What is the time complexity of Number of Steps to Reduce a Number in Binary Representation to One?

The optimal solution for Number of Steps to Reduce a Number in Binary Representation to One runs in O(n) time and uses O(1) space. The time complexity is O(n) because we iterate through the binary string once, and the space complexity is O(1) since we only use a few variables.

What is the brute force solution for Number of Steps to Reduce a Number in Binary Representation to One?

The brute force approach for Number of Steps to Reduce a Number in Binary Representation to One is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach simulates each step of reducing the binary number to 1. We check if the number is odd or even and apply the corresponding operation until we reach 1. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Number of Steps to Reduce a Number in Binary Representation to One?

Number of Steps to Reduce a Number in Binary Representation to One uses the following concepts: String, Bit Manipulation, Simulation. The recommended patterns to study are: Bit Manipulation, Simulation.

What are the interview tips for Number of Steps to Reduce a Number in Binary Representation to One?

Always consider the properties of the data structure you're working with (like binary strings). Explain your thought process clearly, especially when transitioning from a brute-force to an optimal solution. Practice simulating operations on strings or numbers to build intuition for similar problems.

What are common mistakes when solving Number of Steps to Reduce a Number in Binary Representation to One?

Not recognizing that the binary string can be manipulated directly without conversion to decimal. Overcomplicating the logic by trying to handle each operation separately instead of counting steps efficiently.

#1404

Number of Steps to Reduce a Number in Binary Representation to One

Medium

StringBit ManipulationSimulationBit ManipulationSimulation

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Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(1)

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Intuition

Time O(n)Space O(1)

The optimal solution leverages the properties of binary numbers. Instead of converting to decimal and simulating the steps, we can directly manipulate the binary string to count the steps more efficiently.

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Algorithm

5 steps

1Step 1: Initialize a counter for steps.
2Step 2: Iterate over the binary string from the end to the beginning.
3Step 3: For each character, if it's '0', increment the step counter (divide by 2).
4Step 4: If it's '1', increment the step counter (add 1) and then add 1 more step for the subsequent division.
5Step 5: Return the total steps.

solution.py10 lines

1# Full working Python code
2
3def num_steps(s):
4    steps = 0
5    for i in range(len(s) - 1, 0, -1):
6        if s[i] == '0':
7            steps += 1  # divide by 2
8        else:
9            steps += 2  # add 1 and then divide by 2
10    return steps

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Complexity note: The time complexity is O(n) because we iterate through the binary string once, and the space complexity is O(1) since we only use a few variables.

1Understanding binary representation helps in optimizing the solution.
2Simulating the process directly on the binary string can save time and space.

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