Time Complexity: Graph & Machine Learning Algorithms

Summary of the time complexity of some common optimization and machine learning algorithms

Overview

Time complexity (or worst case scenario for the duration of execution given a number of elements) is commonly used in computing science. However, you will be hard pressed to find a comparison of machine learning algorithms using their asymptotic execution time.

Summary

The following summary list the time complexity of some of the most common algorithms used in machine learning including, recursive computation for dynamic programming, linear algebra, optimization and discriminative supervised training.

Algorithm	Time complexity	Description
Recursion with one element reduction	N2	N: Number of elements
Recursion with halving reduction	LogN
Recursion with halving reduction and processing	N
Nearest neighbors search	M.logk.N.LogN	M: number of features k: number of neighbors N: number of observations
Matrix multiplication (m,n)x(n,d)	m.n.d	m,n,d matrices dimension
Matrix multiplication (n,n)	n3	n matrix dimension
Matrix multiplication (n,n) Strassen	n2.8	n matrix dimension
Partial eigenvalues extraction (n,n)	e.N2	e: number of eigenvalues N: number of observations
Complete eigenvalues extraction (n,n)	N3	N: number of observations
Minimum spanning tree Prim linear queue	V2	V: number vertices
Minimum spanning tree Prim binary heap	(E + V).LogV	E: number of edges V: number vertices
Minimum spanning tree Prim Fibonacci heap	V.LogV	E: number of edges V: number vertices
Shortest paths Graph Dijkstra linear sorting	V2	V: number of vertices
Shortest paths Graph Dijkstra binary heap	(E + V).logV	V: number of vertices
Shortest paths Graph Dijkstra Fibonacci heap	V.log	E: number of edges V: number of vertices
Shortest paths kNN Graph - Dijkstra	(k + logN).N2	k: number of neighbors N: number of observations
Shortest paths kNN Graph - Floyd-Warshall	N3	N: number of observations
Fast Fourier transform	N.LogN	N: Number of observations
Batched gradient descent	N.P.I	N: Number of observations P: number of parameters I: number of iterations
Stochastic gradient descent	N.P.I	N: number of observations P: Number of variables I: number of epochs
Newton with Hessian computation	N3.I	N: number of observations I: number iterations
Conjugate gradient	N.m.sqrt(k)	N: number of observations m: number of non-zero k condition of the matrix
L-BFGS	N.M.I	N: number of observations M: estimated number of grads I: number of iterations
K-means (*)	C.N.M.I	C: Number of clusters M: Dimension N: number observations I: number of iterations
Lasso regularization - LARS(*)	N.M.min(N,M)	M: Dimension N: number observations
Hidden Markov model Forward-backward pass	N2.M	N: number of states M: number of observations
Multilayer Perceptron (*)	n.M.P.N.e	n: input variables M: number hidden neurons P: number output values N: number of observations e: Number of epochs
Support vector machine (*) using Newton	N3	N: number of observations
Support vector machine (*) using Cholesky	N2	N: number of observations
Support vector machine (*) Sequential minimal optimization	N2	N: number of observations
Principal Components Analysis (*)	M2N + N3	N: Number of observations M: number of features
Expectation-Maximization (*)	M2N	N: Number of observations M: number of variables
Laplacian Eigenmaps	M.logk.N.logN + m.N.k2 + d.N2	N: Number of observations M: number of variables
Genetic algorithms	P.logP.I.C	C: number of genes/chromosome P: population size I: Number of iterations

(*): Training

References

Introduction to Algorithms T. Cormen, C. Leiserson, R. Rivest - MIT Press 1990
Machine Learning: A probabilistic Perspective K. Murphy - MIT Press, 2012
Big O cheat sheet