Introduction:
Solving the task of determining the initial number by toggling K distinct bits presents an intriguing challenge in C++. This scenario revolves around deciphering the binary representation of a number and altering specific bits within it. In the realm of digital computing, integers are represented in binary form, consisting of binary digits 0s and 1s. Flipping a bit involves toggling from 0 to 1 or 1 to 0, with the objective of transforming the binary encoding. The core issue lies in deducing the original number after executing a specified number of bit flip operations.
Suppose you have three values denoted as X, Y, and Z, believed to be derived from an unknown base number by altering K distinct bits. The task at hand is to decipher the mystery and unveil the original number. This puzzle is tackled through C++ coding, a strong and extensively employed programming language renowned for tackling computational enigmas.
The process involves utilizing bitwise operations, which allow for the manipulation of individual bits found in the binary representation of a number. The procedure checks the positions of the altered bits in the binary representations of two specified numbers. By analyzing the connections between these numbers, it becomes possible to identify the shared patterns of the modified positions and subsequently reconstruct the initial number.
This challenge holds importance beyond just a computational enigma. It holds practical relevance across various fields such as computer science and cryptography, where the comprehension and manipulation of binary formats are essential. The resolution involves a structured method of analytical reasoning and the application of programming principles to navigate the realm of binary data.
Program:
#include <iostream>
#include <unordered_set>
// Helper function to find the flipped positions between
// two numbers
std::unordered_set<int> helper(int X, int Y) {
std::unordered_set<int> flippedPositions;
int ans = X ^ Y;
// Find the positions where bits are flipped
for (int i = 0; i < 32; i++) {
if (ans & (1 << i)) {
flippedPositions.insert(i);
}
}
return flippedPositions;
}
int findOriginalNumber(int X, int Y, int Z, int K) {
// Use helper function to find the flipped positions
// in the binary representation of x and y
std::unordered_set<int> A = helper(X, Y);
// Use helper function to find the flipped positions
// in the binary representation of x and z
std::unordered_set<int> B = helper(X, Z);
// Find the common flipped positions in both sets
std::unordered_set<int> commonPositions;
for (int position : A) {
if (B.count(position)) {
commonPositions.insert(position);
}
}
// Flip the bits at the common positions to
// reconstruct the original number
for (int p : commonPositions) {
int t = 1 << p;
X = X ^ t;
}
return X;
}
int main() {
std::cout << findOriginalNumber(9, 17, 29, 1) << std::endl;
return 0;
}
Output:
Explanation:
- Helper Function:
The auxiliary function accepts two integer parameters, X and Y, and aims to identify the positions where the binary digits of X and Y differ. By using the bitwise XOR (^) operation, it calculates a value (ans) that indicates the specific bit locations where the two numbers have contrasting bits.
For every bit that is enabled (1) in the variable "ans," the function iterates over the 32-bit variable and includes the position in the set named "flippedPositions."
- locateOriginalValue Method:
The findOriginalNumber function takes in three integer variables (X, Y, and Z) and an integer K as arguments. It utilizes a helper function to identify the positions where bits are reversed in X and Y (referred to as set A) and in X and Z (referred to as set B). It iterates through the overlapping positions, which are locations where bits are changed in both sets A and B, and stores them in a set named commonPositions.
The function proceeds by iterating over the shared positions and toggles the corresponding bits in the initial number X by employing bitwise XOR. The end result is the updated value of X.
- Main Process:
Within the primary function, the findOriginalNumber function is called, passing in the values 9, 17, 29, and I as parameters. The result is displayed on the console.
- Printout:
The provided code, upon execution, displays the output of the findOriginalNumber function with the specified parameters (9, 17, 29, 1) on the console.
Overall, the code aims to determine the initial number (X) from three numbers (X, Y, and Z). Here, X, Y, and Z are derived by altering K distinct bits in the original number. The process involves a helper function that identifies the positions of the altered bits and a findOriginalNumber function that reconstructs the original number using this information.
Complexity Analysis:
The provided code exhibits time and space complexities that scale in accordance with the binary size of the numbers.
Time Complexity:
Bit Flipping (helper function):
The helperfunction utilizes the bitwise XOR operation to manipulate the individual bits of both X and Y, enabling the identification of differing bit positions. Iterating through each of the 32 bits results in a time complexity of O(32), which simplifies to O(1) due to the constant number of processed bits. The findOriginalNumber function identifies the shared positions within sets A and B.
Time complexity remains at O(32) or O(1) because the number of iterations through the bits is fixed.
Overall Time Complexity:
Considering each step individually, the time complexity is primarily influenced by the fixed number of bits (32). As a result, the time complexity can be expressed as O(1).
Space Complexity:
The auxiliary function saves the variables (flippedPositions) in a set format, representing the flipped bits. The storage space for this set scales linearly with the count of flip positions, capped at K, the overall bit count. Consequently, the spatial complexity amounts to O(32) or O(1).
The findOriginalNumbers method makes use of a set named commonPositions to keep track of the shared positions between sets A and B. Additionally, the memory usage is inversely related to the occurrence frequency, which is capped at 32. Consequently, the computational complexity is O(32) or effectively O(1).
Overall Space Complexity:
The total space complexity is determined by the highest storage requirement of flippedPositions and commonPositions, which amounts to O(32) or O(1).
