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Concepts158

Groups

📐Linear Algebra15📈Calculus & Differentiation10🎯Optimization14🎲Probability Theory12📊Statistics for ML9📡Information Theory10🔺Convex Optimization7🔢Numerical Methods6🕸Graph Theory for Deep Learning6🔵Topology for ML5🌐Differential Geometry6∞Measure Theory & Functional Analysis6🎰Random Matrix Theory5🌊Fourier Analysis & Signal Processing9🎰Sampling & Monte Carlo Methods10🧠Deep Learning Theory12🛡️Regularization Theory11👁️Attention & Transformer Theory10🎨Generative Model Theory11🔮Representation Learning10🎮Reinforcement Learning Mathematics9🔄Variational Methods8📉Loss Functions & Objectives10⏱️Sequence & Temporal Models8💎Geometric Deep Learning8

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🔷All∑Math⚙️Algo🗂️DS📚Theory

Level

AllBeginnerIntermediate
🗂️Data StructureAdvanced

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.

#segment tree#lazy propagation#affine update+12
⚙️AlgorithmIntermediate

Bitset Optimization

Bitset optimization exploits word-level parallelism so one CPU instruction processes 64 bits at once on typical 64-bit machines.

#bitset
12345
Advanced
Filtering by:
#competitive programming
#word-level parallelism
#bitmask
+12
∑MathAdvanced

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.

#floor sum#atcoder library#euclidean algorithm+12
⚙️AlgorithmIntermediate

Tree Isomorphism

Tree isomorphism asks whether two trees have exactly the same shape, ignoring vertex names.

#tree isomorphism#ahu algorithm#canonical form+12
∑MathIntermediate

Harmonic Lemma

The Harmonic Lemma says that the values of \lfloor n/i \rfloor only change about 2\sqrt{n} times, so you can iterate those value blocks in O(\sqrt{n}) instead of O(n).

#harmonic lemma#integer division trick#block decomposition+12
⚙️AlgorithmAdvanced

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.”

#plug dp#broken profile#hamiltonian path+12
⚙️AlgorithmAdvanced

Matrix Exponentiation - Advanced

Matrix exponentiation turns repeated linear transitions into fast O(n^{3} log k) computation using exponentiation by squaring.

#matrix exponentiation#adjacency matrix#walk counting+12
🗂️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
∑MathIntermediate

Game Theory - Nim

Nim is a two-player impartial game with several piles where a move removes any positive number of stones from exactly one pile.

#nim#game theory#xor+11
∑MathIntermediate

Game Theory - Calculation Techniques

Sprague–Grundy theory converts any impartial, normal-play game into an equivalent Nim heap using a Grundy number.

#sprague-grundy#grundy numbers#nim-sum+12
∑MathIntermediate

Linearity of Expectation Applications

Linearity of expectation says the expected value of a sum equals the sum of expected values, even if the variables are dependent.

#linearity of expectation#indicator variables#expected inversions+12
∑MathIntermediate

Expected Value

Expected value is the long-run average outcome of a random variable if you could repeat the experiment many times.

#expected value#linearity of expectation#indicator variables+12