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Concepts172

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🗂️Data StructureAdvanced

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.

#top tree#dynamic tree#link cut+12
⚙️AlgorithmAdvanced

Voronoi Diagram and Delaunay

Voronoi diagrams partition the plane so each region contains points closest to one site, while the Delaunay triangulation connects sites whose Voronoi cells touch.

#voronoi diagram
7891011
Advanced
#delaunay triangulation
#fortune algorithm
+12
∑MathAdvanced

Generating Functions - EGF

Exponential generating functions (EGFs) encode a sequence (a_n) as A(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!}, which naturally models labeled combinatorial objects.

#exponential generating function#egf#binomial convolution+11
∑MathAdvanced

Pólya Enumeration

Pólya Enumeration Theorem generalizes Burnside’s Lemma by turning counting under symmetry into a polynomial substitution problem.

#pólya enumeration#cycle index#burnside lemma+12
∑MathAdvanced

Burnside's Lemma

Burnside's Lemma says the number of distinct objects up to a symmetry group equals the average number of objects fixed by each symmetry.

#burnside's lemma#cauchy-frobenius#polya enumeration+12
∑MathAdvanced

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.

#partition function#integer partitions#euler pentagonal theorem+11
∑MathAdvanced

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.

#ordinary generating function#ogf#coefficient extraction+12
∑MathAdvanced

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.

#stirling numbers of the first kind#unsigned cycle numbers#signed stirling numbers+12
∑MathAdvanced

Stirling Numbers of Second Kind

Stirling numbers of the second kind S(n,k) count how many ways to split n labeled items into k non-empty, unlabeled groups.

#stirling numbers of the second kind#set partitions#bell numbers+12
∑MathAdvanced

Primitive Roots

A primitive root modulo n is a number g that cycles through all units modulo n when you repeatedly multiply by g, so its multiplicative order equals \(\varphi(n)\).

#primitive root#multiplicative order#euler totient+10
∑MathAdvanced

Discrete Logarithm

The discrete logarithm problem asks for x such that g^x ≡ h (mod p) in a multiplicative group modulo a prime p.

#discrete logarithm#baby-step giant-step#pollard rho dlp+12
∑MathAdvanced

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.

#pollard's rho#integer factorization#cycle detection+10