Algorithm classification

1. Vector operations
   1.  Pairwise summation
      1.  Pairwise summation of numbers
      2.  Parallel prefix scan algorithm using pairwise summation
   2.  Uniform norm of a vector: Real version, serial-parallel variant
   3.  Dot product
   4.  The serial-parallel summation method
2. Matrix-vector operations
   1.  Dense matrix-vector multiplication
3. Matrix operations
   1.  Dense matrix multiplication
4. Matrix decompositions
   1. Triangular decompositions
      1. Gaussian elimination (finding the LU decomposition)
      2. Gaussian elimination
         1. 1. LU decomposition via Gaussian elimination
            2. Compact scheme for Gaussian elimination and its modifications
               1. Compact scheme for Gaussian elimination: Dense matrix
               2.  Compact scheme for Gaussian elimination and its modifications: Tridiagonal matrix
                  1. Compact scheme for Gaussian elimination: Tridiagonal matrix, serial variant
                  2.  Stone doubling algorithm for the LU decomposition of a tridiagonal matrix
                  3.  Serial-parallel algorithm for the LU decomposition of a tridiagonal matrix
         2. Gaussian elimination with pivoting
            1. Gaussian elimination with column pivoting
            2. Gaussian elimination with row pivoting
            3. Gaussian elimination with diagonal pivoting
            4. Gaussian elimination with complete pivoting
      3.  Cholesky method
         1.  Cholesky decomposition
      4. Available triangular decompositions for matrices of special form
   2. Unitary-triangular decompositions
      1.  Givens method
      2. Householder method (reflection method) for the QR-decomposition of a matrix
   3. Unitary Hessenberg decompositions
      1. Householder method (reflection method) for reducing a matrix to Hessenberg (bidiagonal) form
      2. Givens method (rotation method) for reducing a matrix to Hessenberg (bidiagonal) form
   4. Unitary-diagonal decompositions
      1.  Eigenvalue decomposition (finding eigenvalues and eigenvectors)
      2.  Singular value decomposition (finding singular values and singular vectors)
5. Solving systems of linear algebraic equations
   1. Direct methods
      1. Linpack benchmark)
      2. Matrices of a special form
         1. Triangular matrices
            1.  Forward substitution
            2.  Backward substitution
         2. Bidiagonal matrices
            1. 1. Forward and backward substitution for bidiagonal matrices
               2. Stone doubling algorithm for solving bidiagonal matrices
               3. Serial-parallel variant of the backward substitution
         3.  Methods for solving tridiagonal SLAEs
            1. Methods based on the conventional LU decomposition
               1. Thomas algorithm
                  1.  Thomas algorithm, pointwise version
                  2.  Repeated Thomas algorithm, pointwise version
               2.  Stone doubling algorithm
               3.  Serial-parallel method for solving tridiagonal matrices based on the LU decomposition and backward substitutions
            2. Other methods
               1. Reduction method
                  1. Complete reduction method
                  2. Reduction method repeated for a new right-hand side
               2. Two-sided Thomas algorithm
                  1.  Two-sided Thomas algorithm, pointwise version
                  2.  Repeated two-sided Thomas algorithm, pointwise version
               3. Cyclic reduction
                  1.  Complete cyclic reduction
                  2. Cyclic reduction repeated for a new right-hand side
               4. Bordering method
         4. Methods for solving block triangular matrices
            1. Block forward substitution (real version)
            2. Block backward substitution (real version))
            3. Methods for solving block bidiagonal matrices
               1. Forward and backward substitution for block bidiagonal matrices
               2. Stone doubling algorithm for solving block bidiagonal matrices
               3. Serial-parallel variant of the block backward substitution for solving block bidiagonal matrices
         5. Methods for solving block tridiagonal matrices
            1. Methods based on the conventional LU decomposition
               1.  Block Thomas algorithm
               2. Serial-parallel method for solving block systems of linear algebraic equations based on the LU decomposition and backward substitutions
