Algorithms: A Guide to Fundamental and Advanced Techniques

This section covers the most important areas of algorithms: mathematical algorithms, data structures, sorting, recursion, dynamic programming, graphs, and advanced techniques. The material is intended for high school students, competitive programmers, and anyone who wants a deeper understanding of algorithmic principles.

What You Can Expect from This Tutorial

The tutorial is structured to gradually guide you from basic concepts to advanced techniques. Each subsection includes:

• Theory — clear explanations of key concepts and complexity analysis (time and memory).
• Illustrated examples — small step-by-step examples showing how an algorithm works.
• Implementations — ready-to-use code (pseudocode, Python / C++ / Java) that can be tested and modified.
• Practice problems — tasks of varying difficulty with commented solutions and optimizations.
• Tips & usage — when to use a specific approach, common mistakes, and optimization techniques.

Note: some topics (especially advanced topics and full coverage of graphs and trees) are planned but not yet fully implemented — they will be added gradually.

Mathematical Algorithms

• Introduction to Mathematical Algorithms
• Fibonacci Sequence
• Prime Numbers and Factorization
• Euclidean Algorithm (GCD)
• Fast Exponentiation

Data Structures

• Introduction to Data Structures
• Vectors and Lists
• DSU – Disjoint Set Union
• Basic Data Structures

Sorting

• Sorting — Introduction
• Quick Sort
• Merge Sort

Recursion and Dynamic Programming

• Recursion — Introductory Lessons
• Tower of Hanoi
• DP Fibonacci
• Knapsack Problem
• Longest Common Subsequence (LCS)

Graphs and Trees

• Introduction to Graphs
• BFS and DFS
• Trees and Hierarchies
• Dijkstra’s Algorithm
• Minimum Spanning Tree (MST)

Advanced Techniques

• Binary Search on the Answer
• Two Pointers Technique
• Prefix and Suffix Arrays
• Fenwick Tree (Binary Indexed Tree)
• Segment Tree

Quick Examples (To Get the Big Picture)

Below are two short, concrete examples with explanations and implementations, so you can immediately see how theory is applied in practice:

Example 1 — Euclidean Algorithm for Greatest Common Divisor (GCD)

Idea: GCD(a, b) = GCD(b, a % b). The algorithm is simple, converges quickly, and works in O(log min(a, b)).

// Pseudocode (Python-style)
def gcd(a, b):
    while b != 0:
        a, b = b, a % b
    return a

# Example: gcd(48, 18) -> 6
    

Example 2 — DFS (Graph Traversal) and Counting Connected Components

Idea: use recursion or an explicit stack to visit all nodes in a component. Time complexity is O(N + M) for a graph with N vertices and M edges.

// Pseudocode (Python-style)
# graph represented as adjacency list: graph[v] = [neighbors]
visited = set()

def dfs(v):
    stack = [v]
    while stack:
        x = stack.pop()
        if x in visited:
            continue
        visited.add(x)
        for nei in graph[x]:
            if nei not in visited:
                stack.append(nei)

# Count connected components
count = 0
for v in all_vertices:
    if v not in visited:
        dfs(v)
        count += 1
    

If you want, I can add more examples (e.g. Merge Sort, Dijkstra, DP strategies), or prepare problem sets with solutions for each subsection.

← Back to Home