Given positive integers N, L, and R, we have to find the maximum value of N ⊕ X, where X ∈ [L, R].
Input : N = 7
L = 2
R = 23
Output : 23
Explanation : When X = 16, we get 7 ⊕ 16 = 23 which is the maximum value for all X ∈ [2, 23].
Input : N = 10
L = 5
R = 12
Output : 15
Explanation : When X = 5, we get 10 ⊕ 5 = 15 which is the maximum value for all X ∈ [5, 12].
Brute force approach: We can solve this problem using brute force approach by looping over all integers over the range [L, R] and taking their XOR with N, while keeping a record of the maximum result encountered so far. The complexity of this algorithm will be O(R – L), and it is not feasible when the input variables approach high values such as 109.
Efficient approach: Since the XOR of two bits is 1 if and only if they are complementary to each other, we need X to have complementary bits to that of N to have the maximum value. We will iterate from the largest bit (log2(R)th bit) to the lowest (0th bit). The following two cases can arise for each bit:
- If the bit is not set, i.e. 0, we will try to set it in X. If setting this bit to 1 results in X exceeding R, then we will not set it.
- If the bit is set, i.e. 1, then we will try to unset it in X. If the current value of X is already greater than or equal to L, then we can safely unset the bit. In the other case, we will check if setting all of the next bits is enough to keep X >= L. If not, then we are required to set the current bit. Observe that setting all of the next bits is equivalent to adding (1 << b) – 1, where b is the current bit.
The time complexity of this approach is O(log2(R)).
The output is 23
- Maximum XOR value of a pair from a range
- Maximum Bitwise AND pair from given range
- Maximum Bitwise OR pair from a range
- Maximum possible Bitwise OR of the two numbers from the range [L, R]
- Number whose sum of XOR with given array range is maximum
- Smallest number whose set bits are maximum in a given range
- Maximum XOR value of maximum and second maximum element among all possible subarrays
- XOR of all the elements in the given range [L, R]
- Bitwise and (or &) of a range
- Bitwise OR (or | ) of a range
- Count of numbers having only 1 set bit in the range [0, n]
- Unset bits in the given range
- Toggle bits in the given range
- Check whether XOR of all numbers in a given range is even or odd
- Set bits in N equals to M in the given range.
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