Gunadarma Complexity Analysis Study Guide, Study Guides, Projects, Research of Data Structures and Algorithms

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Complexity Analysis

Data Structures

Informatics Engineering

Gunadarma University

Introduction to Complexity Analysis: Complexity analysis, often represented using Big O notation, is a method to analyze the efficiency of algorithms. It helps in understanding how an algorithm's runtime or space requirements grow as the input size increases. Big O notation provides an upper bound on the growth rate of a function, allowing us to compare the efficiency of algorithms. Understanding Big O Notation: Big O notation describes the worst-case scenario for the time or space complexity of an algorithm in relation to the size of its input. It provides a simplified way to express the efficiency of an algorithm.

• O(1) : Constant time complexity. The runtime of the algorithm remains constant, regardless of the size of the input.
• O(log n) : Logarithmic time complexity. The runtime of the algorithm grows logarithmically as the size of the input increases.
• O(n) : Linear time complexity. The runtime of the algorithm grows linearly with the size of the input.
• O(n log n) : Linearithmic time complexity. Common in efficient sorting algorithms like Merge Sort and Quick Sort.
• O(n^2 ) : Quadratic time complexity. The runtime of the algorithm grows quadratically with the size of the input.
• O(2n) : Exponential time complexity. The runtime of the algorithm doubles with each additional element in the input.
• O(n!) : Factorial time complexity. The runtime of the algorithm grows factorially with the size of the input.

Tips for Analyzing Complexity:

• Identify Loops and Recursive Calls: Look for iterations and recursive functions within the algorithm.
• Consider the Dominant Term: Focus on the term that grows the fastest to determine the overall complexity.
• Ignore Constants: In Big O notation, constants and lower-order terms are disregarded. Practice Problems:

1. Analyze the time complexity of various sorting algorithms like Bubble Sort, Insertion Sort, and Merge Sort.
2. Determine the time complexity of recursive algorithms such as Fibonacci sequence generation or binary tree traversal.
3. Analyze the space complexity of algorithms that involve data structures like arrays, linked lists, or trees. Conclusion: Understanding complexity analysis, particularly Big O notation, is crucial for analyzing and comparing the efficiency of algorithms. By mastering this concept, you'll be better equipped to design and optimize algorithms for various problem-solving scenarios. Practice analyzing the complexity of different algorithms to reinforce your understanding.