The Ultimate Topic List

The Ultimate Topic List

A comprehensive list of topics in Competitive Programming

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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

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.

Euclidean Algorithm

259.

Euclidean Algorithm

260.

Extended Euclid

261.

Bezout's Identity

262.

Chicken McNugget Theorem

Modular Arithmetic

263.

Linear Congruence Equation

264.

Chinese Remainder Theorem (CRT)

265.

266.

Multiplicative Order and Carmichael's Lambda Function

267.

Discrete Log / Baby Step Giant Step

268.

269.

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

270.

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

271.

Number of Distinct Kth Powers Modulo N

272.

Number of Solutions to X^2 = 1 mod M

Linear Diophantine Equations

273.

Linear Diophantine Equation with Two Variables

274.

Trivariable Linear Diophantine Equation with Nonnegative Solutions

275.

Multivariable Linear Diophantine Equation with Nonnegative Solutions

276.

Linear Diophantine With N Unknowns and Two Equations

Solutions to Equations

277.

Number of Solutions to a Basic Linear Algebraic Equation

278.

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

279.

Number of Nonnegative Integer Solutions to ax + by ≤ c

280.

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

281.

Two / Three / Four Squares Theorem

282.

Pell's Equation

Floor Sum

283.

284.

Floor Sum and Mod Sum of Arithmetic Progression

285.

Generalized Floor Sum of Arithmetic Progression

Fibonacci Numbers

286.

Fibonacci Numbers

287.

[Problem] LCM of Fibonacci Numbers

288.

289.

Miscellaneous

290.

Pythagorean Triplets

291.

Min of Mod of Arithmetic Progression

292.

Factorial Number System

293.

Stern-Brocot Tree and Rational Approximation

294.

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

295.

[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

296.

Contribution Technique

Catalan

297.

Catalan Numbers

298.

Catalan Convolution

Inclusion Exclusion

299.

Principle of Inclusion and Exclusion (PIE)

300.

301.

Inclusion Exclusion on Multiples

Factorials and Binomial Coefficients (nCr)

302.

303.

nCr Modulo Any Mod

304.

Large Factorials and Binomial Coefficients

305.

Legendre's Formula

306.

Prefix Sum Queries of Binomial Coefficients

307.

Prefix Sum of Binomial Coefficients for a Fixed Large N

308.

Sum of nCr Over a Fixed Congruence Class

309.

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

Stirling Numbers

310.

Stirling Numbers (Basic)

311.

Stirling Number of the First Kind for Fixed N

312.

Stirling Number of the First Kind for Fixed K

313.

Stirling Number of the Second Kind for Fixed N

314.

Stirling Number of the Second Kind for Fixed K

Partitions

315.

Partitions of an Integer, Partition Function and Pentagonal Number Theorem

316.

Partitions of a Set / Bell Number

Miscellaneous

317.

Burnside's Lemma / Polya Enumeration Theorem

318.

Bertrand's Ballot Theorem

319.

Young Tableaus and the Hook Length Formula

Copy Link to Math

Convolution

320.

Fast Fourier Transform (FFT)

321.

Number Theoretic Transform (NTT)

322.

323.

Online FFT / NTT

324.

Fast Walsh Hadamard Transform (FWHT) and its Variations

325.

GCD Convolution

326.

LCM Convolution

Polynomial

327.

Generating Functions

328.

Polynomial Inverse

329.

330.

331.

332.

Polynomial Sqrt

333.

Polynomial Composition

334.

Multipoint Evaluation

335.

Chirp Z Transform

336.

Polynomial Interpolation

337.

Polynomial Shift

338.

Shift of Sampling Points of Polynomial

339.

Polyomial Division and Remainder

340.

Polynomial Roots Under a Modulo

341.

Resultant of Two Polynomials and Half-GCD Algorithm

342.

Lagrange Interpolation

343.

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

344.

Polynomial with Binomial Coefficients

345.

Subset Sum Problem using Generating Functions

346.

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

347.

Schwartz–Zippel Lemma

348.

Lagrange Inversion Theorem

Matrices

349.

Matrix Exponentiation

350.

Permanent of a Matrix

351.

Freivald's Algorithm

352.

Vandermonde Matrix

353.

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

354.

Frobenius Form of a Matrix and Fast Matrix Power

Linear Recurrence

355.

Linear Recurrence Relations

356.

