The Ultimate Topic List

The Ultimate Topic List

A comprehensive list of topics in Competitive Programming

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Completed: ‌
out of 566

Copy Link to Basics

💡 Complete the basic topics in the given order to get started with Competitive Programming.

Intro to Programming

1.

What is Programming?

Learn a Language

💡 Learning C++ is recommended.

2.

Intro to C++ and Setting up the Environment

3.

Data Types, Variables and Input Output

4.

5.

Conditional Statements

6.

7.

8.

9.

10.

Pointers and References

11.

Intro to Competitive Programming

12.

What is Competitive Programming?

Bitwise Thinking

13.

Bitwise Operations and Bitmasks

Complexity Analysis

14.

Big O Notation and Time and Space Complexity Analysis

15.

Fast Input Output

Recursion

16.

Very Basic Math

17.

18.

19.

Harmonic Number

Range Queries Without Updates

20.

21.

Standard Template Library (STL)

22.

Intro to STL, Containers and Iterators

23.

Pairs and Tuples

24.

25.

26.

27.

28.

29.

Set, Unordered Set and Multiset

30.

Map and Unordered Map

31.

32.

Leveraging Monotonicity

33.

34.

Basic Sorting Algorithms

35.

36.

37.

38.

39.

40.

41.

42.

C++ STL Sort and Custom Comparator

Divide and Conquer

43.

Divide and Conquer

Greedy and Constructive

44.

Basic Greedy Algorithms

45.

Basic Constructive Algorithms

2D Prefix Sum

46.

More Techniques

47.

Difference Array

48.

Sliding Window Technique

Bit Manipulation

49.

Bit Manipulation

50.

Basic Modular Arithmetic

51.

Binary Exponentiation and Basic Modular Arithmetic

52.

Fermat's Little Theorem and Modular Inverse

Basic Counting

53.

Permutations, Combinations and Basic Counting Problems

54.

55.

Pigeonhole Principle

Very Basic Graphs

56.

Introduction to Graphs | Definitions and Representations

Copy Link to Data Structures (DS)

Segment Tree

57.

Segment Tree (Point Update Range Query)

58.

Segment Tree with Lazy Propagation

59.

Persistent Segment Tree

60.

Persistent Segment Tree with Lazy Propagation

61.

2D Segment Tree

62.

Dynamic Segment Tree

63.

2D Dynamic Segment Tree

64.

Merge Sort Tree

65.

Iterative Segment Tree

66.

Segment Tree Merging

67.

Segment Tree Beats

68.

XOR Segment Tree

69.

Historic Information on Segment Trees

Binary Indexed Tree (BIT)

70.

BIT / Fenwick Tree

71.

Lower bound on BIT

72.

BIT with Range Update and Range Query

73.

74.

2D BIT with Range Update and Range Query

Sparse Table

75.

76.

Sparse Table 2D

77.

Disjoint Sparse Table

Trie

78.

Trie on Strings and Bits

79.

Persistent Trie

Disjoint Set Union (DSU)

80.

DSU / Union Find

81.

DSU with Rollbacks

82.

Reachability Tree / DSU Tree / Kruskal Reconstruction Tree (KRT)

83.

Partially Persistent DSU

84.

85.

Augmented DSU / Weighted DSU

86.

DSU on Tree and Small to Large Merging

87.

[Trick] Maintaining Log N DSU / Range Parallel DSU

Square Root Decomposition

88.

Basic SQRT Decomposition

89.

Splitting Objects into Light and Heavy / Case Processing

90.

Batch Processing / SQRT Decomposition on Queries

91.

SQRT Decomposition Split and Build Technique

92.

SQRT Fragmented Tree / Block Tree / SQRT Decomposition on Trees

93.

94.

Rollback Mo and Mo's with DSU

95.

96.

97.

Mo's with Update / 3D Mo (Offline)

98.

Mo's with Update / 3D Mo (Online)

99.

Heavy Light Decomposition (HLD)

100.

Heavy Light Decomposition (HLD) ft Subtrees and Path Query

Centroid Decomposition

101.

Centroid Decomposition

102.

Persistent Centroid Decomposition

Long Path Decomposition

103.

Long Path Decomposition

BST

104.

Binary Search Tree (BST)

Balanced Binary Search Tree (BBST)

105.

