# Minimum bitwise operations to convert given a into b.

Given two positive integer a and b you have to change a to b by applying any of the three operations on binary form of a. You can select ai and aj (any two bits where i!=j) from binary form of a and then perform operation as:

- AND operation as : temp = ai & aj, ai = temp & ai, aj = temp & aj
- OR operation as : temp = ai | aj, ai = temp | ai, aj = temp | aj
- XOR operation as : temp = ai ^ aj, ai = temp ^ ai, aj = temp ^ aj

where & = bitwise AND, | = bitwiese OR and ^ = bitwise XOR.

Find the minimum operation required for conversion of a to b. Also, if conversion of a to b is not possible then print -1.

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Examples:

Input : a = 12 (1100), b = 10 (1010) Output : 1 Explanation : select a2 and a3 and perform XOR Input : a = 15 (1111), b = 10 (1010) Output : -1 Explanation : Conversion from a to b is not possible

**Explanation :** First of all let’s understand the working of all three operation.

**AND operation**as : temp = ai & aj, ai = temp & ai, aj = temp & ajIf any of ai or aj is 0 then it makes both as 0 otherwise no effect on ai and aj.

**OR operation**as : temp = ai | aj, ai = temp | ai, aj = temp | ajIf any of ai or aj is 1 then it makes both as 1 otherwise no effect on ai and aj.

**XOR operation**as : temp = ai ^ aj, ai = temp ^ ai, aj = temp ^ ajSimply interchange value of ai and aj.

Some conclusion on basis of working of operations :

- If all bits of a are 1 or 0 then we can not change value of a.
- If a equals to b then no operation required.
- Let n be number of indices i, where ai = 0 and bi = 1.

Let m be number of indices i, where ai = 1 and bi = 0.Let us think about the n elements, where ai = 0 and bi = 1. We have to change all of these zeros into ones. Note that this will require at least n operations.

Similarly for all the m elements, where ai = 1 and bi = 0. We have to change all of these ones into zeros. Note that this will require at least m operations.Let res = max(n, m). We can make the a an b equal in res operations as follows.

Let n >= m. Take m 1’s and n 0’s in A and apply the XOR operation to swap 0’s with 1’s. After that you will be left with total n-m zeros elements to change to one. That you can do by taking each of these zeros with some single one and applying the OR operation.

Let m >= n. Take m 1’s and n 0’s in A and apply the XOR operation to swap 0’s with 1’s. After that you will be left with total m-n ones elements to change to zero. That you can do by taking each of these ones with some single zero and applying the OR operation.

`// Cpp program to find min operation to convert a to b` `#include <bits/stdc++.h>` `using` `namespace` `std;` ` ` `// function to return min operation` `int` `minOp(bitset<32> a1, bitset<32> b1)` `{` ` ` `// if a1 == b1 return 0` ` ` `if` `(a1 == b1)` ` ` `return` `0;` ` ` ` ` `// if all bits of a = 0` ` ` `if` `(a1 == 0)` ` ` `return` `-1;` ` ` ` ` `// if all bits of a =1` ` ` `// first convert a1 to int and then cal a1 & a1+1` ` ` `if` `(((` `int` `)a1.to_ulong() & ((` `int` `)a1.to_ulong() + 1)) ` ` ` `== 0)` ` ` `return` `-1;` ` ` ` ` `// convert a and b to binary string` ` ` `string a = a1.to_string();` ` ` `string b = b1.to_string();` ` ` ` ` `// check where ai and bi are different` ` ` `// and count n where ai = 1 and m where ai = 0` ` ` `int` `n = 0, m = 0;` ` ` `for` `(` `int` `i = 0; i < b.size(); i++) {` ` ` `if` `(b[i] != a[i]) {` ` ` `if` `(a[i] == ` `'1'` `)` ` ` `n++;` ` ` `else` ` ` `m++;` ` ` `}` ` ` `}` ` ` ` ` `// return result` ` ` `return` `max(n, m);` `}` ` ` `// driver program` `int` `main()` `{` ` ` `bitset<32> a = 14, b = 1;` ` ` `cout << minOp(a, b);` ` ` `return` `0;` `}` |

Output:

3

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