The complexity ladder, Data Structure & Algorithms

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

........

 

Posted Date: 4/4/2013 6:21:54 AM | Location : United States







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