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

357.

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

358.

Linear Recurrence with Polynomial Coefficients

359.

Linear Recurrence on Matrices

360.

Generating Function of a Linear Recurrence

361.

Berlekamp Messey

362.

Reeds-Sloane Algorithm

Linear Algebra

363.

Gaussian Elimination

364.

Gaussian Elimination Modulo 2

365.

Determinant of a Matrix

366.

Determinant under Composite Modulo

367.

Characteristic Polynomial and Hesserberg Matrix

368.

[Problem] Determinant of Product Matrix

369.

Determinant of Sparse Matrix and Black Box Linear Algebra

370.

[Problem] Determinant of Permutant and Circulant Matrix

371.

GCD Determinant / Smith's Determinant

372.

Cauchy–Binet formula

373.

Inverse of a Matrix

374.

Inverse of a Matrix modulo 2

375.

Thomas Algorithm

376.

Vector Space and XOR Basis

377.

[Technique] XOR Basis ft Reduced Row Echelon Form

378.

[Technique] XOR Basis ft Lexicographically Largest Basis

379.

Formulas

380.

Gauss's Eureka Theorem

381.

382.

Lifting-the-exponent(LTE) Lemma

383.

Faulhaber's Formula and Bernoulli Numbers

Calculus

384.

Differentiation and Integration

385.

Probabilities and Expected Values

386.

Probabilities and Expected Values

387.

Expected Value Powers Technique

388.

The Slime Trick

389.

Expected Occurrence Time as Substring

390.

391.

Mathematical Optimization

392.

Lagrange Multiplier

393.

Linear Programming / Simplex Algorithm

394.

Duality in Linear Programming (LP <> Flow)

Miscellaneous

395.

Continued Fractions

396.

Polynomial Real Roots

397.

Finite Field Arithmetic Binary

Copy Link to Strings

String Matching

398.

Knuth Morris Pratt (KMP) and Prefix Automaton

399.

400.

String Matching using Bitsets

401.

String Matching with FFT ft Wildcard Matching

Hashing

402.

403.

2D String Hashing

404.

405.

Multiset Hashing

Aho Corasick

406.

407.

Dynamic Aho Corasick

Suffix Structures

408.

409.

Isomorphic Suffix Array

410.

Dynamic Suffix Array

411.

Suffix Automaton

412.

[Problem] Distinct Substring Queries in Range

413.

Palindromes

414.

Palindromic Tree

415.

Persistent Palindromic Tree

416.

Manachers Algorithm

417.

[Problem] Minimum Palindrome Factorization

418.

[Problem] Number of Palindromes in Range

Longest Common Subsequence (LCS)

419.

Longest Common Subsequence (LCS)

420.

All Substring Longest Common Subsequence

421.

422.

423.

LCS on RLE compressed string

Miscellaneous

424.

Balanced Brackets

425.

Expression Parsing

426.

Lyndon Factorization / Duval Algorithm

427.

Finding Repetitions / Main-Lorentz Algorithm

428.

De Bruijn Sequence

Copy Link to Dynamic Programming (DP)

Intro to DP

429.

Knapsack and Basic Dynamic Programming

430.

Knapsack and Subset Sum Optimizations

Digit DP

431.

DP Optimizations

432.

Divide and Conquer Optimization

433.

Knuth Optimization

434.

Aliens Trick / WQS Binary Search

435.

1D1D DP Optimization

436.

DP Optimization using Data Structures [Some Practice Problems]

Convex Hull Trick (CHT)

437.

Convex Hull Trick (CHT) and Dynamic CHT

438.

Li Chao Tree

439.

440.

Persistent Li Chao Tree

441.

Extended Li Chao tree

Bit Related DP

442.

443.

Sum Over Subsets DP (SOS DP)

444.

[Problem] Subset Union of Bitsets

445.

Subset Convolution

446.

[Trick] Dynamic Submask Count

447.

[Problem] XOR Equation

More Tricks

448.

Open and Close Interval Trick

449.

450.

451.

Combining Subtrees Technique

Classic DP

452.

Longest Increasing Subsequence (LIS)

453.

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

454.

Dp on Different Structures

455.

DP on Trees and DAGs

456.

DP over Divisors

457.

458.

DP on Convex Hulls

Miscellaneous

459.

Connected Component DP

460.

Broken Profile DP / Plug DP

461.

