What is an algorithm?
An algorithm is a set of step-by-step instructions designed to solve a specific problem or perform a particular task. It is a finite sequence of well-defined instructions or rules that, when followed, lead to a solution for a problem in a finite amount of time.
Why do you need to evaluate an algorithm?
Evaluating an algorithm is essential to understand its efficiency and performance characteristics. It helps in determining whether the algorithm is suitable for solving a particular problem efficiently, especially when dealing with large inputs or datasets. By evaluating algorithms, you can compare different approaches, optimize them, and choose the most appropriate one for a given scenario.
Counting number of instructions:
Counting the number of instructions involves analyzing the algorithm to determine the total number of elementary operations (such as assignments, comparisons, arithmetic operations, etc.) it performs. This helps in understanding the computational complexity of the algorithm and estimating its runtime behavior.
Asymptotic behavior:
Asymptotic behavior refers to how the performance of an algorithm scales with the size of the input. It focuses on understanding the behavior of the algorithm as the input size approaches infinity. Common asymptotic behaviors include constant time, logarithmic time, linear time, polynomial time, exponential time, etc.
How will you compare algorithms?
Algorithms can be compared based on various factors such as time complexity, space complexity, simplicity, readability, maintainability, and practical performance. However, the most common approach to comparing algorithms is by analyzing their time complexity and space complexity, which provide insights into their efficiency and resource requirements.
Big O Notation:
Big O notation is a mathematical notation used to describe the upper bound or worst-case scenario of the time complexity or space complexity of an algorithm. It provides a way to classify algorithms based on how their runtime or memory usage grows relative to the size of the input. For example, O(n) represents linear time complexity, O(log n) represents logarithmic time complexity, and O(1) represents constant time complexity.
Rules of thumb for calculating complexity of algorithm:
- Focus on the dominant terms: When calculating the complexity, focus on the terms that grow the fastest with input size.
- Ignore constants and lower-order terms: Big O notation simplifies the complexity analysis by ignoring constants and lower-order terms, as they become insignificant for large inputs.
- Use specific rules for common operations: For example, iterating over an array of size n has a time complexity of O(n), while nested loops may result in O(n^2) or O(n^3) depending on the depth.
Logarithmic complexity:
Logarithmic complexity (O(log n)) occurs when the algorithm's performance scales logarithmically with the size of the input. This means that as the input size increases, the algorithm's runtime or space requirements grow at a rate proportional to the logarithm of the input size.
Exercise:
An exercise to reinforce understanding could involve analyzing the time complexity of various algorithms (e.g., searching, sorting, graph traversal) and comparing their efficiency using Big O notation. Additionally, implementing and benchmarking different algorithms for solving the same problem can provide hands-on experience in evaluating algorithm performance.