106.

107.

Persistent Treap

108.

109.

Link Cut Tree (LCT)

110.

Link Cut Tree (LCT)

111.

Euler Tour Tree (ETT)

112.

Top Tree / AAA Tree

113.

Link Cut Cactus

Policy Based Data Structures

114.

115.

More Tree Data Structures

116.

117.

118.

K-Dimensional Tree (KD Tree)

119.

120.

Permutation Tree

121.

Dynamic Connectivity

122.

Dynamic Connectivity Problem (Offline)

123.

Dynamic Connectivity Problem (Online) / HDLT Algorithm

Miscellaneous

124.

Monotonic Queue

125.

Monotonic Queue 2D

126.

Monotonic Deque

127.

128.

Persistent Array

129.

Persistent Queue

130.

Persistent Meldable Heap / Leftist Heap

Techniques

131.

Venice Technique

132.

Offline Queries

133.

Static to Dynamic Trick / Log Decomposition

134.

Divide and Conquer on Queries

135.

CDQ Divide and Conquer

136.

Queue Undo Trick

137.

Priority Queue Undo Trick

Copy Link to Graph Theory

💡 Complete the "Very Basic Graphs" section under the "Basics" category first.

Graph Traversal

138.

Depth First Search (DFS)

139.

Breadth First Search (BFS)

140.

141.

Topological Sorting

142.

143.

Euler Tour Technique

144.

Lowest Common Ancestor (LCA)

145.

Binary Lifting and LCA

146.

Graph Connectivity

147.

Strongly Connected Components (SCC)

148.

Incremental SCC

149.

Articulation Bridges and Bridge Tree

150.

Online Articulation Bridges

151.

Articulation Points and Block Cut Tree

152.

Three Edge Connectivity

153.

Four Edge Connectivity

154.

Dynamic K-Connectivity

155.

Ear Decomposition

Orientation

156.

Strong Orientation

157.

ST-numbering / Bipolar Orientation

Minimum Spanning Tree (MST)

158.

Minimum Spanning Tree (Prim's and Kruskal's)

159.

Steiner Tree Problem

160.

Boruvka's Algorithm

161.

Minimum Diameter Spanning Tree

162.

163.

164.

165.

166.

Kirchoffs Theorem ft Number of MSTs

Shortest Paths

167.

Dijkstra's Algorithm

168.

[Trick] Dijkstra on Segment Tree

169.

170.

171.

Shortest Path Faster Algorithm (SPFA)

172.

Johnson's Alogrithm

173.

174.

Dial's algorithm

175.

Eppsteins Algorithm

176.

Suurballe's Algorithm

177.

Cycles

178.

Cycle Detection

179.

[Problem] Minimum Weight Cycle For Each Vertex

180.

[Problem] Minimum Weight Cycle For Each Edge

181.

Minimum Mean Weight Cycle

182.

Number of 3 and 4 length Cycles

More Tree Stuff

183.

Virtual Tree / Auxiliary Tree

184.

Rerooting Technique

185.

Union of Two Paths on a Tree

186.

Intersection of Two Paths on a Tree

187.

[Problem] Number of Paths of Each Length in a Tree

188.

Dynamic Online Tree Diameter

189.

[Problem] Tree Orientation to Maximize Pairs of Reachable Nodes

Satisfiability

190.

191.

192.

System Of Difference Constraints

Eulerian Path

193.

Eulerian Path and Cycle

Hamiltonian Path

194.

Hamiltonian Path and Cycle

195.

Hamiltonian Path Heuristic Algorithm

Cliques and Independent Sets

196.

197.

Maximum Independent Set (MIS)

Graph Coloring

198.

Chromatic Number

199.

Welsh-Powell Algorithm

200.

Chromatic Polynomial ft Number of DAGs

201.

Edge Coloring of Simple Graph

202.

Edge Coloring of Bipartite Graph

Graph Counting

203.

Counting Labeled Graphs

204.

205.

[Problem] Number of Arborescences with n Nodes

206.

Tuttes Theorem ft Arborescences in a Graph

207.

Dynamic Acyclic Graphs (DAG)

208.

DAG Reachability

209.

Incremental DAG Reachability

Dominators

210.

Special Graphs

211.