            2. Other methods
               1.  Two-sided Thomas algorithm, block variant
               2. Block cyclic reduction
               3. Block bordering method
      3. Solving systems of linear algebraic equations with coefficient matrices of special form whose inverses are known
   2. Iterative methods for solving systems of linear algebraic equations
      1.  High Performance Conjugate Gradient (HPCG) benchmark
6. Solving eigenvalue problems
   1.  Eigenvalue decomposition (finding eigenvalues and eigenvectors)
      1.  QR algorithm
      2. Givens method (rotation method) for solving the symmetric eigenvalue problem
   2.  Singular value decomposition (finding singular values and singular vectors)
7. Computer benchmarks
   1.  High Performance Conjugate Gradient (HPCG) benchmark
   2.  Linpack benchmark
8. Fourier transform
   1.  Cooley–Tukey Fast Fourier Transform, radix-2 case
9. Algebra of polynomials
   1.  Horners method
10. Numerical quadrature
    1.  Cubature rules
    2.  Numerical quadrature (cubature) rules on an interval (for a multidimensional cube)
       1. Rectangle rule
       2. Trapezoid rule
       3. Simpson rule
       4. Gauss rule
11. Graph algorithms
    1. Graph traversal
       1.  Breadth-first search (BFS)
       2.  Depth-first search (DFS)
    2.  Single Source Shortest Path (SSSP)
       1.  Breadth-first search (BFS) (for unweighted graphs)
       2.  Dijkstra's algorithm
       3.  Bellman-Ford algorithm
       4.  Δ-stepping algorithm
    3.  All Pairs Shortest Path (APSP)
       1.  Johnson's algorithm
       2.  Floyd-Warshall algorithm
    4.  Transitive closure of a directed graph
       1.  Purdom's algorithm
    5.  Longest shortest path
    6.  Construction of the minimum spanning tree (MST)
       1. Boruvka's akgorithm
       2.  Kruskal's algorithm
       3.  Prim's algorithm
       4.  GHS algorithm
    7.  Search for isomorphic subgraphs
       1.  Ullman's algorithm
       2.  VF2 algorithm
    8.  Graph connectivity
       1.  Shiloach-Vishkin algorithm for finding the connected components
       2.  Disjoint set union
       3.  Tarjan's strongly connected components algorithm
       4.  DCSC algorithm for finding the strongly connected components
       5.  Tarjan's biconnected components algorithm
       6.  Tarjan-Vishkin biconnected components algorithm
       7.  Tarjan's algorithm for finding the bridges of a graph
       8.  Vertex connectivity of a graph
       9.  Gabow's edge connectivity algorithm
    9.  Finding maximal flow in a transportation network
       1.  Ford–Fulkerson algorithm
       2.  Preflow-Push algorithm
    10.  Finding minimal-cost flow in a transportation network
    11.  Assignment problem
        1.  Hungarian algorithm
        2.  Auction algorithm
        3.  Hopcroft–Karp algorithm
    12. Betweenness centrality algorithm
12. Search algorithms
    1.  Binary search: Finding the position of a target value within a sorted array, [math]O(log(n))[/math]
13. Sorting algorithms
    1. Binary tree sort
    2. Bubble sort
    3. Merge sort (serial and parallel variants)
14. Computational geometry
    1. Finding the diameter of a point set
    2. Finding the convex hull of a point set
    3. Delaunay triangulation
    4. Voronoi diagram
    5. Point-in-polygon problem
    6. Convex polygon intersection - complexity [math]O(n_1 + n_2)[/math]
    7. Star-shaped polygon intersection - complexity [math]O(n_1 * n_2)[/math]
15. Computer graphics
    1. Line drawing algorithms: Approximating a line segment on discrete graphical media
    2. Determining visible parts of a three-dimensional scene
    3. Ray tracing: Rendering realistic images
    4. Global illumination: Regarding direct illumination and reflection from other objects
16. Cryptographic algorithms
17. Neural networks
18. Optimization algorithms
    1. Linear programming
    2. Simplex algorithm
    3. Branch and bound method (serial and parallel variants)
    4. Genetic algorithms
    5. Ant colony algorithms
    6. Hybrid algorithms
    7. Finding extrema of a function
19. Game theory algorithms
20. Algorithms of quantum system simulation
    1. Algorithms of quantum computation simulation
       1.  Single-qubit transform of a state vector
       2.  Two-qubit transform of a state vector
       3. Quantum Fourier transform simulation
21. Algorithms for solving equations of mathematical physics
    1.  Poisson equation, solving with DFT
22. Other algorithms