Concepts52
Sqrt Tree
A sqrt tree is a layered block-decomposition data structure that answers range queries in O(1) time after O(n \log \log n) preprocessing.
Segment Tree with Range Affine Transformation
A segment tree with lazy propagation can support range updates of the form x → a·x + b (affine transformations) and range-sum queries in O(log n) per operation.
Floor Sum Formula
The floor sum computes S(n,m,a,b) = sum_{i=0}^{n-1} floor((a i + b)/m) efficiently in O(log(min(a,m))) time.
DP on Broken Profile - Plug DP
Plug DP (DP on broken profile with plugs) sweeps a grid cell by cell while remembering how partial path segments cross the frontier as labeled “plugs.”
Matrix Exponentiation - Advanced
Matrix exponentiation turns repeated linear transitions into fast O(n^{3} log k) computation using exponentiation by squaring.
Top Tree
Top trees are dynamic tree data structures that represent a forest as a hierarchy of clusters, allowing O(log n) amortized time for link, cut, path queries/updates, and many subtree queries.
Partition Function
The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.
Generating Functions - OGF
An ordinary generating function (OGF) encodes a sequence (a_n) as a formal power series A(x) = \sum_{n \ge 0} a_n x^n.
Stirling Numbers of First Kind
Stirling numbers of the first kind count permutations by their number of cycles and connect power polynomials to rising/falling factorials.
Discrete Logarithm
The discrete logarithm problem asks for x such that g^x ≡ h (mod p) in a multiplicative group modulo a prime p.
Pollard's Rho Factorization
Pollard's Rho is a randomized algorithm that finds a non-trivial factor of a composite integer by walking a pseudorandom sequence modulo n and extracting a factor with a gcd.
Quadratic Residues
A quadratic residue modulo an odd prime p is any a for which x^2 ≡ a (mod p) has a solution; exactly half of the nonzero classes are residues.