Given two positive integers A and B such that A != B, the task is to find a positive integer X which maximizes the expression (A AND X) * (B AND X).
Input: A = 9 B = 8
(9 AND 8) * (8 AND 8) = 8 * 8 = 64 (maximum possible)
Input: A = 11 and B = 13
Naive approach: One can run a loop from 1 to max(A, B) and can easily find X which maximizes the given expression.
Efficient approach: It is known that,
(a – b)2 ≥ 0
which implies (a + b)2 – 4*a*b ≥ 0
which implies a * b ≤ (a + b)2 / 4
Hence, it concludes that a * b will be maximum when a * b = (a + b)2 / 4
which implies a = b
From the above result, (A AND X) * (B AND X) will be maximum when (A AND X) = (B AND X)
Now X can be found as:
A = 11 = 1011
B = 13 = 1101
X = ? = abcd
At 0th place: (1 AND d) = (1 AND d) implies d = 0, 1 but to maximize (A AND X) * (B AND X) d = 1
At 1st place: (1 AND d) = (0 AND d) implies c = 0
At 2nd place: (0 AND d) = (1 AND d) implies b = 0
At 3rd place: (1 AND d) = (1 AND d) implies a = 0, 1 but to maximize (A AND X) * (B AND X) a = 1
Hence, X = 1001 = 9
Below is the implementation of the above approach:
- Maximize the Expression | Bit Manipulation
- Maximize the value of the given expression
- Array Manipulation and Sum
- Bits manipulation (Important tactics)
- Bit manipulation | Swap Endianness of a number
- Maximize the value of x + y + z such that ax + by + cz = n
- Expression Evaluation
- Maximize the number of subarrays with XOR as zero
- Maximize the bitwise OR of an array
- Maximize a value for a semicircle of given radius
- Find the minimum value of X for an expression
- Find all possible outcomes of a given expression
- Maximize big when both big and small can be exchanged
- Find Range Value of the Expression
- What is an Expression and What are the types of Expressions?
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.