Algorithms & data structures project

Algorithms and data structures are fundamental to efficient code and good software design. Creating and designing excellent algorithms is required for being an exemplary programmer. This repository's goal is to demonstrate how to correctly implement common data structures and algorithms in the simplest and most elegant ways.

Contributing

This repository is contribution friendly 😃. If you'd like to add or improve an algorithm, your contribution is welcome! Please be sure to checkout the Wiki for instructions.

Data Structures

Dynamic Programming

Coin change problem - O(nW)
Edit distance - O(nm)
🎥Knapsack 0/1 - O(nW)
Knapsack unbounded (0/∞) - O(nW)
Maximum contiguous subarray - O(n)
Longest Common Subsequence (LCS) - O(nm)
Longest Increasing Subsequence (LIS) - O(n²)
Longest Palindrome Subsequence (LPS) - O(n²)
🎥Traveling Salesman Problem (dynamic programming, iterative) - O(n²2ⁿ)
Traveling Salesman Problem (dynamic programming, recursive) - O(n²2ⁿ)
Minimum Weight Perfect Matching (iterative, complete graph) - O(n²2ⁿ)

Geometry

Angle between 2D vectors - O(1)
Angle between 3D vectors - O(1)
Circle-circle intersection point(s) - O(1)
Circle-line intersection point(s) - O(1)
Circle-line segment intersection point(s) - O(1)
Circle-point tangent line(s) - O(1)
Closest pair of points (line sweeping algorithm) - O(nlog(n))
Collinear points test (are three 2D points on the same line) - O(1)
Convex hull (Graham Scan algorithm) - O(nlog(n))
Convex hull (Monotone chain algorithm) - O(nlog(n))
Convex polygon area - O(n)
Convex polygon cut - O(n)
Convex polygon contains points - O(log(n))
Coplanar points test (are four 3D points on the same plane) - O(1)
Line class (handy infinite line class) - O(1)
Line-circle intersection point(s) - O(1)
Line segment-circle intersection point(s) - O(1)
Line segment to general form (ax + by = c) - O(1)
Line segment-line segment intersection - O(1)
Longitude-Latitude geographic distance - O(1)
Point is inside triangle check - O(1)
Point rotation about point - O(1)
Triangle area algorithms - O(1)
[UNTESTED] Circle-circle intersection area - O(1)
[UNTESTED] Circular segment area - O(1)

Graph theory

Tree algorithms

Tree isomorphism - O(Elog(E))
Tree center(s) - O(V+E)
Tree diameter - O(V+E)

Network flow

Bipartite graph verification (adjacency list) - O(V+E)
🎥Max flow & Min cut (Ford-Fulkerson with DFS, adjacency list) - O(fE)
Max flow & Min cut (Ford-Fulkerson with DFS, adjacency matrix) - O(fV²)
🎥Max flow & Min cut (Edmonds-Karp, adjacency list) - O(VE²)
🎥Max flow & Min cut (Capacity scaling, adjacency list) - O(E²log₂(U))
🎥Max flow & Min cut (Dinic's, adjacency list) - O(EV²) or O(E√V) for bipartite graphs
Maximum Cardinality Bipartite Matching (augmenting path algorithm, adjacency list) - O(VE)
Min Cost Max Flow (Bellman-Ford, adjacency list) - O(E²V²)
Min Cost Max Flow (Johnson's algorithm, adjacency list) - O(E²Vlog(V))

