Minimize given flips required to reduce N to 0

Given an integer N, the task is to reduce the value of N to 0 by performing the following operations minimum number of times:

• Flip the rightmost (0^th) bit in the binary representation of N.
• If (i – 1)^th bit is set, then flip the i^th bit and clear all the bits from (i – 2)^th to 0^th bit.

Examples:

Input: N = 3 
Output: 2 
Explanation:
The binary representation of N (= 3) is 11 
Since 0^th bit in binary representation of N(= 3) is set, flipping the 1^st bit of binary representation of N modifies N to 1(01). 
Flipping the rightmost bit of binary representation of N(=1) modifies N to 0(00). 
Therefore, the required output is 2

Input: N = 4 
Output: 7

Approach: The problem can be solved based on the following observations:

1 -> 0 => 1 
10 -> 11 -> 01 -> 00 => 2 + 1 = 3 
100 -> 101 -> 111 -> 110 -> 010 -> … => 4 + 2 + 1 = 7 
1000 -> 1001 -> 1011 -> 1010 -> 1110 -> 1111 -> 1101 -> 1100 -> 0100 -> … => 8 + 7 = 15 
Therefore, for N = 2^N total (2^{(N + 1)} – 1) operations required.
If N is not a power of 2, then the recurrence relation is: 
MinOp(N) = MinOp((1 << cntBit) – 1) – MinOp(N – (1 << (cntBit – 1)))
cntBit = total count of bits in binary representation of N. 
MinOp(N) denotes minimum count of operations required to reduce N to 0. 

Follow the steps below to solve the problem:

• Calculate the count of bits in binary representation of N using log₂(N) + 1.
• Use the above recurrence relation and calculate the minimum count of operations required to reduce N to 0.

Below is the implementation of the above approach.

C++

// C++ program to implement
// the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum count of
// operations required to Reduce N to 0
int MinOp(int N)
{

    if (N <= 1)
        return N;

    // Stores count of
    // bits in N
    int bit = log2(N) + 1;

    // Recurrence relation
    return ((1 << bit) - 1)
           - MinOp(N - (1 << (bit - 1)));
}

// Driver Code
int main()
{

    int N = 4;
    cout << MinOp(N);
    return 0;
}

Java

// Java program to implement
// the above approach
class GFG{

// Function to find the minimum count of
// operations required to Reduce N to 0
public static int MinOp(int N)
{
    if (N <= 1)
        return N;

    // Stores count of
    // bits in N
    int bit = (int)(Math.log(N) /
                    Math.log(2)) + 1;

    // Recurrence relation
    return ((1 << bit) - 1) - MinOp(
        N - (1 << (bit - 1)));
}

// Driver code
public static void main(String[] args)
{
    int N = 4;

    System.out.println(MinOp(N));
}
}

// This code is contributed by divyeshrabadiya07

Python3

# Python program to implement
# the above approach

# Function to find the minimum count of
# operations required to Reduce N to 0
import math
def MinOp(N):
    if (N <= 1):
        return N;

    # Stores count of
    # bits in N
    bit = (int)(math.log(N) / math.log(2)) + 1;

    # Recurrence relation
    return ((1 << bit) - 1) - MinOp(N - (1 << (bit - 1)));

# Driver code
if __name__ == '__main__':
    N = 4;

    print(MinOp(N));

# This code is contributed by 29AjayKumar

C#

// C# program to implement
// the above approach
using System;

class GFG{

// Function to find the minimum count of
// operations required to Reduce N to 0
public static int MinOp(int N)
{
    if (N <= 1)
        return N;

    // Stores count of
    // bits in N
    int bit = (int)(Math.Log(N) /
                    Math.Log(2)) + 1;

    // Recurrence relation
    return ((1 << bit) - 1) - MinOp(
        N - (1 << (bit - 1)));
}

// Driver code
public static void Main()
{
    int N = 4;

    Console.WriteLine(MinOp(N));
}
}

// This code is contributed by sanjoy_62

Javascript

<script>
// javascript program to implement
// the above approach

// Function to find the minimum count of
// operations required to Reduce N to 0
function MinOp(N)
{
    if (N <= 1)
        return N;

    // Stores count of
    // bits in N
    let bit = (Math.log(N) /
                    Math.log(2)) + 1;

    // Recurrence relation
    return ((1 << bit) - 1) - MinOp(
        N - (1 << (bit - 1)));
}

// Driver code
    let N = 4;
    document.write(MinOp(N));

    // This code is contributed by souravghosh0416.
</script>

Output:

7

Time Complexity: O(log(N)) //since the logarithm function is used, hence the time complexity is logarithmic
Auxiliary Space: O(1) // since no extra variable is used hence the space is taken by the algorithm is constant