Introduction to Problem Solving

Introduction to Problem Solving

Overview

Algorithm design is the art and science of creating efficient solutions to computational problems. This foundational course introduces you to the core principles of algorithmic thinking.

What is an Algorithm?

An algorithm is a well-defined sequence of steps that transforms input into output. It must be:

• Precise: Each step is clearly defined
• Finite: Terminates after a finite number of steps
• Effective: Each step is executable
• Correct: Produces the correct output for all valid inputs

Problem-Solving Framework

1. Problem Understanding

• Read the problem carefully
• Identify constraints and requirements
• Understand input/output format
• Clarify edge cases

2. Solution Design

• Break down into subproblems
• Identify patterns
• Choose appropriate data structures
• Design algorithm steps

3. Implementation

• Write clear, readable code
• Handle edge cases
• Add comments for clarity

4. Analysis

• Verify correctness
• Analyze time and space complexity
• Optimize if needed

Example: Finding Maximum Element

def find_max(arr):
    """
    Find the maximum element in an array.
    Time: O(n), Space: O(1)
    """
    if not arr:
        return None
    
    max_val = arr[0]
    for i in range(1, len(arr)):
        if arr[i] > max_val:
            max_val = arr[i]
    
    return max_val

Key Concepts

Correctness

An algorithm is correct if it:

• Terminates for all valid inputs
• Produces the correct output

Efficiency

Efficiency is measured by:

• Time complexity: How fast the algorithm runs
• Space complexity: How much memory it uses

Practice Problems

1. Find the minimum element in an array
2. Count occurrences of an element
3. Check if array is sorted
4. Reverse an array
5. Find the second largest element