Maximum XOR value of a pair from a range
Given a range [L, R], we need to find two integers in this range such that their XOR is maximum among all possible choices of two integers. More Formally,
given [L, R], find max (A ^ B) where L <= A, B
Input : L = 8 R = 20 Output : 31 31 is XOR of 15 and 16. Input : L = 1 R = 3 Output : 3
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A simple solution is to generate all pairs, find their XOR values and finally return the maximum XOR value.
An efficient solution is to consider pattern of binary values from L to R. We can see that first bit from L to R either changes from 0 to 1 or it stays 1 i.e. if we take the XOR of any two numbers for maximum value their first bit will be fixed which will be same as first bit of XOR of L and R itself.
After observing the technique to get first bit, we can see that if we XOR L and R, the most significant bit of this XOR will tell us the maximum value we can achieve i.e. let XOR of L and R is 1xxx where x can be 0 or 1 then maximum XOR value we can get is 1111 because from L to R we have all possible combination of xxx and it is always possible to choose these bits in such a way from two numbers such that their XOR becomes all 1. It is explained below with some examples,
Examples 1: L = 8 R = 20 L ^ R = (01000) ^ (10100) = (11100) Now as L ^ R is of form (1xxxx) we can get maximum XOR as (11111) by choosing A and B as 15 and 16 (01111 and 10000) Examples 2: L = 16 R = 20 L ^ R = (10000) ^ (10100) = (00100) Now as L ^ R is of form (1xx) we can get maximum xor as (111) by choosing A and B as 19 and 20 (10011 and 10100)
So the solution of this problem depends on the value of (L ^ R) only. We will calculate the L^R value first and then from most significant bit of this value, we will add all 1s to get the final result.
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