+        - 0 XOR 1 = 1
+        - 1 XOR 0 = 1
+        - 1 XOR 1 = 0
 
-- Example Calculation
+    - Example Calculation:
+    ----------------------
 
-Consider the bitwise XOR of the binary numbers `1010` and `1100`:
+    Consider the bitwise XOR of the binary numbers `1010` and `1100`:
 
-    1010
-XOR 1100
----------
-    0110
+            1010
+        XOR 1100
+        ---------
+            0110
 
-- The first bit: 1 XOR 1 = 0
-- The second bit: 0 XOR 1 = 1
-- The third bit: 1 XOR 0 = 1
-- The fourth bit: 0 XOR 0 = 0
+    - The first bit: 1 XOR 1 = 0
+    - The second bit: 0 XOR 1 = 1
+    - The third bit: 1 XOR 0 = 1
+    - The fourth bit: 0 XOR 0 = 0
 
-The result is `0110`.
+    The result is `0110`.
 
-- Properties of Bitwise XOR
+    - Properties of Bitwise XOR:
+    ----------------------------
 
-1. Commutative: `A XOR B = B XOR A`
-2. Associative: `(A XOR B) XOR C = A XOR (B XOR C)`
-3. Identity: `A XOR 0 = A`
-4. Self-inverse: `A XOR A = 0`
+        1. Commutative: `A XOR B = B XOR A`
+        2. Associative: `(A XOR B) XOR C = A XOR (B XOR C)`
+        3. Identity: `A XOR 0 = A`
+        4. Self-inverse: `A XOR A = 0`
 
-- Applications
+    - Applications:
+    ---------------
 
-1. Swapping Values: XOR can be used to swap two values without using a temporary variable:
+        1. Swapping Values: XOR can be used to swap two values without using a temporary variable:
 
-    a = a XOR b
-    b = a XOR b
-    a = a XOR b
+                a = a XOR b
+                b = a XOR b
+                a = a XOR b
 
-2. Encryption and Decryption: In some simple encryption algorithms, a key is XORed with the plaintext to produce ciphertext.
-3. Checksum and Error Detection: XOR is used in generating parity bits for error detection in data transmission.
+        2. Encryption and Decryption: In some simple encryption algorithms, a key is XORed with the plaintext to produce
+            ciphertext.
+        3. Checksum and Error Detection: XOR is used in generating parity bits for error detection in data transmission.
 
-Bitwise XOR is a versatile and efficient operation widely used in low-level programming and computer science for its
-unique properties and simplicity.
+    Bitwise XOR is a versatile and efficient operation widely used in low-level programming and computer science for its
+    unique properties and simplicity.
 
-Certainly! Here’s a high-level pseudocode for performing a Bitwise XOR (exclusive OR) operation between two integers:
+    - Implementation:
+    -----------------
 
-- Pseudocode
+    In Java, the bitwise XOR operator is represented by the caret symbol (`^`). It operates on the individual bits of
+    its operands and returns a new integer whose bits are set according to the XOR logic.
 
-Function BitwiseXOR(a, b):
-    Initialize result to 0
-    Initialize bitPosition to 0
+        * Syntax
 
-    While a is not 0 or b is not 0:
-        // Extract the least significant bit of a and b
-        bitA = a AND 1
-        bitB = b AND 1
+        result = operand1 ^ operand2;
 
-        // Perform XOR on the extracted bits
-        xorBit = bitA XOR bitB
+        * Example Usage
 
-        // Place the xorBit in the correct bit position in the result
-        result = result OR (xorBit << bitPosition)
+        Here's an example demonstrating the use of the bitwise XOR operator in Java:
 
-        // Shift a and b to the right to process the next bit
-        a = a >> 1
-        b = b >> 1
+        public class BitwiseXORExample {
+            public static void main(String[] args) {
+                int a = 10; // in binary: 1010
+                int b = 12; // in binary: 1100
 
-        // Move to the next bit position
-        bitPosition = bitPosition + 1
+                // Perform bitwise XOR
+                int result = a ^ b;
 
