Here’s a list of 100 chapter titles for Fast Fourier Transform (FFT) tailored for competitive programming, organized from beginner to advanced levels:
- Introduction to Fast Fourier Transform (FFT) in Competitive Programming
- Understanding Polynomials and Their Representations
- Coefficient Representation of Polynomials
- Point-Value Representation of Polynomials
- Converting Between Coefficient and Point-Value Representations
- Introduction to Complex Numbers
- Properties of Complex Numbers
- Roots of Unity: Definition and Properties
- Discrete Fourier Transform (DFT): Definition and Applications
- Naive DFT Algorithm: Implementation and Limitations
- Introduction to Divide and Conquer in Polynomial Multiplication
- Fast Fourier Transform (FFT): Intuition and Overview
- Recursive FFT Algorithm: Implementation
- Iterative FFT Algorithm: Implementation
- Inverse FFT: Recovering Coefficients from Point-Values
- Multiplying Polynomials Using FFT
- Comparing FFT with Naive Polynomial Multiplication
- Understanding the Time Complexity of FFT
- Handling Edge Cases in FFT
- FFT for Convolution: Introduction
- FFT for Cross-Correlation: Introduction
- FFT for Signal Processing: Basic Concepts
- FFT for Image Processing: Basic Concepts
- FFT for Audio Processing: Basic Concepts
- FFT for String Matching: Basic Concepts
- FFT for Number Theory: Basic Concepts
- FFT for Combinatorics: Basic Concepts
- FFT for Geometry: Basic Concepts
- FFT for Probability: Basic Concepts
- Basic Problems on FFT in Competitive Programming
- Optimizing FFT: Bit-Reversal Permutation
- Optimizing FFT: In-Place Computation
- Optimizing FFT: Precomputing Roots of Unity
- Optimizing FFT: Using SIMD Instructions
- Optimizing FFT: Parallelization Techniques
- FFT with Arbitrary Moduli: Introduction
- Number Theoretic Transform (NTT): Definition and Applications
- Implementing NTT for Polynomial Multiplication
- Comparing FFT and NTT
- FFT for Large Integers: Multiplication
- FFT for Large Integers: Division
- FFT for Large Integers: Exponentiation
- FFT for Large Integers: Factorials
- FFT for Large Integers: Binomial Coefficients
- FFT for Large Integers: Fibonacci Numbers
- FFT for Large Integers: Catalan Numbers
- FFT for Large Integers: Eulerian Numbers
- FFT for Large Integers: Stirling Numbers
- FFT for Large Integers: Bell Numbers
- FFT for Large Integers: Partition Numbers
- FFT for Large Integers: Harmonic Numbers
- FFT for Large Integers: Bernoulli Numbers
- FFT for Large Integers: Generating Functions
- FFT for Large Integers: Recurrence Relations
- FFT for Large Integers: Linear Recurrences
- FFT for Large Integers: Matrix Exponentiation
- FFT for Large Integers: Dynamic Programming
- FFT for Large Integers: Graph Theory
- FFT for Large Integers: Game Theory
- Intermediate Problems on FFT in Competitive Programming
- Advanced Techniques for FFT: Mixed-Radix FFT
- Advanced Techniques for FFT: Bluestein's Algorithm
- Advanced Techniques for FFT: Rader's Algorithm
- Advanced Techniques for FFT: Cooley-Tukey Algorithm
- Advanced Techniques for FFT: Good-Thomas Algorithm
- Advanced Techniques for FFT: Bruun's Algorithm
- Advanced Techniques for FFT: Winograd's Algorithm
- Advanced Techniques for FFT: Prime Factor Algorithm
- Advanced Techniques for FFT: Split-Radix Algorithm
- Advanced Techniques for FFT: Vector-Radix Algorithm
- Advanced Techniques for FFT: Multidimensional FFT
- Advanced Techniques for FFT: Non-Power-of-Two FFT
- Advanced Techniques for FFT: Real-Valued FFT
- Advanced Techniques for FFT: Half-Complex FFT
- Advanced Techniques for FFT: Fast Hartley Transform
- Advanced Techniques for FFT: Fast Cosine Transform
- Advanced Techniques for FFT: Fast Sine Transform
- Advanced Techniques for FFT: Fast Walsh-Hadamard Transform
- Advanced Techniques for FFT: Fast Wavelet Transform
- Advanced Techniques for FFT: Fast Gabor Transform
- Advanced Techniques for FFT: Fast Chirplet Transform
- Advanced Techniques for FFT: Fast Curvelet Transform
- Advanced Techniques for FFT: Fast Ridgelet Transform
- Advanced Techniques for FFT: Fast Contourlet Transform
- Advanced Techniques for FFT: Fast Shearlet Transform
- Advanced Techniques for FFT: Fast Radon Transform
- Advanced Techniques for FFT: Fast Hough Transform
- Advanced Techniques for FFT: Fast Sparse FFT
- Advanced Techniques for FFT: Fast Approximate FFT
- Advanced Problems on FFT in Competitive Programming
- Designing Custom FFT Algorithms for Specific Applications
- Designing Custom FFT Algorithms for Real-Time Systems
- Designing Custom FFT Algorithms for Large-Scale Systems
- Designing Custom FFT Algorithms for Parallel Systems
- Designing Custom FFT Algorithms for Distributed Systems
- Designing Custom FFT Algorithms for Secure Systems
- Designing Custom FFT Algorithms for High-Performance Systems
- Open Problems in FFT and Competitive Programming
- Research Trends in FFT
- Future Directions in FFT
This structured progression ensures a comprehensive understanding of FFT, from foundational concepts to advanced techniques, all tailored for competitive programming.