Generate largest number with given x active and y inactive bits

The problem at hand involves generating the largest possible number with a specific count of active (set) and inactive (unset) bits. We are given two integers, x and y, representing the counts of active and inactive bits, respectively. The goal is to find the largest number that satisfies these bit conditions.

Problem Statement and Description

Given integers x and y, representing the counts of active and inactive bits respectively, the task is to find the largest number that satisfies these bit conditions. In other words, we want to create a binary number with x active bits followed by y inactive bits, such that the number is as large as possible.

Example

For example, if x is 5 and y is 2, we want to create the largest possible number that has 5 active bits followed by 2 inactive bits. In binary representation, this number would be 1111100, which is equivalent to 124 in decimal. Therefore, the output for this example should be "Largest Number is: 124".

1. For x = 5 and y = 2:

   • The largest possible number with 5 active bits and 2 inactive bits in binary is 1111100, which is 124 in decimal.
   • The output is "Given x: 5, y: 2\nLargest Number is: 124".

2. For x = 1 and y = 2:

   • The largest possible number with 1 active bit and 2 inactive bits in binary is 100, which is 4 in decimal.
   • The output is "Given x: 1, y: 2\nLargest Number is: 4".

3. For x = 4 and y = 4:

   • The largest possible number with 4 active bits and 4 inactive bits in binary is 11110000, which is 240 in decimal.
   • The output is "Given x: 4, y: 4\nLargest Number is: 240".

Idea to Solve the Problem

The core idea to solve this problem is to construct the largest possible number using bitwise operations. We'll utilize the property of bitwise XOR to set specific bits in the number. To generate a binary number with x active bits and y inactive bits, we'll perform the following steps:

1. Calculate result as follows:
   • ((1 << (x + y)) - 1) generates a number with x + y active bits. This covers the first x positions.
   • ((1 << y) - 1) generates a number with y active bits. This covers the last y positions.
   • XOR the two results together to get a number with the desired count of active and inactive bits.

Pseudocode

Here's the pseudocode for the algorithm:

function largestNum(x, y):
        if x < 0 or y < 0 or (x + y) > 31:
            // x and y not in valid range
            return
        
        result = ((1 << (x + y)) - 1) ^ ((1 << y) - 1)
        print "Given x:", x, ", y:", y
        print "Largest Number is:", result

Algorithm Explanation

1. The function largestNum(x, y) takes two integers x and y as inputs.
2. It first checks whether x and y are within valid ranges. If not, the function returns without processing.
3. The function calculates result as explained earlier.
4. It prints the values of x and y to show the input.
5. Finally, the function prints the calculated result, which is the largest number with the specified counts of active and inactive bits.

Code Solution

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Time Complexity

The time complexity of this algorithm is constant, as it involves basic arithmetic and bitwise operations that execute in constant time regardless of the input values. The complexity is not dependent on the magnitude of x and y.