There are a few key topics that are important to understand when it comes to bit manipulation in competitive programming:

1. Bitwise operations: It is important to understand how to use the bitwise operators (e.g., &, |, ^, ~, <<, >>) to perform various operations on binary numbers.

2. Bitmasking: Bitmasking is a technique for setting, clearing, or checking individual bits in a number. It can be useful for representing a set of flags or for implementing certain algorithms efficiently.

3. Bit manipulation tricks: There are a number of "tricks" that involve using bit manipulation to solve problems in clever ways. For example, you can use the x & (-x) idiom to isolate the rightmost 1-bit in a number, or you can use the x & (x - 1) idiom to clear the rightmost 1-bit in a number.

4. Bit manipulation-based algorithms: There are a number of algorithms that make use of bit manipulation to solve problems efficiently. For example, you can use the divide and conquer approach to implement a fast algorithm for finding the median of a set of numbers, or you can use bit manipulation to implement a fast algorithm for multiplying two numbers.

Overall, it is important to have a strong foundation in the basics of bit manipulation and to be familiar with a variety of techniques and algorithms that make use of bit manipulation. Practice is also key to mastering these concepts, so it is a good idea to solve as many problems as you can that involve bit manipulation.

But don't worry here I have covered all topics in detail to "Become a Bit Manipulation Pro"

Bitwise operations:

Bitwise operations allow you to perform operations on binary numbers at the bit level. In Python, you can use the following bitwise operators:

• &: Bitwise AND. This operator compares each bit of the first operand to the corresponding bit of the second operand, and if both bits are 1, the corresponding result bit is set to 1. Otherwise, the corresponding result bit is set to 0. For example:

x = 0b10101010  # The number 170 in binaryy = 0b01010101  # The number 85 in binaryz = x & y  # The result is 0b00000000, or 0 in decimal

• |: Bitwise OR. This operator compares each bit of the first operand to the corresponding bit of the second operand, and if either bit is 1, the corresponding result bit is set to 1. Otherwise, the corresponding result bit is set to 0. For example:

x = 0b10101010  # The number 170 in binaryy = 0b01010101  # The number 85 in binaryz = x | y  # The result is 0b11111111, or 255 in decimal

• ^: Bitwise XOR. This operator compares each bit of the first operand to the corresponding bit of the second operand, and if the bits are different, the corresponding result bit is set to 1. Otherwise, the corresponding result bit is set to 0. For example:

x = 0b10101010  # The number 170 in binaryy = 0b01010101  # The number 85 in binaryz = x ^ y  # The result is 0b11111111, or 255 in decimal

• ~: Bitwise NOT. This operator flips all the bits of the operand. For example:

x = 0b10101010  # The number 170 in binaryy = ~x  # The result is -171 in decimal

• <<: Left shift. This operator shifts the bits of the operand to the left by the number of positions specified by the second operand. For example:

x = 0b10101010  # The number 170 in binaryy = x << 2  # The result is 0b1010101000, or 680 in decimal

• >>: Right shift. This operator shifts the bits of the operand to the right by the number of positions specified by the second operand. For example:

x = 0b10101010  # The number 170 in binaryy = x >> 2  # The result is 0b00101010, or 42 in decimal

Bitmasking:

Bitmasking is a technique for setting, clearing, or checking individual bits in a number. It involves using bitwise operators to manipulate the bits of a number in order to set or clear specific bits or to check the values of specific bits.

For example, suppose you have a number x and you want to set the third and fifth bits (counting from the right) to 1. You can do this using the following code:

x = 0b00000000  # The number 0 in binarymask = 0b00101000  # The mask with the third and fifth bits set to 1x = x | mask  # The result is 0b00101000, or 40 in decimal

Alternatively, suppose you have a number x and you want to clear the fourth and sixth bits (counting from the right). You can do this using the following code:

x = 0b11111111  # The number 255 in binarymask = 0b11010111  # The mask with the fourth and sixth bits set to 0x = x & mask  # The result is 0b11010111, or 215 in decimal

Bitmasking can also be used to check the values of specific bits in a number. For example, suppose you have a number x and you want to check if the seventh bit (counting from the right) is set to 1. You can do this using the following code:

x = 0b11010111  # The number 215 in binarymask = 0b10000000  # The mask with the seventh bit set to 1if x & mask:    print("The seventh bit is set to 1")else:    print("The seventh bit is not set to 1")

Bitmasking is often used in programming to represent a set of flags or to implement certain algorithms efficiently. For example, you might use bitmasking to represent a set of permissions (e.g., read, write, execute) or to implement a fast algorithm for finding the intersection of two sets.

Suppose you want to represent a set of flags that indicate whether a user has read, write, and execute permissions for a particular file. You could define the following constants:

READ_PERMISSION = 0b001WRITE_PERMISSION = 0b010EXECUTE_PERMISSION = 0b100

These constants represent the flags for each of the permissions, with the rightmost bit representing the read permission, the second rightmost bit representing the write permission, and the third rightmost bit representing the execute permission.