-    Return result
+                // Print the result
+                System.out.println("a ^ b = " + result); // Outputs 6, which is 0110 in binary
+            }
+        }
 
-- Explanation
+        * Detailed Breakdown
 
-1. Function Signature: Define a function `BitwiseXOR` that takes two integers, `a` and `b`.
-2. Initialize Variables:
-   - `result` to store the final result of the XOR operation.
-   - `bitPosition` to keep track of the current bit position being processed.
-3. Loop Until Both Numbers Are Zero: Use a `while` loop to process each bit of `a` and `b` until both numbers become zero.
-4. Extract the Least Significant Bits: Use the bitwise AND operation to extract the least significant bit of `a` and `b`.
-5. Perform XOR on the Extracted Bits: Use the XOR operation to compute the XOR of the extracted bits.
-6. Update the Result: Place the `xorBit` in the correct position in the `result` using bitwise OR and left shift.
-7. Shift `a` and `b` Right: Move to the next bit by shifting `a` and `b` to the right.
-8. Update the Bit Position: Increment the `bitPosition` to reflect the shift.
-9. Return the Result: After processing all bits, return the final `result`.
+        Let's break down the example:
 
-This pseudocode outlines the process of performing a bitwise XOR operation bit by bit. It ensures that each bit of the
-two input numbers is XORed and the result is constructed correctly.
+        - `a = 10` which is `1010` in binary.
+        - `b = 12` which is `1100` in binary.
+        - Performing `a ^ b` (bitwise XOR):
 
-- Implementation
+          1010
+        ^ 1100
+        ------
+          0110
 
-In Java, the bitwise XOR operator is represented by the caret symbol (`^`). It operates on the individual bits of its operands
-and returns a new integer whose bits are set according to the XOR logic.
+        - The result is `6`, which is `0110` in binary.
 
-- Syntax
+        * Swapping Two Numbers Using XOR
 
-result = operand1 ^ operand2;
+        One of the classic uses of the XOR operator is swapping two numbers without using a temporary variable:
 
-- Example Usage
+            public class XORSwapExample {
+                public static void main(String[] args) {
+                    int x = 5; // in binary: 0101
+                    int y = 9; // in binary: 1001
 
-Here's an example demonstrating the use of the bitwise XOR operator in Java:
+                    // Display original values
+                    System.out.println("Before swapping: x = " + x + ", y = " + y);
 
-public class BitwiseXORExample {
-    public static void main(String[] args) {
-        int a = 10; // in binary: 1010
-        int b = 12; // in binary: 1100
+                    // Swap using XOR
+                    x = x ^ y;
+                    y = x ^ y;
+                    x = x ^ y;
 
-        // Perform bitwise XOR
-        int result = a ^ b;
+                    // Display swapped values
+                    System.out.println("After swapping: x = " + x + ", y = " + y);
+                }
+            }
 
-        // Print the result
-        System.out.println("a ^ b = " + result); // Outputs 6, which is 0110 in binary
-    }
-}
+    - Practical Applications:
+    -------------------------
 
-- Detailed Breakdown
+        1. Toggle Bits: XOR can be used to toggle specific bits in a number.
+        2. Checksum Calculation: XOR is used in algorithms to compute checksums for error detection.
+        3. Encryption/Decryption: XOR is used in simple encryption algorithms where the same key is used to encrypt and
+            decrypt data.
 
-Let's break down the example:
+    - Conclusion:
+    -------------
 
-- `a = 10` which is `1010` in binary.
-- `b = 12` which is `1100` in binary.
-- Performing `a ^ b` (bitwise XOR):
+        The bitwise XOR operator in Java is a powerful tool for bit-level manipulation, enabling efficient and concise
+            operations that are fundamental in many areas of programming and computer science.
 
-  1010
-^ 1100
-------
-  0110
+- Bitwise XOR Example LeetCode Question: 1835. Find XOR Sum of All Pairs Bitwise AND
+====================================================================================
 
-- The result is `6`, which is `0110` in binary.
+    To solve the problem efficiently, let's break down the operations and utilize properties of the XOR and AND bitwise
+    operations to reduce the computational complexity.
 
