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Big-O Notation

Analysis of Algorithms

(how fast does an algorithm grow with respect to N)

(Note: Best recollection is that a good bit of this document comes from C++ For You++, by Litvin & Litvin)

The time efficiency of almost all of the algorithms we have discussed can be characterized by only a few growth rate functions:

I. O(l) - constant time

This means that the algorithm requires the same fixed number of steps regardless of the size of the task.

Examples (assuming a reasonable implementation of the task):

A. Push and Pop operations for a stack (containing n elements);

B. Insert and Remove operations for a queue.

II. O(n) - linear time

This means that the algorithm requires a number of steps proportional to the size of the task.

Examples (assuming a reasonable implementation of the task):

A. Traversal of a list (a linked list or an array) with n elements;

B. Finding the maximum or minimum element in a list, or sequential search in an unsorted list of n elements;

C. Traversal of a tree with n nodes;

D. Calculating iteratively n-factorial; finding iteratively the nth Fibonacci number.

III. O(n2) - quadratic time

The number of operations is proportional to the size of the task squared.

Examples:

A. Some more simplistic sorting algorithms, for instance a selection sort of n elements;

B. Comparing two two-dimensional arrays of size n by n;

C. Finding duplicates in an unsorted list of n elements (implemented with two nested loops).

IV. O(log n) - logarithmic time

Examples:

A. Binary search in a sorted list of n elements;

B. Insert and Find operations for a binary search tree with n nodes;

C. Insert and Remove operations for a heap with n nodes.

V. O(n log n) - "n log n " time

Examples:

A. More advanced sorting algorithms - quicksort, mergesort

VI. O(an) (a > 1) - exponential time

Examples:

A. Recursive Fibonacci implementation

B. Towers of Hanoi

C. Generating all permutations of n symbols

The best time in the above list is obviously constant time, and the worst is exponential time which, as we have seen, quickly

overwhelms even the fastest computers even for relatively small n. Polynomial growth (linear, quadratic, cubic, etc.) is

considered manageable as compared to exponential growth.

Order of asymptotic behavior of the functions from the above list:

Using the "<" sign informally, we can say that

O(l) < O(log n) < O(n) < O(n log n) < O(n2) < O(n3) < O(an)

A word about O(log n) growth:

As we know from the Change of Base Theorem, for any a, b > 0, and a, b ¹ 1

logb n =

b

n

a

...