Non-negative Integers without Consecutive Ones

The given problem requires us to find the count of numbers from 0 to ( n ) (inclusive) whose binary representations do not contain consecutive ones. Here’s how we can approach this problem:

Step 1: Understand the Pattern

Observe the pattern in the number of valid numbers without consecutive ones for different lengths:

  • Length 1: 2 ways (0, 1)
  • Length 2: 3 ways (00, 01, 10)
  • Length 3: 5 ways
  • Length 4: 8 ways
  • …

The pattern follows the Fibonacci sequence.

Step 2: Use Dynamic Programming

Build a lookup table that will store the count of valid numbers for each length, following the Fibonacci sequence.

Step 3: Check Binary Representation

Check the binary representation of ( n ) from the most significant bit. If we find two consecutive ones, we know that the count for the remaining bits doesn’t need to be considered as they will have consecutive ones. If we find a zero after a one, we will add the valid counts for the remaining bits, considering the pattern.

Code

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class Solution:
    def findIntegers(self, n: int) -> int:
        # Generate Fibonacci numbers
        fib = [1, 2]
        for i in range(2, 31):
            fib.append(fib[i - 1] + fib[i - 2])

        # Convert n to binary and reverse it
        binary = bin(n)[2:][::-1]

        # Initialize variables to keep track of results
        result = 0
        prev_bit = 0

        # Iterate through the binary representation
        for i in range(len(binary) - 1, -1, -1):
            if binary[i] == '1':
                result += fib[i]
                if prev_bit == 1:
                    # If previous bit was also 1, no need to continue
                    break
            prev_bit = int(binary[i])

        if binary.count('11') == 0:
            # If there were no consecutive ones, add one to the result
            result += 1

        return result

Key Takeaways

  • The problem can be solved using dynamic programming, following the Fibonacci sequence pattern.
  • By analyzing the binary representation of ( n ), we can find the count of numbers without consecutive ones.
  • The solution leverages both pattern recognition and a logical approach to the binary representation.

10 Prerequisite LeetCode Problems

This involves knowledge of bit manipulation, dynamic programming and Fibonacci numbers. Here are some problems to prepare:

  1. LeetCode 338. Counting Bits

    • This problem helps you understand how to manipulate bits and gives an introduction to dynamic programming.

  2. LeetCode 198. House Robber

    • This problem is a fundamental problem for dynamic programming. Understanding how to solve this can help build intuition on dynamic programming concepts.

  3. LeetCode 70. Climbing Stairs

    • This problem is a good introduction to dynamic programming, it also introduces you to Fibonacci sequence which is related to the main problem.

  4. LeetCode 509. Fibonacci Number

    • This problem directly deals with Fibonacci numbers. Understanding this will be helpful as the non-consecutive ones problem is related to Fibonacci numbers.

  5. LeetCode 136. Single Number

    • This problem involves bit manipulation and will help you understand the XOR operation.

  6. LeetCode 191. Number of 1 Bits

    • This problem involves counting the number of one bits in an integer, this will be helpful in understanding how to manipulate bits.

  7. LeetCode 279. Perfect Squares

    • This problem is a dynamic programming problem which involves finding an optimal solution by breaking down the problem into simpler sub-problems.

  8. LeetCode 300. Longest Increasing Subsequence

    • This problem is also a dynamic programming problem that gives you practice in understanding how to find a solution by considering previous states.

  9. LeetCode 746. Min Cost Climbing Stairs

    • This is another dynamic programming problem which introduces the concept of calculating minimum cost which can be beneficial in the main problem.

  10. LeetCode 371. Sum of Two Integers

    • This problem helps you understand the manipulation of bits to calculate the sum of two integers without using arithmetic operators.

These cover dynamic programming, bit manipulation, and Fibonacci numbers which are important for tackling the problem “600. Non-negative Integers without Consecutive Ones”.

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class Solution:
    def findIntegers(self, n: int) -> int:
        f = [1, 2]
        for i in range(2, 30):
            f.append(f[-1]+f[-2])

        ans, last_seen = 0, 0
        for i in reversed(range(30)):
            if (1 << i) & n: 
                ans += f[i]
                if last_seen: 
                    ans -= 1
                    break
                last_seen = 1
            else:
                last_seen = 0
        return ans+1

Problem Classification

The given problem statement falls under the domain of Number Theory and Bit Manipulation. The task is to find the count of numbers within a specified range that fulfill a certain condition related to their binary representation.

The ‘What’ components of the problem are:

  1. Input: A positive integer n, which defines the range [0, n].
  2. Output: The number of integers within the given range whose binary representation do not contain consecutive ones.

We can further classify the problem into the following categories:

  1. Number Theory: We are dealing with positive integers and need to understand the binary representation of these numbers. This involves an understanding of base conversions and binary number theory.