Other graph theory

🎥Articulation points/cut vertices (adjacency list) - O(V+E)
Bellman-Ford (edge list, negative cycles, fast & optimized) - O(VE)
🎥Bellman-Ford (adjacency list, negative cycles) - O(VE)
Bellman-Ford (adjacency matrix, negative cycles) - O(V³)
🎥Breadth first search (adjacency list) - O(V+E)
Breadth first search (adjacency list, fast queue) - O(V+E)
🎥Bridges/cut edges (adjacency list) - O(V+E)
Find connected components (adjacency list, union find) - O(Elog(E))
Find connected components (adjacency list, DFS) - O(V+E)
Depth first search (adjacency list, iterative) - O(V+E)
Depth first search (adjacency list, iterative, fast stack) - O(V+E)
🎥Depth first search (adjacency list, recursive) - O(V+E)
🎥Dijkstra's shortest path (adjacency list, lazy implementation) - O(Elog(V))
🎥Dijkstra's shortest path (adjacency list, eager implementation + D-ary heap) - O(Elog_E/V(V))
🎥Eulerian Path (directed edges) - O(E+V)
🎥Floyd Warshall algorithm (adjacency matrix, negative cycle check) - O(V³)
Graph diameter (adjacency list) - O(VE)
Kruskal's min spanning tree algorithm (edge list, union find) - O(Elog(E))
🎥Kruskal's min spanning tree algorithm (edge list, union find, lazy sorting) - O(Elog(E))
🎥Prim's min spanning tree algorithm (lazy version, adjacency list) - O(Elog(E))
Prim's min spanning tree algorithm (lazy version, adjacency matrix) - O(V²)
🎥Prim's min spanning tree algorithm (eager version, adjacency list) - O(Elog(V))
Steiner tree (minimum spanning tree generalization) - O(V³ + V² * 2^T + V * 3^T)
🎥Tarjan's strongly connected components algorithm (adjacency list) - O(V+E)
Tarjan's strongly connected components algorithm (adjacency matrix) - O(V²)
🎥Topological sort (acyclic graph, adjacency list) - O(V+E)
Topological sort (acyclic graph, adjacency matrix) - O(V²)
Traveling Salesman Problem (brute force) - O(n!)
🎥Traveling Salesman Problem (dynamic programming, iterative) - O(n²2ⁿ)
Traveling Salesman Problem (dynamic programming, recursive) - O(n²2ⁿ)

Linear algebra

Freivald's algorithm (matrix multiplication verification) - O(kn²)
Gaussian elimination (solve system of linear equations) - O(cr²)
Gaussian elimination (modular version, prime finite field) - O(cr²)
Linear recurrence solver (finds nth term in a recurrence relation) - O(m³log(n))
Matrix determinant (Laplace/cofactor expansion) - O((n+2)!)
Matrix inverse - O(n³)
Matrix multiplication - O(n³)
Matrix power - O(n³log(p))
Square matrix rotation - O(n²)

Mathematics

[UNTESTED] Chinese remainder theorem
Prime number sieve (sieve of Eratosthenes) - O(nlog(log(n)))
Prime number sieve (sieve of Eratosthenes, compressed) - O(nlog(log(n)))
Totient function (phi function, relatively prime number count) - O(n^1/4)
Totient function using sieve (phi function, relatively prime number count) - O(nlog(log(n)))
Extended euclidean algorithm - ~O(log(a + b))
Greatest Common Divisor (GCD) - ~O(log(a + b))
Fast Fourier transform (quick polynomial multiplication) - O(nlog(n))
Fast Fourier transform (quick polynomial multiplication, complex numbers) - O(nlog(n))
Primality check - O(√n)
Primality check (Rabin-Miller) - O(k)
Least Common Multiple (LCM) - ~O(log(a + b))
Modular inverse - ~O(log(a + b))
Prime factorization (pollard rho) - O(n^1/4)
Relatively prime check (coprimality check) - ~O(log(a + b))

Other

Bit manipulations - O(1)
List permutations - O(n!)
🎥Power set (set of all subsets) - O(2ⁿ)
Set combinations - O(n choose r)
Set combinations with repetition - O((n+r-1) choose r)
Sliding Window Minimum/Maximum - O(1)
Square Root Decomposition - O(1) point updates, O(√n) range queries
Unique set combinations - O(n choose r)

Search algorithms

Binary search (real numbers) - O(log(n))
Interpolation search (discrete discrete) - O(n) or O(log(log(n))) with uniform input
Ternary search (real numbers) - O(log(n))
Ternary search (discrete numbers) - O(log(n))

Sorting algorithms

Bubble sort - O(n²)
Bucket sort - Θ(n + k)
Counting sort - O(n + k)
Heapsort - O(nlog(n))
Insertion sort - O(n²)
Mergesort - O(nlog(n))
Quicksort (in-place, Hoare partitioning) - Θ(nlog(n))
Selection sort - O(n²)

String algorithms

Booth's algorithm (finds lexicographically smallest string rotation) - O(n)
Knuth-Morris-Pratt algorithm (finds pattern matches in text) - O(n+m)
Longest Common Prefix (LCP) array - O(nlog(n)) bounded by SA construction, otherwise O(n)
🎥Longest Common Substring (LCS) - O(nlog(n)) bounded by SA construction, otherwise O(n)
🎥Longest Repeated Substring (LRS) - O(nlog(n))
Manacher's algorithm (finds all palindromes in text) - O(n)
Rabin-Karp algorithm (finds pattern matches in text) - O(n+m)
Substring verification with suffix array - O(nlog(n)) SA construction and O(mlog(n)) per query

License

This repository is released under the MIT license. In short, this means you are free to use this software in any personal, open-source or commercial projects. Attribution is optional but appreciated.