Hirschberg's Algorithm

462.

[Problem] Number of Subsequences Having Product at least K

463.

[Problem] Range LIS Query

Copy Link to Game Theory

Grundy Number / Nim

464.

Grundy Number and Basic Game Theory

465.

Hackenbush

466.

Green Hackenbush on Trees and Graphs

467.

Blue Red HackenBush

Graph Games

468.

Games on Arbitrary Graphs

469.

[Problem] Matching Game On A Graph

Miscellaneous

470.

Alpha Beta Pruning

Copy Link to Geometry

Euclidean Geometry

471.

Euclidean Geometry

Geometry 2D

472.

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

473.

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

474.

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

475.

476.

Circles of Apollonius(2D)

477.

Circle Union(2D)

478.

479.

Centroid of a Polygon(2D)

480.

Geometric Median(2D)

481.

Convex Hull(2D)

482.

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

483.

Winding Number and Point in Simple Polygon(2D)

484.

Extreme Vertex(2D)

485.

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

486.

Minimum Enclosing Circle(2D)

487.

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

488.

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

489.

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

490.

Maximum Distance from Convex to Convex(2D)

491.

Polygon Union(2D)

492.

Polygon Triangulation(2D)

493.

Maximum Circle Cover(2D)

494.

Maximum Inscribed Circle in a Convex(2D)

495.

Triangle Circle and Polygon Circle Intersection(2D)

Geometry 3D

496.

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

497.

498.

Plane Based Coordinate System(3D)

499.

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

500.

Triangles and Distance from Triangle to Triangle(3D)

501.

Spherical Geometry and Winding Number(3D)

502.

Minimum Enclosing Sphere(3D)

503.

Polyhedron, Pyramid, Cone and Cylinder(3D)

504.

Convex Hull(3D)

Delaunay

505.

Delaunay Triangulation and Voronoi Diagram

Half Plane

506.

Half Plane Intersection

507.

Dynamic Half Plane Intersection

Convex Polygon

508.

509.

Dynamic Convex Hull

510.

Onion Decomposition

Lattice Points

511.

512.

Lattice Points inside a Non-Lattice Polygon

Sweep Line

513.

Sweep Line Algorithm and Radial Sweep

514.

Closest Pair of Points

515.

Rectangle Union

516.

All Pair Segment Intersection.

517.

Vertical decomposition

Planar Geometry

518.

Faces in Planar Graph

519.

Miscellaneous

520.

Hill Climbing and Simulated Annealing

521.

[Trick] Linear Combination of Vectors

522.

Reflections on a Rectangle

523.

[Problem] Maximum Area of a Triangle from Given Lengths

524.

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

525.

[Problem] Circle/Polygon Inclusion Tree

Copy Link to Miscellaneous

Greedy Algorithms

526.

Activity Selection Problem

527.

Fractional Knapsack Problem

528.

529.

Exchange Arguments

Search Techniques

530.

531.

Parallel Binary Search

532.

Fraction Binary Search

533.

Fracturing Search

Interactive Problems

534.

Interactive Problems

535.

Interactive Problems on Centroids

Permutations

536.

Permutations and Permutation Cycles

537.

[Problem] K-th Root of a Permutation

538.

Schreier–Sims Algorithm

Matroids

539.

Matroid Intersection

Chess

540.

[Problem] Knight Moves in Infinity Grid

541.

[Problem] Knight's Tour Problem

542.

[Problem] Bishop Placement

Techniques

543.

Coordinate Compression

544.

Direction Array

545.

Meet in the Middle

546.

Invariants and Monovariants

547.

Manhattan <> Chebyshev

548.

LCM OR Technique

549.

[Trick] Logarithmic Subarray Aggregator

550.

Bitwise Disjoint Partitioning

Min Plus Convolution

551.

Min Plus Convolution (Convex and Convex)

552.

SMAWK Algorithm and Min Plus Convolution (Convex and Arbitrary)

Randomized

553.

Randomized Algorithms

554.

Birthday Paradox

Bigint

555.

556.

Miscellaneous

557.

558.

Josephus Problem

559.

560.

All Nearest Smaller Values and All Subarray Maximum/Minimum

561.

562.

[Problem] MEX of all Subarrays

563.

Lindstrom Gessel Viennot(LGV) Lemma

564.

Dynamic Bitsets

565.

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