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Introduction to Algorithm Analysis

Part 1
Table of Contents
algorithm-fundamentals

Introduction to Algorithm Analysis

Understanding Algorithm Efficiency

Before diving into specific algorithms, it’s crucial to understand how we measure their efficiency. This forms the foundation of algorithm analysis.

What is Algorithm Analysis?

Algorithm analysis is the process of determining the amount of resources (such as time and storage) necessary to execute an algorithm. We focus on two main aspects:

  • Time Complexity: How long does it take to run?
  • Space Complexity: How much memory does it use?

Big O Notation

Basic Concepts

Big O notation is the language we use to talk about how long an algorithm takes to run or how much memory it needs. It’s a way of expressing the upper bound of growth rate of algorithms.

Common Time Complexities

Here are the most common time complexities you’ll encounter:

  1. O(1) - Constant Time
  2. O(log n) - Logarithmic Time
  3. O(n) - Linear Time
  4. O(n log n) - Linearithmic Time
  5. O(n²) - Quadratic Time
  6. O(2ⁿ) - Exponential Time

Practical Examples

def linear_search(arr, target):
    for i in range(len(arr)):  # O(n)
        if arr[i] == target:
            return i
    return -1
def binary_search(arr, target):
    left = 0
    right = len(arr) - 1
    
    while left <= right:  # O(log n)
        mid = (left + right) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    return -1

Practice Problems

To reinforce your understanding, try analyzing the time complexity of these operations:

  1. Finding the maximum element in an unsorted array
  2. Checking if a number is prime
  3. Computing Fibonacci numbers recursively

Summary

In this introduction to algorithm analysis, we’ve covered:

  1. The importance of analyzing algorithms
  2. Big O notation basics
  3. Common time complexities
  4. Practical examples with code
  5. Practice problems for self-assessment

Next Steps

In the next part, we’ll dive deeper into:

  • Space complexity analysis
  • Amortized analysis
  • Best, worst, and average case scenarios
  • More complex algorithm examples