Convolution is a mathematical operation used in signal processing, image processing, and other fields to combine two functions in order to produce a third function. It is defined as the integral of the product of two functions, one of which is flipped and shifted over time. It is often represented using the symbol “*” and is useful for filtering, smoothing, and other operations on signals and images.
Fast Convolution:
 Fast convolution for 64bit integers in competitive programming and provide an overview of the different algorithms and techniques that are commonly used.
 Additionally, it will explore the advantages and limitations of each method, as well as the tradeoffs that must be made when choosing a particular approach. So let us move forward and unlock the power of lightningfast calculations with the ultimate guide to Fast convolution for 64bit integers in competitive programming.
 Fast convolution is a technique used to efficiently calculate the convolution of two sequences, a, and b, which is defined as the sum of the products of the corresponding elements of a and b, shifted by different amounts.
The convolution operation is commonly represented by the symbol ‘*’ and can be represented mathematically as:
c[n] = sum(a[m] * b[n – m]) for m = 0 to n
Steps involved in the implementation of the code:
 Define the two functions (signal and filter)
 Flip the filter function.
 Shift the filter function over time.
 Calculate the product of the two functions.
 Integrate the product over time.
 Plot the result.
Below is the code for the above approach:
C++

Convolution result: 2.2
Popular Algorithms for Fast convolution are:
 Fast Fourier Transform (FFT) algorithm
 Karatsuba algorithm
Fast Fourier Transform (FFT) algorithm:
 This algorithm uses the properties of complex numbers and trigonometric functions to convert the convolution operation into a pointwise multiplication operation in the frequency domain.
 This greatly reduces the computational complexity of the operation and makes it possible to perform convolutions of large sequences in a relatively short amount of time.
 However, the FFT algorithm can be difficult to implement and may not be suitable for all types of problems.
Karatsuba algorithm:
 This algorithm is based on a divideandconquer approach and is often used to perform the multiplication of large integers.
 The Karatsuba algorithm can also be used for convolution by treating the input sequences as two large integers and then applying the multiplication algorithm to the sequences.
 The Karatsuba algorithm is relatively simple to implement and is often used as a fallback option when other algorithms are not suitable.
Karatsuba vs FFT algorithm:
 The Karatsuba algorithm is an efficient algorithm for multiplying large integers. It reduces the number of multiplications required by breaking the integers into smaller chunks and using a recursive approach.
 FFT (Fast Fourier Transform) is an efficient algorithm for calculating the discrete Fourier transform of a signal. It is widely used in signal processing and other fields to analyze signals and images.
 In terms of performance, FFT is generally considered to be faster than Karatsuba for large inputs. FFT algorithms take advantage of the symmetry and periodicity of the input signal to reduce the number of calculations required. However, the Karatsuba algorithm is more efficient for small inputs.
Conclusion:
 Fast convolution is a technique used to efficiently calculate the convolution of two sequences which is a fundamental operation in many areas of computer science, including competitive programming.
 For large integers, different algorithms such as FFT, Karatsuba, and ToomCook can be used, each with its own advantages and limitations.
 Additionally, techniques such as modulus operation, sliding window approach, and the use of efficient libraries can be used to optimize performance and reduce computational complexity.
 In competitive programming, it’s important to consider the specific requirements of the problem and choose the best combination of algorithms, techniques and optimizations accordingly.