Concepts158
Linear Sieve
The linear sieve builds all primes up to n in O(n) time by ensuring each composite is marked exactly once by its smallest prime factor (SPF).
Prime Factorization
Prime factorization expresses any integer greater than 1 as a product of primes raised to powers, uniquely up to ordering.
Extended Euclidean Algorithm
The Extended Euclidean Algorithm finds integers x and y such that ax + by = gcd(a, b) while also computing gcd(a, b).
Polynomial Operations
Fast polynomial operations treat coefficients like numbers but use FFT/NTT to multiply in O(n \log n) time instead of O(n^2).
Sieve of Eratosthenes
The Sieve of Eratosthenes marks multiples of each prime to find all primes up to n in O(n log log n) time.
GCD and Euclidean Algorithm
The greatest common divisor (gcd) of two integers is the largest integer that divides both without a remainder.
Convolution Applications
Convolution turns local pairwise combinations (like matching characters or adding two dice) into a single fast transform–multiply–inverse pipeline.
NTT (Number Theoretic Transform)
The Number Theoretic Transform (NTT) is an FFT-like algorithm that performs discrete convolutions exactly using modular arithmetic instead of floating-point numbers.
Matrix Exponentiation
Matrix exponentiation turns repeated linear transitions into a single fast power of a matrix using exponentiation by squaring.
Mo's Algorithm - With Updates
Mo's algorithm with updates treats array modifications as a third dimension called time and answers range queries on the correct version of the array.
DSU on Tree (Sack)
DSU on Tree (also called the Sack technique) answers many subtree queries in O(n \log n) by keeping data from the heavy child and temporarily re-adding light subtrees.
Small-to-Large Merging
Small-to-large merging is a technique where you always merge the smaller container into the larger one to guarantee low total work.