212.

Tournament Graph

213.

Flow and Min Cut

214.

Maximum Flow / Minimum Cut

215.

Min Cost Max Flow (MCMF)

216.

Min Cost Max Flow with Negative Cycles

217.

Maximum Closure Problem

218.

219.

220.

Gomory-Hu Tree of a Planar Graph

221.

Stoer Wagner Algorithm

222.

Chinese Postman Problem

223.

Uniqueness of Min Cut

224.

Maximum Density Subgraph

225.

Min Cut in a Planar Graph

226.

Max Cut in a Planar Graph

Matching

227.

Bipartite Matching (HopCroft Karp Algorithm and Kuhn's Algorithm)

228.

Hungarian Algorithm / Bipartite Weighted Matching

229.

Blossom Algorithm Unweighted / General Matching / Randomized General Matching

230.

Blossom Algorithm Weighted / General Weighted Matching

231.

Stable Marriage Problem

232.

Konig's Theorem and Maximum Independent Set in Bipartite Graphs

233.

234.

Matching using Determinants

Partially Ordered Set (POSET)

235.

POSET ft Dilworth's and Mirsky's Theorem

Miscellaneous

236.

Functional Graphs

237.

Degree Sequences

238.

Tree Isomorphism

239.

Binarizing a Tree

240.

Planarity Check

Copy Link to Number Theory

Primes and Divisors

241.

Sieve, Prime Factorization and Divisors

242.

243.

Segmented Sieve

244.

245.

Miller-Rabin Primality Test

246.

247.

Prime Counting Function

248.

[Problem] Counting Numbers Having Exactly K Divisors

249.

[Problem] Smallest Number Having Exactly K Divisors

250.

Sum of The Number of Divisors in cbrt(n)

251.

Wilson's Theorem

252.

Implicit Prime Factorization

Multiplicative Functions

253.

Number of Divisors / Sum of Divisors / Sigma Function

254.

Euler's Totient Function / Phi Function

255.

Power Tower / Generalized Euler theorem

256.

Mobius Function and Mobius Inversion

257.

Dirichlet convolution

258.

259.

Powerful Number Sieve / PN Sieve

Euclidean Algorithm

260.

Euclidean Algorithm

261.

Extended Euclid

262.

Bezout's Identity

263.

Chicken McNugget Theorem

Modular Arithmetic

264.

Linear Congruence Equation

265.

Chinese Remainder Theorem (CRT)

266.

267.

Multiplicative Order and Carmichael's Lambda Function

268.

Discrete Log / Baby Step Giant Step

269.

270.

Discrete Root in O(P^(1/4))

271.

Discrete Square Root, Tonelli Shanks Algorithm, Cipolla's Algorithm

272.

Number of Distinct Kth Powers Modulo N

273.

Number of Solutions to X^2 = 1 mod M

Linear Diophantine Equations

274.

Linear Diophantine Equation with Two Variables

275.

Trivariable Linear Diophantine Equation with Nonnegative Solutions

276.

Multivariable Linear Diophantine Equation with Nonnegative Solutions

277.

Linear Diophantine With N Unknowns and Two Equations

Solutions to Equations

278.

Number of Solutions to a Basic Linear Algebraic Equation

279.

Number of Solutions to a Basic Linear Algebraic Equation using Meet in the Middle

280.

Number of Nonnegative Integer Solutions to ax + by ≤ c

281.

Number of Integer Solutions to 1/x + 1/y = 1/k

282.

Two / Three / Four Squares Theorem

283.

Pell's Equation

Floor Sum

284.

285.

Floor Sum and Mod Sum of Arithmetic Progression

286.

Generalized Floor Sum of Arithmetic Progression

Fibonacci Numbers

287.

Fibonacci Numbers

288.

[Problem] LCM of Fibonacci Numbers

289.

290.

Miscellaneous

291.

Pythagorean Triplets

292.

Min of Mod of Arithmetic Progression

293.

Factorial Number System

294.

Stern-Brocot Tree and Rational Approximation

295.

[Problem] Number of a * x mod p in a Range

296.

[Problem] Smallest Nonnegative Integer x s.t. l ≤ a * x mod p ≤ r

Copy Link to Combinatorics

💡 Complete the "Basic Counting" section under the "Basics" category first.

