create a new runnable csharp console app

dotnet new console -n csharp

Resources

• Big O Cheat Sheet{:target="_blank"}
• Halting Problem{:target="_blank"}
• Big O Notation{:target="_blank"}
• Asymptotic Analysis{:target="_blank"}
• Polynomial{:target="_blank"}
• Stack Overflow: Big O Calculation{:target="_blank"}

Finding the time complexity of an algorithmn

Time complexity is a measure of the computational complexity that describes the amount of time an algorithm takes to run as a function of the size of its input. Here’s a step-by-step guide to determine the time complexity of an algorithm:

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1. Understand the Algorithm

• Carefully analyze the algorithm or code.
• Identify the input size n.
• Determine what the algorithm does with the input (loops, function calls, etc.).

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2. Identify Basic Operations

• Count the most frequently executed operation, such as arithmetic operations, comparisons, data assignments, or function calls.
• Focus on the part of the code that has the largest impact on the running time.

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3. Analyze Loops

• Simple Loops: A loop running n times contributes ( O(n) ).
• Nested Loops: Multiply the iterations. For example:
  • A loop inside another loop, each running n times, results in ( O(n^2) ).
• Logarithmic Loops: A loop that halves the input size in each iteration contributes ( O(\log n) ).

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4. Analyze Recursion

• Use recurrence relations to model recursive calls.
• Solve the recurrence using methods like the Master Theorem or Substitution Method.
• Example: A recursive algorithm splitting the problem into two halves (like merge sort) often has a complexity of ( O(n \log n) ).

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5. Consider All Code Blocks

• If the algorithm has multiple independent steps, sum their complexities.
  • Example: ( O(n) + O(n^2) ) simplifies to ( O(n^2) ) (higher-order term dominates).
• For conditional branches (if, else), analyze the worst-case complexity.

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6. Drop Constants and Lower-Order Terms

• Time complexity focuses on the growth rate, so constants and non-dominant terms can be omitted.
  • Example: ( O(2n + 10) ) simplifies to ( O(n) ).

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7. Express the Result in Big-O Notation

• Big-O notation describes the upper bound of an algorithm’s growth rate.
• Examples:
  • ( O(1) ): Constant time.
  • ( O(\log n) ): Logarithmic time.
  • ( O(n) ): Linear time.
  • ( O(n^2) ): Quadratic time.

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8. Verify with Examples

• Test the algorithm with various input sizes.
• Ensure the theoretical complexity matches the observed behavior.

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Common Time Complexities

Notation	Example Algorithms	Growth Rate
( O(1) )	Accessing an array element	Constant
( O(\log n) )	Binary search	Logarithmic
( O(n) )	Linear search	Linear
( O(n \log n) )	Merge sort, Quick sort (avg)	Log-linear
( O(n^2) )	Bubble sort, Selection sort	Quadratic
( O(2^n) )	Recursive Fibonacci	Exponential

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By following these steps, you can systematically determine the time complexity of any algorithm.