The complexity ladder, Data Structure & Algorithms

Assignment Help:

The complexity Ladder:

  • T(n) = O(1). It is called constant growth. T(n) does not raise at all as a function of n, it is a constant. For illustration, array access has this characteristic. A[i] takes the identical time independent of the size of the array A.
  • T(n) = O(log2 (n)). It is called logarithmic growth. T(n) raise proportional to the base 2 logarithm of n. In fact, the base of logarithm does not matter. For instance, binary search has this characteristic.
  • T(n) = O(n). It is called linear growth. T(n) linearly grows with n. For instance, looping over all the elements into a one-dimensional array of n elements would be of the order of O(n).
  • T(n) = O(n log (n). It is called nlogn growth. T(n) raise proportional to n times the base 2 logarithm of n. Time complexity of Merge Sort contain this characteristic. Actually no sorting algorithm that employs comparison among elements can be faster than n log n.
  • T(n) = O(nk). It is called polynomial growth. T(n) raise proportional to the k-th power of n. We rarely assume algorithms which run in time O(nk) where k is bigger than 2 , since such algorithms are very slow and not practical. For instance, selection sort is an O(n2) algorithm.
  • T(n) = O(2n) It is called exponential growth. T(n) raise exponentially.

In computer science, Exponential growth is the most-danger growth pattern. Algorithms which grow this way are fundamentally useless for anything except for very small input size.

Table 1 compares several algorithms in terms of their complexities.

Table 2 compares the typical running time of algorithms of distinct orders.

The growth patterns above have been tabulated in order of enhancing size. That is,   

  O(1) <  O(log(n)) < O(n log(n)) < O(n2)  < O(n3), ... , O(2n).

Notation

Name

Example

O(1)

Constant

Constant growth. Does

 

 

not grow as a function

of n. For example, accessing array for one element A[i]

O(log n)

Logarithmic

Binary search

O(n)

Linear

Looping over n

elements, of an array of size n (normally).

O(n log n)

Sometimes called

"linearithmic"

Merge sort

O(n2)

Quadratic

Worst time case for

insertion sort, matrix multiplication

O(nc)

Polynomial,

sometimes

 

O(cn)

Exponential

 

O(n!)

Factorial

 

 

              Table 1: Comparison of several algorithms & their complexities

 

 

 

Array size

 

Logarithmic:

log2N

 

Linear: N

 

Quadratic: N2

 

Exponential:

2N

 

8

128

256

1000

100,000

 

3

7

8

10

17

 

8

128

256

1000

100,000

 

64

16,384

65,536

1 million

10 billion

 

256

3.4*1038

1.15*1077

1.07*10301

........

 


Related Discussions:- The complexity ladder

Enumerate about the carrier set members, Enumerate about the carrier set me...

Enumerate about the carrier set members Ruby is written in C, so carrier set members (which is, individual symbols) are implemented as fixed-size arrays of characters (which is

Explain about hubs, Hubs - In reality a multiport repeater - Connect...

Hubs - In reality a multiport repeater - Connects stations in a physical star topology - As well may create multiple levels of hierarchy to remove length limitation of 10

Explain floyds algorithm, Explain Floyd's algorithm It is convenient to...

Explain Floyd's algorithm It is convenient to record the lengths of shortest paths in an n by n matrix D known as the  distance matrix: the element d ij   in the i th   row an

A full binary tree with 2n+1 nodes, A full binary tree with 2n+1 nodes have...

A full binary tree with 2n+1 nodes have n non-leaf nodes

Define techniques of dry running of flowcharts, Explain the term- Dry runni...

Explain the term- Dry running of flowcharts  Dry running of flowcharts is essentially a technique to: Determine output for a known set of data to check it carries out th

#, write an algorithm to search a particular node in linked list which retu...

write an algorithm to search a particular node in linked list which returns " FOUND" or "NOT FOUND" as outcome.

Algorithm, Example of worse case of time

Example of worse case of time

Stacks, Q. Explain w hat are the stacks? How can we use the stacks  to chec...

Q. Explain w hat are the stacks? How can we use the stacks  to check whether an expression is correctly parentheses or not. For example (()) is well formed but (() or )()( is not w

Define complete binary tree, Define Complete Binary Tree Complete Binar...

Define Complete Binary Tree Complete Binary Tree:- A whole binary tree of depth d is that strictly binary tree all of whose leaves are at level D.

Searhing and sorting algorithms, how I can easily implement the bubble,sele...

how I can easily implement the bubble,selection,linear,binary searth algorithms?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd