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README.md
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README.md

Thanos Level

1. Algorithmic Complexity Theory and Computability

  • Introduction to Complexity Classes:

    • Overview of P, NP, NP-Complete, and NP-Hard problems.
    • Understanding computational feasibility and intractability.
    • Polynomial Time (P): Problems that can be solved in polynomial time.
    • Non-deterministic Polynomial Time (NP): Problems verifiable in polynomial time.
    • NP-Complete: Importance and significance in real-world problems.
    • Reduction: Polynomial-time reductions and transforming problems into NP-Complete form.

  • Intractability and Approximation Algorithms:

    • Strategies for solving NP-Hard problems through approximation.
    • Examples: Traveling Salesman Problem, Vertex Cover, Set Cover, Approximate Matching.

  • Computability and Undecidability:

    • Turing Machines: Formalization of computation.
    • Halting Problem: Example of an undecidable problem.
    • Church-Turing Thesis: The theoretical limits of computation.

  • Resources:

2. Advanced Data Structures

  • Self-Balancing Trees (Advanced):

    • Splay Trees: Tree rebalancing through splaying, amortized complexity, and use cases.
    • B-Trees: Optimized for reading and writing large blocks of data, commonly used in databases and file systems.
    • B+ Trees: Extending B-Trees for range queries in databases.

  • Segment Trees and Fenwick Trees:

    • Segment Trees: Efficient range queries and point updates.
    • Fenwick Trees (Binary Indexed Trees): More memory-efficient alternative to segment trees, with focus on cumulative frequency tables.

  • Persistent Data Structures:

    • Introduction to Persistent Trees: How to make data structures immutable and keep previous versions (useful for undo functionality).
    • Trie (Radix Trees): Efficient storage for dynamic sets of strings, applications in IP routing, dictionaries, and autocomplete features.

  • Suffix Trees and Arrays:

    • Suffix Trees: Linear-time pattern matching, construction, and usage in text-based search algorithms.
    • Suffix Arrays: Space-efficient alternative to suffix trees with focus on solving string problems.

  • Skip Trees:

    • Combining the efficiency of Skip Lists with the hierarchical structure of trees to achieve balanced search structures.

  • Resources:

3. Advanced Graph Algorithms and Theoretical Concepts

  • Graph Theory Concepts:

    • Graph Isomorphism: Definition and significance, and current open questions in complexity theory.
    • Planar Graphs: Special properties, algorithms for planarity testing, and applications.

  • Advanced Pathfinding Algorithms:

    • Johnson's Algorithm: All-pairs shortest path problem, using Bellman-Ford and Dijkstra’s algorithms in combination.
    • Edmonds-Karp Algorithm: Implementation of the Ford-Fulkerson method for solving the max-flow problem.

  • Matching and Network Flow:

    • Bipartite Graph Matching: Hungarian algorithm for maximum matching in bipartite graphs.
    • Min-Cut Max-Flow Theorem: The relation between network flow and cut problems, applications in network routing.

  • Approximation Algorithms for Hard Graph Problems:

    • Approximate Vertex Cover: Using greedy techniques for NP-hard graph problems.
    • Approximate Traveling Salesman Problem: Christofides’ algorithm for near-optimal solutions.

  • Graph Embedding:

    • Understanding and applying graph embeddings in machine learning and computational geometry.

  • Resources:

4. Parallel and Distributed Algorithms

  • Introduction to Parallel Computing:

    • Understanding Parallelism vs Concurrency.
    • Parallel algorithms for sorting, searching, and matrix multiplication (e.g., Parallel Merge Sort, Parallel Matrix Multiply).
    • MapReduce Paradigm: Decomposing problems for distributed systems.

  • Distributed Data Structures:

    • Data consistency models in distributed systems (e.g., CAP Theorem).
    • Designing scalable, fault-tolerant distributed hash tables (DHTs), such as Chord and CAN (Content Addressable Networks).

  • Consensus Algorithms:

    • Introduction to Paxos and Raft algorithms for achieving consensus in distributed systems.
    • Fault-tolerance and leader election.

  • Blockchain and Distributed Ledger Algorithms:

    • Deep dive into consensus protocols used in blockchain (e.g., Proof of Work, Proof of Stake).
    • Optimizing distributed systems for decentralized networks.

  • Parallel Dynamic Programming:

    • Efficient parallelization strategies for solving dynamic programming problems (e.g., parallelized knapsack problem).

  • Resources:

5. Advanced Optimization Techniques

  • Linear and Non-Linear Programming:

    • Introduction to Linear Programming (LP): Applications, simplex algorithm, duality theory.
    • Non-linear Optimization: Techniques for solving optimization problems where the objective function or constraints are non-linear.

  • Integer Programming:

    • Branch and Bound: General method for solving NP-hard problems in integer programming.

  • Convex Optimization:

    • Understanding the properties of convex functions and why convex optimization problems are efficiently solvable.
    • Applications of convex optimization in machine learning, signal processing, and economics.

  • Metaheuristics:

    • Genetic Algorithms: Evolutionary approaches to optimization problems.
    • Simulated Annealing: Probabilistic optimization technique inspired by annealing in metallurgy.
    • Ant Colony Optimization: Bio-inspired algorithms for solving combinatorial optimization problems.

  • Dynamic Graph Algorithms:

    • Techniques for efficiently updating graph algorithms when the graph changes dynamically (e.g., edge insertions/deletions).

  • Resources:

6. Quantum Algorithms

  • Introduction to Quantum Computing:

    • Overview of Quantum Gates and Quantum Circuits.
    • Quantum bits (qubits), superposition, and entanglement as quantum computing foundations.

  • Shor’s Algorithm:

    • Quantum algorithm for integer factorization, crucial for breaking RSA encryption.

  • Grover’s Algorithm:

    • Quadratically faster search algorithm for unsorted databases, applications in optimization problems.

  • Quantum Error Correction:

    • Ensuring fault tolerance in quantum computations through error-correcting codes.

  • Adiabatic Quantum Computing:

    • Alternative quantum computation model based on evolving quantum systems.

  • Quantum Approximation Algorithms:

    • Variational quantum algorithms for solving combinatorial problems on near-term quantum computers.

  • Resources:

7. Machine Learning and Algorithms

  • Optimization in Machine Learning:

    • Stochastic Gradient Descent (SGD): Optimization for large-scale machine learning models.
    • Conjugate Gradient Method: Optimization for convex problems with many variables.

  • Deep Learning Algorithms:

    • Understanding backpropagation, gradient descent, and optimization for neural networks.
    • Implementing deep neural networks with real-world datasets.

  • Reinforcement Learning Algorithms:

    • Q-Learning, Policy Gradient, and Monte Carlo methods for decision-making problems.

  • Graph Neural Networks (GNN):

    • Neural networks applied to graph-structured data, used for node classification and link prediction in graph-based systems.

  • Resources:

8. Advanced Computational Geometry

  • Geometric Data Structures:

    • Voronoi Diagrams: Efficiently dividing space for nearest neighbor queries.
    • Convex Hull Algorithms: Jarvis March, Graham Scan, and Quickhull.

  • Computational Geometry Applications:

    • Range searching, nearest neighbor search, and their applications in graphics, robotics, and GIS systems.

  • High-Dimensional Geometry:

    • Understanding the curse of dimensionality in high-dimensional data structures.
    • Algorithms for nearest-neighbor searches in high dimensions, such as locality-sensitive

hashing (LSH).