  2. Bit Manipulation: The requirement to count numbers that do not have consecutive 1s in their binary representation suggests we’ll need to manipulate and analyze the bits of each number. This involves a solid understanding of bitwise operations.

  3. Counting: Ultimately, the problem is about counting the numbers that fulfill a specific condition. This requires an understanding of combinatorial counting techniques and possibly dynamic programming if we are to avoid checking every number in the range individually.

  4. Range Analysis: Since the problem defines a range from 0 to n, there’s an element of analyzing this range and understanding how the binary representation of numbers changes within this range.

This problem requires an understanding of binary numbers and bit manipulation, as well as some level of combinatorial counting or dynamic programming to efficiently solve the problem.

Language Agnostic Coding Drills

The code is a solution to find the number of non-negative integers below a given number n that do not contain consecutive 1s in their binary representation.

Here’s a breakdown of the main concepts involved:

  1. List Operations: Understanding how to create and manipulate lists. This includes appending elements to a list and accessing elements by index.

  2. Loops: Understanding the usage of for loop for repeating certain operations multiple times.

  3. Bitwise Operations: This involves understanding bitwise shift and bitwise AND operations. Bitwise operations are functions that work on the bit level, which is useful for optimizing certain problems.

  4. Conditionals: Utilizing if-else statements to control the flow of the code.

  5. Arithmetic Operations: Basic understanding of addition, subtraction and incrementing values.

  6. Understanding Binary: The solution works at binary level so a good understanding of binary numbers, their representation and properties is required.

Here is the high level logic of the problem:

The code works by examining the binary representation of n from the most significant bit to the least significant bit. For each bit, if it is 1, it adds the number of valid integers of length equal to the bit position to the answer. If it encounters two consecutive 1 bits, it stops the process and subtracts 1 from the answer, because all of the numbers with the current prefix have a pair of consecutive 1’s and are thus invalid. At the end, it adds 1 to the answer to include the number n itself (provided it doesn’t have consecutive 1’s).

Note that the list f stores the Fibonacci sequence shifted by one, which represents the count of binary strings of length i (for f[i]) without consecutive ones. This is because for a binary string of length i without consecutive ones, its last two bits can either be “00” or “01”, but not “11” (since it’s not allowed). Therefore, the number of such strings is the sum of the two previous lengths’ counts, which gives the Fibonacci sequence.

Targeted Drills in Python

1. List Operations: Creating, appending to and accessing elements from a list.

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# Creating a list with two elements
f = [1, 2]

# Appending elements to a list
for i in range(2, 10):
    f.append(f[-1]+f[-2])

# Accessing elements in a list
print(f[0])  # Outputs: 1
print(f[-1])  # Outputs: Last element of the list

2. Loops: Understanding for loops and range() function.

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# Using for loop to repeat an operation multiple times
for i in range(5):
    print(i)  # Outputs: 0, 1, 2, 3, 4

3. Bitwise Operations: Understanding bitwise operations such as shift (<<) and bitwise AND (&).

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# Shifting bits to left
print(1 << 3)  # Outputs: 8 because 1 shifted 3 places to left is 1000 (which is 8 in decimal)

# Bitwise AND operation
print(10 & 7)  # Outputs: 2 because bitwise AND of 1010 (10) and 0111 (7) is 0010 (2)

4. Conditionals: Utilizing if-else statements to control the flow of the code.

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# Using if-else statements
a = 10
if a > 5:
    print("a is greater than 5")  # This will be executed
else:
    print("a is not greater than 5")

5. Arithmetic Operations: Basic arithmetic operations like addition, subtraction and incrementing values.

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# Basic arithmetic operations
a = 5
b = 3

print(a + b)  # Addition: Outputs 8
print(a - b)  # Subtraction: Outputs 2

6. Understanding Binary: Working with binary numbers.

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# Converting a decimal number to binary and vice versa
print(bin(10))  # Outputs: '0b1010'
print(int('1010', 2))  # Outputs: 10

After mastering these drills, you can combine these concepts to understand and implement the solution to the given problem.

Problem Classification

Problem Statement: Given a positive integer n, return the number of the integers in the range [0, n] whose binary representations do not contain consecutive ones.

Example 1:

Input: n = 5 Output: 5 Explanation: Here are the non-negative integers <= 5 with their corresponding binary representations: 0 : 0 1 : 1 2 : 10 3 : 11 4 : 100 5 : 101 Among them, only integer 3 disobeys the rule (two consecutive ones) and the other 5 satisfy the rule. Example 2:

Input: n = 1 Output: 2 Example 3:

Input: n = 2 Output: 3

Constraints:

1 <= n <= 109

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Given the problem , identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.