-- Swapping Two Numbers Using XOR
+    * Problem Explanation:
+    ----------------------
 
-One of the classic uses of the XOR operator is swapping two numbers without using a temporary variable:
+    Given two arrays `arr1` and `arr2`, we need to consider every possible pair `(i, j)` where `0 <= i < arr1.length`
+    and `0 <= j < arr2.length`. For each pair, we calculate `arr1[i] AND arr2[j]`, and then we need to find the XOR sum
+    of all these results.
 
-public class XORSwapExample {
-    public static void main(String[] args) {
-        int x = 5; // in binary: 0101
-        int y = 9; // in binary: 1001
+    * Observations:
+    ---------------
 
-        // Display original values
-        System.out.println("Before swapping: x = " + x + ", y = " + y);
+        1. AND and XOR properties:
+           - XOR is both commutative and associative, which means the order of operations does not matter.
+           - AND operation is distributive over XOR.
 
-        // Swap using XOR
-        x = x ^ y;
-        y = x ^ y;
-        x = x ^ y;
+        2. Decomposing the problem:
+           - Instead of calculating every `arr1[i] AND arr2[j]` and then taking the XOR, we can leverage the distributive
+           property of the AND and XOR operations.
 
-        // Display swapped values
-        System.out.println("After swapping: x = " + x + ", y = " + y);
-    }
-}
+    * Efficient Calculation:
+    ------------------------
 
-- Practical Applications
+    Let's consider how the bitwise operations interact:
 
-1. Toggle Bits: XOR can be used to toggle specific bits in a number.
-2. Checksum Calculation: XOR is used in algorithms to compute checksums for error detection.
-3. Encryption/Decryption: XOR is used in simple encryption algorithms where the same key is used to encrypt and decrypt data.
+    For any bit position ( k ):
 
-- Conclusion
+        - For each number in `arr1`, determine how many numbers in `arr2` have the ( k )-th bit set and how many do not.
+        - The XOR sum for a bit position ( k ) is determined by whether the number of set bits in `arr1` and `arr2` at
+            position ( k ) are odd or even.
 
-The bitwise XOR operator in Java is a powerful tool for bit-level manipulation, enabling efficient and concise operations that
-are fundamental in many areas of programming and computer science.
+    If we denote:
+
+        - `count1_k` as the number of elements in `arr1` where the ( k )-th bit is set.
+        - `count2_k` as the number of elements in `arr2` where the ( k )-th bit is set.
+
+    The contribution to the XOR sum from the \( k \)-th bit is significant only if `count1_k * count2_k` is odd.
+    This is because an even number of `1` bits in a position will cancel out in XOR, whereas an odd number will result in a `1`.
+
+    * Implementation:
+    -----------------
+
+    Let's implement this idea in Java:
+
+        public class Solution {
+            public int getXORSum(int[] arr1, int[] arr2) {
+                int xor1 = 0, xor2 = 0;
+
+                // Compute the XOR of all elements in arr1
+                for (int num : arr1) {
+                    xor1 ^= num;
+                }
+
+                // Compute the XOR of all elements in arr2
+                for (int num : arr2) {
+                    xor2 ^= num;
+                }
+
+                // The result is the AND of the two XORs
+                return xor1 & xor2;
+            }
+        }
+
+    * Explanation:
+    --------------
+
+        1. Compute `xor1`: This is the XOR of all elements in `arr1`.
+        2. Compute `xor2`: This is the XOR of all elements in `arr2`.
+        3. Result: The XOR sum of all `arr1[i] AND arr2[j]` pairs is equivalent to `(xor1 & xor2)`.
+
+    This approach ensures that we compute the required XOR sum in O(n + m) time complexity, where ( n ) and ( m ) are the
+    lengths of `arr1` and `arr2`, respectively. This is efficient and avoids the need to explicitly compute and store all
+    possible pairs.