Techniques

297.

Contribution Technique

Catalan

298.

Catalan Numbers

299.

Catalan Convolution

Inclusion Exclusion

300.

Principle of Inclusion and Exclusion (PIE)

301.

302.

Inclusion Exclusion on Multiples

Factorials and Binomial Coefficients (nCr)

303.

304.

nCr Modulo Any Mod

305.

Large Factorials and Binomial Coefficients

306.

Legendre's Formula

307.

Prefix Sum Queries of Binomial Coefficients

308.

Prefix Sum of Binomial Coefficients for a Fixed Large N

309.

Sum of nCr Over a Fixed Congruence Class

310.

[Problem] Sum of nCr(a[i], K) for each K from 1 to N

Stirling Numbers

311.

Stirling Numbers (Basic)

312.

Stirling Number of the First Kind for Fixed N

313.

Stirling Number of the First Kind for Fixed K

314.

Stirling Number of the Second Kind for Fixed N

315.

Stirling Number of the Second Kind for Fixed K

Partitions

316.

Partitions of an Integer, Partition Function and Pentagonal Number Theorem

317.

Partitions of a Set / Bell Number

Miscellaneous

318.

Burnside's Lemma / Polya Enumeration Theorem

319.

Bertrand's Ballot Theorem

320.

Young Tableaus and the Hook Length Formula

Copy Link to Math

Convolution

321.

Fast Fourier Transform (FFT)

322.

Number Theoretic Transform (NTT)

323.

324.

Online FFT / NTT

325.

Fast Walsh Hadamard Transform (FWHT) and its Variations

326.

GCD Convolution

327.

LCM Convolution

Polynomial

328.

Generating Functions

329.

Polynomial Inverse

330.

331.

332.

333.

Polynomial Sqrt

334.

Polynomial Composition

335.

Multipoint Evaluation

336.

Chirp Z Transform

337.

Polynomial Interpolation

338.

Polynomial Shift

339.

Shift of Sampling Points of Polynomial

340.

Polyomial Division and Remainder

341.

Polynomial Roots Under a Modulo

342.

Resultant of Two Polynomials and Half-GCD Algorithm

343.

Lagrange Interpolation

344.

[Problem] Sum of r^i * poly(i)

345.

Polynomial with Binomial Coefficients

346.

Subset Sum Problem using Generating Functions

347.

Polynomial Factorization of (x^n - 1) and Cyclotomic Polynomials

348.

Schwartz–Zippel Lemma

349.

Lagrange Inversion Theorem

Matrices

350.

Matrix Exponentiation

351.

Permanent of a Matrix

352.

Freivald's Algorithm

353.

Vandermonde Matrix

354.

Hafnian of a Matrix / Number of Perfect Matchings in a Graph

355.

Frobenius Form of a Matrix and Fast Matrix Power

Linear Recurrence

356.

Linear Recurrence Relations

357.

Linear Recurrence using Cayley-Hamilton Theorem in O(N^2 log K)

358.

Linear Recurrence using Generating Functions in O(N log N log K)

359.

Linear Recurrence with Polynomial Coefficients

360.

Linear Recurrence on Matrices

361.

Generating Function of a Linear Recurrence

362.

Berlekamp Messey

363.

Reeds-Sloane Algorithm

Linear Algebra

364.

Gaussian Elimination

365.

Gaussian Elimination Modulo 2

366.

Determinant of a Matrix

367.

Determinant under Composite Modulo

368.

Characteristic Polynomial and Hesserberg Matrix

369.

[Problem] Determinant of Product Matrix

370.

Determinant of Sparse Matrix and Black Box Linear Algebra

371.

[Problem] Determinant of Permutant and Circulant Matrix

372.

GCD Determinant / Smith's Determinant

373.

Cauchy–Binet formula

374.

Inverse of a Matrix

375.

Inverse of a Matrix modulo 2

376.

Thomas Algorithm

377.

Vector Space and XOR Basis

378.

[Technique] XOR Basis ft Reduced Row Echelon Form

379.

[Technique] XOR Basis ft Lexicographically Largest Basis

380.

Formulas

381.

Gauss's Eureka Theorem

382.

383.

Lifting-the-exponent(LTE) Lemma

384.

