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

What is bubble sort, What is bubble sort? Bubble Sort: The basic ide...

What is bubble sort? Bubble Sort: The basic idea in bubble sort is to scan the array to be sorted sequentially various times. Every pass puts the largest element in its corr

Determine the complexity, 1)    The set of the algorithms whose order is O ...

1)    The set of the algorithms whose order is O (1) would run in the identical time.  True/False 2)    Determine the complexity of the following program into big O notation:

State z-buffer algorithm, Z-Buffer Algorithm Also known as the Depth-Bu...

Z-Buffer Algorithm Also known as the Depth-Buffer algorithm, this image-space method simply selects for  display the polygon or portion of a polygon that is nearest to the view

Assignment, How do I submit a three page assignment

How do I submit a three page assignment

Calculate address of an element in an array., Q. Explain the technique to c...

Q. Explain the technique to calculate the address of an element in an array. A  25 × 4  matrix array DATA is stored in memory in 'row-major order'. If base  address is 200 and

Complexity of quick sort, Q. What do you mean by the best case complexity o...

Q. What do you mean by the best case complexity of quick sort and outline why it is so. How would its worst case behaviour arise?

First class Abstract data type , 3. A function to convert a complex number ...

3. A function to convert a complex number in algebraic form to a complex number in phasor form

If, 1. Start 2. Get h 3. If h T=288.15+(h*-0.0065) 4. else if h T=2...

1. Start 2. Get h 3. If h T=288.15+(h*-0.0065) 4. else if h T=216.65 5. else if h T=216.65+(h*0.001) 6. else if h T=228.65+(h*0.0028) 7. else if h T=270.65 8.

Relative and direct files, Each data record contains a fixed place in a rel...

Each data record contains a fixed place in a relative file. Each record ought to have associated with it in integer key value which will help identify this slot. Therefore, this ke

Program of implementation of stack using arrays, include int choice, st...

include int choice, stack[10], top, element; void menu(); void push(); void pop(); void showelements(); void main() { choice=element=1; top=0; menu()

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