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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N as the result of input size n for each function. In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm.
Created independently in 1977 by W. Eddy and in 1978 by A. Bykat. Just like the quicksort algorithm, it has the expected time complexity of O(n log n), but may degenerate to O(n 2) in the worst case. Divide and conquer, a.k.a. merge hull — O(n log n) Another O(n log n) algorithm, published in 1977 by Preparata and Hong. This algorithm is also ...
In other words, a problem with input size n is in NC if there exist constants c and k such that it can be solved in time O((log n) c) using O(n k) parallel processors. Stephen Cook [1] [2] coined the name "Nick's class" after Nick Pippenger, who had done extensive research [3] on circuits with polylogarithmic depth and polynomial size. [4]
The worst-case complexity is the maximum of the complexity over all inputs of size n, and the average-case complexity is the average of the complexity over all inputs of size n (this makes sense, as the number of possible inputs of a given size is finite). Generally, when "complexity" is used without being further specified, this is the worst ...
The analysis of the former and the latter algorithm shows that it takes at most log 2 n and n check steps, respectively, for a list of size n. In the depicted example list of size 33, searching for "Morin, Arthur" takes 5 and 28 steps with binary (shown in cyan) and linear (magenta) search, respectively.
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.
It is possible to find the maximum clique, or the clique number, of an arbitrary n-vertex graph in time O (3 n/3) = O (1.4422 n) by using one of the algorithms described above to list all maximal cliques in the graph and returning the largest one. However, for this variant of the clique problem better worst-case time bounds are possible.
To show that NL is contained in C, we simply take an NL algorithm and choose a random computation path of length n, and execute this 2 n times. Because no computation path exceeds length n, and because there are 2 n computation paths in all, we have a good chance of hitting the accepting one (bounded below by a constant).