Faulhaber's Formula and Bernoulli Numbers

Calculus

385.

Differentiation and Integration

386.

Probabilities and Expected Values

387.

Probabilities and Expected Values

388.

Expected Value Powers Technique

389.

The Slime Trick

390.

Expected Occurrence Time as Substring

391.

392.

Mathematical Optimization

393.

Lagrange Multiplier

394.

Linear Programming / Simplex Algorithm

395.

Duality in Linear Programming (LP <> Flow)

Miscellaneous

396.

Continued Fractions

397.

Polynomial Real Roots

398.

Finite Field Arithmetic Binary

Copy Link to Strings

String Matching

399.

Knuth Morris Pratt (KMP) and Prefix Automaton

400.

401.

String Matching using Bitsets

402.

String Matching with FFT ft Wildcard Matching

Hashing

403.

404.

2D String Hashing

405.

406.

Multiset Hashing

Aho Corasick

407.

408.

Dynamic Aho Corasick

Suffix Structures

409.

410.

Isomorphic Suffix Array

411.

Dynamic Suffix Array

412.

Suffix Automaton

413.

[Problem] Distinct Substring Queries in Range

414.

Palindromes

415.

Palindromic Tree

416.

Persistent Palindromic Tree

417.

Manachers Algorithm

418.

[Problem] Minimum Palindrome Factorization

419.

[Problem] Number of Palindromes in Range

Longest Common Subsequence (LCS)

420.

Longest Common Subsequence (LCS)

421.

All Substring Longest Common Subsequence

422.

423.

424.

LCS on RLE compressed string

Miscellaneous

425.

Balanced Brackets

426.

Expression Parsing

427.

Lyndon Factorization / Duval Algorithm

428.

Finding Repetitions / Main-Lorentz Algorithm

429.

De Bruijn Sequence

Copy Link to Dynamic Programming (DP)

Intro to DP

430.

Knapsack and Basic Dynamic Programming

431.

Knapsack and Subset Sum Optimizations

Digit DP

432.

DP Optimizations

433.

Divide and Conquer Optimization

434.

Knuth Optimization

435.

Aliens Trick / WQS Binary Search

436.

1D1D DP Optimization

437.

DP Optimization using Data Structures [Some Practice Problems]

Convex Hull Trick (CHT)

438.

Convex Hull Trick (CHT) and Dynamic CHT

439.

Li Chao Tree

440.

441.

Persistent Li Chao Tree

442.

Extended Li Chao tree

Bit Related DP

443.

444.

Sum Over Subsets DP (SOS DP)

445.

[Problem] Subset Union of Bitsets

446.

Subset Convolution

447.

[Trick] Dynamic Submask Count

448.

[Problem] XOR Equation

More Tricks

449.

Open and Close Interval Trick

450.

451.

452.

Combining Subtrees Technique

Classic DP

453.

Longest Increasing Subsequence (LIS)

454.

Interval DP / Range DP / Matrix Chain Multiplication (MCM)

455.

Dp on Different Structures

456.

DP on Trees and DAGs

457.

DP over Divisors

458.

459.

DP on Convex Hulls

Miscellaneous

460.

Connected Component DP

461.

Broken Profile DP / Plug DP

462.

Hirschberg's Algorithm

463.

[Problem] Number of Subsequences Having Product at least K

464.

[Problem] Range LIS Query

Copy Link to Game Theory

Grundy Number / Nim

465.

Grundy Number and Basic Game Theory

466.

Hackenbush

467.

Green Hackenbush on Trees and Graphs

468.

Blue Red HackenBush

Graph Games

469.

Games on Arbitrary Graphs

470.

[Problem] Matching Game On A Graph

Miscellaneous

471.

Alpha Beta Pruning

Copy Link to Geometry

Euclidean Geometry

472.

Euclidean Geometry

Geometry 2D

473.

Points, Vectors, Transformations, Dot Products, Cross Products, Precision Issues and Polar Sort(2D)

474.

Lines, Segments, Rays, Distance and Relation Between Two Structures, Projections and Reflections(2D)

475.

Circles, Circle Relations, Intersections, Inscribed Circles and Circumcircles(2D)

476.

477.

Circles of Apollonius(2D)

478.

Circle Union(2D)

479.

