给定两个数X和Y的总和与异或,st和和异或 ,我们需要找到使X值最小的数字。
例子 :
Input : S = 17
X = 13
Output : a = 2
b = 15
Input : S = 1870807699
X = 259801747
Output : a = 805502976
b = 1065304723
Input : S = 1639
X = 1176
Output : No such numbers exist
Variables Used:
X ==> XOR of two numbers
S ==> Sum of two numbers
X[i] ==> Value of i-th bit in X
S[i] ==> Value of i-th bit in S
一个简单的解决方案是使用给定的XOR生成所有可能的对。要生成所有对,我们可以遵循以下规则。
- 如果X [i]为1,则a [i]和b [i]应该都不同,我们有两种情况。
- 如果X [i]为0,则a [i]和b [i]应该相同。我们有两种情况。
因此,我们生成2 ^ n个可能的对,其中n是X中的位数。然后对于每对,我们检查其总和是否为S。
一个有效的解决方案基于以下事实。
S = X + 2*A
where A = a AND bWe can verify above fact using the sum process. In sum, whenever we see both bits 1 (i.e., AND is 1), we make resultant bit 0 and add 1 as carry, which means every bit in AND is left shifted by 1 OR value of AND is multiplied by 2 and added.
因此我们可以找到A =(S – X)/ 2。
一旦找到A,就可以使用以下规则找到“ a”和“ b”的所有位。
- 如果X [i] = 0且A [i] = 0,则a [i] = b [i] =0。对于该位,只有一种可能性。
- 如果X [i] = 0且A [i] = 1,则a [i] = b [i] = 1。
- 如果X [i] = 1且A [i] = 0,则(a [i] = 1且b [i] = 0)或(a [i] = 0且b [i] = 1),我们可以选择两者中的任何一个。
- 如果X [i] = 1且A [i] = 1,则结果不可能(注X [i] = 1表示不同的位)
令总和为S,XOR为X。
下面是上述方法的实现:
C++
// CPP program to find two numbers with // given Sum and XOR such that value of // first number is minimum. #include
using namespace std; // Function that takes in the sum and XOR // of two numbers and generates the two // numbers such that the value of X is // minimized void compute(unsigned long int S, unsigned long int X) { unsigned long int A = (S - X)/2; int a = 0, b = 0; // Traverse through all bits for (int i=0; i<8*sizeof(S); i++) { unsigned long int Xi = (X & (1 << i)); unsigned long int Ai = (A & (1 << i)); if (Xi == 0 && Ai == 0) { // Let us leave bits as 0. } else if (Xi == 0 && Ai > 0) { a = ((1 << i) | a); b = ((1 << i) | b); } else if (Xi > 0 && Ai == 0) { a = ((1 << i) | a); // We leave i-th bit of b as 0. } else // (Xi == 1 && Ai == 1) { cout << "Not Possible"; return; } } cout << "a = " << a << endl << "b = " << b; } // Driver function int main() { unsigned long int S = 17, X = 13; compute(S, X); return 0; }
Java
// Java program to find two numbers with // given Sum and XOR such that value of // first number is minimum. class GFG { // Function that takes in the sum and XOR // of two numbers and generates the two // numbers such that the value of X is // minimized static void compute(long S, long X) { long A = (S - X)/2; int a = 0, b = 0; final int LONG_FIELD_SIZE = 8; // Traverse through all bits for (int i=0; i<8*LONG_FIELD_SIZE; i++) { long Xi = (X & (1 << i)); long Ai = (A & (1 << i)); if (Xi == 0 && Ai == 0) { // Let us leave bits as 0. } else if (Xi == 0 && Ai > 0) { a = ((1 << i) | a); b = ((1 << i) | b); } else if (Xi > 0 && Ai == 0) { a = ((1 << i) | a); // We leave i-th bit of b as 0. } else // (Xi == 1 && Ai == 1) { System.out.println("Not Possible"); return; } } System.out.println("a = " + a +"\nb = " + b); } // Driver function public static void main(String[] args) { long S = 17, X = 13; compute(S, X); } } // This code is contributed by RAJPUT-JI
Python3
# Python program to find two numbers with # given Sum and XOR such that value of # first number is minimum. # Function that takes in the sum and XOR # of two numbers and generates the two # numbers such that the value of X is # minimized def compute(S, X): A = (S - X)//2 a = 0 b = 0 # Traverse through all bits for i in range(64): Xi = (X & (1 << i)) Ai = (A & (1 << i)) if (Xi == 0 and Ai == 0): # Let us leave bits as 0. pass elif (Xi == 0 and Ai > 0): a = ((1 << i) | a) b = ((1 << i) | b) elif (Xi > 0 and Ai == 0): a = ((1 << i) | a) # We leave i-th bit of b as 0. else: # (Xi == 1 and Ai == 1) print("Not Possible") return print("a = ",a) print("b =", b) # Driver function S = 17 X = 13 compute(S, X) # This code is contributed by ankush_953
C#
// C# program to find two numbers with // given Sum and XOR such that value of // first number is minimum. using System; public class GFG { // Function that takes in the sum and XOR // of two numbers and generates the two // numbers such that the value of X is // minimized static void compute(long S, long X) { long A = (S - X)/2; int a = 0, b = 0; // Traverse through all bits for (int i=0; i<8*sizeof(long); i++) { long Xi = (X & (1 << i)); long Ai = (A & (1 << i)); if (Xi == 0 && Ai == 0) { // Let us leave bits as 0. } else if (Xi == 0 && Ai > 0) { a = ((1 << i) | a); b = ((1 << i) | b); } else if (Xi > 0 && Ai == 0) { a = ((1 << i) | a); // We leave i-th bit of b as 0. } else // (Xi == 1 && Ai == 1) { Console.WriteLine("Not Possible"); return; } } Console.WriteLine("a = " + a +"\nb = " + b); } // Driver function public static void Main() { long S = 17, X = 13; compute(S, X); } } // This code is contributed by RAJPUT-JI
输出
a = 15 b = 2
上述方法的时间复杂度其中b是S中的位数。
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