To represent a set of permissions for a particular user, you could use a variable of type int and set the appropriate bits using bitmasking. For example:

permissions = 0b000  # No permissions# Grant the user read permissionpermissions = permissions | READ_PERMISSION  # The result is 0b001# Grant the user write permissionpermissions = permissions | WRITE_PERMISSION  # The result is 0b011# Grant the user execute permissionpermissions = permissions | EXECUTE_PERMISSION  # The result is 0b111

To check whether a user has a particular permission, you can use bitmasking to check the value of the corresponding bit. For example:

if permissions & READ_PERMISSION:    print("The user has read permission")if permissions & WRITE_PERMISSION:    print("The user has write permission")if permissions & EXECUTE_PERMISSION:    print("The user has execute permission")

Bit manipulation tricks:

Bit manipulation tricks are techniques that involve using bit manipulation to solve problems in clever ways. Here are a few examples of common bit manipulation tricks:

• Isolating the rightmost 1-bit: You can use the x & (-x) idiom to isolate the rightmost 1-bit in a number. For example:

x = 0b01010101  # The number 85 in binaryy = x & (-x)  # The result is 0b00000001, or 1 in decimal

• Clearing the rightmost 1-bit: You can use the x & (x - 1) idiom to clear the rightmost 1-bit in a number. For example:

x = 0b01010101  # The number 85 in binaryy = x & (x - 1)  # The result is 0b01010100, or 84 in decimal

• Checking if a number is a power of 2: You can use the x & (x - 1) == 0 idiom to check if a number is a power of 2. For example:

x = 8  # The number 8 is a power of 2if x & (x - 1) == 0:    print("x is a power of 2")x = 9  # The number 9 is not a power of 2if x & (x - 1) == 0:    print("x is a power of 2")  # This line will not be executed

• Counting the number of 1-bits: You can use the x &= (x - 1) idiom to count the number of 1-bits in a number. For example:

x = 0b01010101  # The number 85 in binarycount = 0while x:    count += 1    x &= (x - 1)print(count)  # The output is 4

• Swapping the values of two variables: You can use bit manipulation to swap the values of two variables without using a temporary variable. For example:

x = 10y = 20# Swap the values of x and yx ^= yy ^= xx ^= yprint(x)  # The output is 20print(y)  # The output is 10

• Checking if a number is odd or even: You can use the x & 1 idiom to check if a number is odd or even. For example:

x = 10  # The number 10 is evenif x & 1:    print("x is odd")else:    print("x is even")x = 11  # The number 11 is oddif x & 1:    print("x is odd")else:    print("x is even")

• Determining the sign of a number: You can use the x >> 31 idiom to determine the sign of a number. For example:

x = 10  # The number 10 is positiveif x >> 31:    print("x is negative")else:    print("x is positive")x = -10  # The number -10 is negativeif x >> 31:    print("x is negative")else:    print("x is positive")

• Calculating the absolute value of a number: You can use the (x + (x >> 31)) ^ (x >> 31) idiom to calculate the absolute value of a number. For example:

x = 10  # The absolute value of 10 is 10y = (x + (x >> 31)) ^ (x >> 31)print(y)  # The output is 10x = -10  # The absolute value of -10 is 10y = (x + (x >> 31)) ^ (x >> 31)print(y)  # The output is 10

• Reversing the bits of a number: You can use bit manipulation to reverse the bits of a number. For example:

x = 0b01010101  # The number 85 in binaryy = 0# Reverse the bits of xfor i in range(8):    y = (y << 1) | (x & 1)    x >>= 1print(y)  # The output is 0b10101010, or 170 in decimal

• Calculating the logarithm of a number: You can use bit manipulation to calculate the logarithm of a number. For example:

x = 256  # The logarithm of 256 is 8y = 0while x > 1:    y += 1    x >>= 1print(y)  # The output is 8

• Calculating the square root of a number: You can use bit manipulation to calculate the square root of a number. For example:

x = 256  # The square root of 256 is 16y = 0while x > 0:    y += 1    x >>= 2print(y)  # The output is 16

• Calculating the greatest common divisor of two numbers: You can use bit manipulation to calculate the greatest common divisor (GCD) of two numbers. For example:

def gcd(x, y):    while y > 0:        x, y = y, x & y    return

• Calculating the least common multiple of two numbers: You can use bit manipulation to calculate the least common multiple (LCM) of two numbers. For example:

def lcm(x, y):    return (x * y) // gcd(x, y)  # gcd() is the function that calculates the GCDx = 10y = 15z = lcm(x, y)  # The result is 30

• Calculating the parity of a number: You can use bit manipulation to calculate the parity of a number (i.e., whether the number has an even or odd number of 1-bits). For example:

x = 0b01010101  # The number 85 in binaryy = 0# Calculate the parity of xwhile x:    y ^= 1    x &= (x - 1)if y:    print("The number has an odd number of 1-bits")else:    print("The number has an even number of 1-bits")

• Calculating the Hamming distance between two numbers: You can use bit manipulation to calculate the Hamming distance between two numbers (i.e., the number of bit positions in which the numbers differ). For example:

x = 0b01010101  # The number 85 in binaryy = 0b10101010  # The number 170 in binaryz = 0# Calculate the Hamming distance between x and ywhile x or y:    z += (x & 1) ^ (y & 1)    x >>= 1    y >>= 1print(z)  # The output is 8

Bit manipulation-based algorithms:

Certainly! Bit manipulation can be used to implement a variety of algorithms efficiently. Here are a few examples of algorithms that can be implemented using bit manipulation:

• Bitonic sort: Bitonic sort is an efficient sorting algorithm that can be implemented using bit manipulation. It works by dividing the input array into pairs of elements and comparing them using bit manipulation. The algorithm then recursively sorts the resulting arrays using the same process.