480.

Centroid of a Polygon(2D)

481.

Geometric Median(2D)

482.

Convex Hull(2D)

483.

Point in Convex Polygon in log(N)(2D)

484.

Winding Number and Point in Simple Polygon(2D)

485.

Extreme Vertex(2D)

486.

Rotating Calipers ft Diameter and Width of a Convex Polygon and Minimum Enclosing Rectangle(2D)

487.

Minimum Enclosing Circle(2D)

488.

Polygon Line and Convex Line Intersection, and Polygon Cutting(2D)

489.

Tangents from Point to Convex and Convex to Convex(2D)

490.

Distance from Point to Convex, Convex to Line and Convex to Convex(2D)

491.

Maximum Distance from Convex to Convex(2D)

492.

Polygon Union(2D)

493.

Polygon Triangulation(2D)

494.

Maximum Circle Cover(2D)

495.

Maximum Inscribed Circle in a Convex(2D)

496.

Triangle Circle and Polygon Circle Intersection(2D)

Geometry 3D

497.

Points, Vectors, Dot Products, Cross Products and Mixed Products(3D)

498.

499.

Plane Based Coordinate System(3D)

500.

Lines, Segments, and Distances, Angles and Intersections Between Lines and Planes(3D)

501.

Triangles and Distance from Triangle to Triangle(3D)

502.

Spherical Geometry and Winding Number(3D)

503.

Minimum Enclosing Sphere(3D)

504.

Polyhedron, Pyramid, Cone and Cylinder(3D)

505.

Convex Hull(3D)

Delaunay

506.

Delaunay Triangulation and Voronoi Diagram

Half Plane

507.

Half Plane Intersection

508.

Dynamic Half Plane Intersection

Convex Polygon

509.

510.

Dynamic Convex Hull

511.

Onion Decomposition

Lattice Points

512.

513.

Lattice Points inside a Non-Lattice Polygon

Sweep Line

514.

Sweep Line Algorithm and Radial Sweep

515.

Closest Pair of Points

516.

Rectangle Union

517.

All Pair Segment Intersection.

518.

Vertical decomposition

Planar Geometry

519.

Faces in Planar Graph

520.

Miscellaneous

521.

Hill Climbing and Simulated Annealing

522.

[Trick] Linear Combination of Vectors

523.

Reflections on a Rectangle

524.

[Problem] Maximum Area of a Triangle from Given Lengths

525.

[Problem] Maximum Area of a Polygon using Given Lengths in CCW Order

526.

[Problem] Circle/Polygon Inclusion Tree

Copy Link to Miscellaneous

Greedy Algorithms

527.

Activity Selection Problem

528.

Fractional Knapsack Problem

529.

530.

Exchange Arguments

Search Techniques

531.

532.

Parallel Binary Search

533.

Fraction Binary Search

534.

Fracturing Search

Interactive Problems

535.

Interactive Problems

536.

Interactive Problems on Centroids

Permutations

537.

Permutations and Permutation Cycles

538.

[Problem] K-th Root of a Permutation

539.

Schreier–Sims Algorithm

Matroids

540.

Matroid Intersection

Chess

541.

[Problem] Knight Moves in Infinity Grid

542.

[Problem] Knight's Tour Problem

543.

[Problem] Bishop Placement

Techniques

544.

Coordinate Compression

545.

Direction Array

546.

Meet in the Middle

547.

Invariants and Monovariants

548.

Manhattan <> Chebyshev

549.

LCM OR Technique

550.

[Trick] Logarithmic Subarray Aggregator

551.

Bitwise Disjoint Partitioning

Min Plus Convolution

552.

Min Plus Convolution (Convex and Convex)

553.

SMAWK Algorithm and Min Plus Convolution (Convex and Arbitrary)

Randomized

554.

Randomized Algorithms

555.

Birthday Paradox

Bigint

556.

557.

Miscellaneous

558.

559.

Josephus Problem

560.

561.

All Nearest Smaller Values and All Subarray Maximum/Minimum

562.

563.

[Problem] MEX of all Subarrays

564.

Lindstrom Gessel Viennot(LGV) Lemma

565.

Dynamic Bitsets

566.

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VisuAlgo — Visualising data structures and algorithms through animation
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