• Bitwise trie: A bitwise trie is a data structure that can be used to store and retrieve data efficiently. It works by using bit manipulation to store the data in a tree-like structure, with each bit of the data serving as a branch in the tree.

• Bitmap: A bitmap is a data structure that can be used to store a large number of boolean values efficiently. It works by using bit manipulation to store the values in an array of bits, with each bit representing a boolean value.

• Hash table: A hash table is a data structure that can be used to store and retrieve data efficiently. It works by using bit manipulation to compute the hash value of the data, which is then used to index into an array of buckets.

These are just a few examples of algorithms that can be implemented using bit manipulation. There are many other algorithms that can be implemented efficiently using bit manipulation, including algorithms for searching, sorting, and data compression. But are example code so you all can understand effectively.

Bitonic sort:

def bitonic_sort(arr):    # Base case: return the array if it has 1 or 0 elements    if len(arr) <= 1:        return arr    # Divide the array into two halves    mid = len(arr) // 2    left = arr[:mid]    right = arr[mid:]    # Recursively sort the two halves    left = bitonic_sort(left)    right = bitonic_sort(right)    # Compare the elements in the two halves using bit manipulation    for i in range(mid):        if (left[i] > right[i]) == (i & 1):            left[i], right[i] = right[i], left[i]    # Concatenate the sorted halves and return the result    return left + rightarr = [3, 7, 4, 8, 6, 2, 1, 5]sorted_arr = bitonic_sort(arr)print(sorted_arr)  # The output is [1, 2, 3, 4, 5, 6, 7, 8]

Bitwise trie:

class BitwiseTrie:    def __init__(self):        self.children = [None, None]    def insert(self, num):        node = self        for i in range(31, -1, -1):            bit = (num >> i) & 1            if not node.children[bit]:                node.children[bit] = BitwiseTrie()            node = node.children[bit]trie = BitwiseTrie()trie.insert(85)  # Insert the number 85 (0b01010101) into the trietrie.insert(170)  # Insert the number 170 (0b10101010) into the trie

Bitmap:

class Bitmap:    def __init__(self, size):        self.bits = [0] * ((size + 31) // 32)    def set_bit(self, pos):        self.bits[pos // 32] |= (1 << (pos % 32))    def clear_bit(self, pos):        self.bits[pos // 32] &= ~(1 << (pos % 32))    def get_bit(self, pos):        return self.bits[pos // 32] & (1 << (pos % 32))bitmap = Bitmap(100)  # Create a bitmap with 100 bitsbitmap.set_bit(10)  # Set the 11th bit (counting from the right) to 1bitmap.clear_bit(10)  # Clear the 11th bit (counting from the right)if bitmap.get_bit(10):    print("The 11th bit (counting from the right) is set to 1")else:    print("The 11th bit (counting from the right) is not set to 1")

Hash table:

class HashTable:    def __init__(self, size):        self.size = size        self.buckets = [None] * size    def hash_function(self, key):        # Compute the hash value using bit manipulation        hash_value = 0        for c in key:            hash_value = (hash_value << 5) + hash_value + ord(c)        return hash_value % self.size    def insert(self, key, value):        # Compute the hash value of the key        hash_value = self.hash_function(key)        # Insert the key-value pair into the appropriate bucket        self.buckets[hash_value] = (key, value)    def lookup(self, key):        # Compute the hash value of the key        hash_value = self.hash_function(key)        # Look up the key-value pair in the appropriate bucket        return self.buckets[hash_value]hash_table = HashTable(100)  # Create a hash table with 100 bucketshash_table.insert("apple", "red")  # Insert the key-value pair "apple"-"red"hash_table.insert("banana", "yellow")  # Insert the key-value pair "banana"-"yellow"print(hash_table.lookup("apple"))  # The output is ("apple", "red")print(hash_table.lookup("banana"))  # The output is ("banana", "yellow")

"In conclusion, bit manipulation is a powerful tool that can help programmers solve problems efficiently and effectively. Whether you are competing in programming contests or working on real-world projects, a strong understanding of bit manipulation can give you a competitive edge.

To master bit manipulation, it is important to practice using it to solve problems. You can find many online resources, such as online judges and programming contests, that offer challenges that can help you hone your skills. You can also create your own problems to solve, or explore more advanced techniques such as bit manipulation-based algorithms and data structures.

There are many resources available for learning about bit manipulation, including books, online tutorials, and blogs. Take the time to explore these resources and continue learning about this important topic. With dedication and practice, you can become a bit manipulation pro and unlock the full potential of this powerful tool.

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