Time Complexity Solving a computational program

Time Complexity

Solving a computational program • Describing the general steps of the solution – Algorithm’s course • Use abstract data types and pseudo code – Data structures course • Implement the solution using a programming language and concrete data structures – Introduction to computer science

Pseudo Code • Pseudo code is a meta language, used to describe an algorithm for a computer program. • We use a notation similar to C or Pascal programming language. • Unlike real code pseudo code uses a free syntax to describe a given problem

Pseudo Code • Pseudo code does not deal with problems regarding a specific programming language, as data abstraction and error checking • Use indentation to distinguish between blocks of code (loop and conditional statements) • Accessing values of an array is expressed with brackets

Pseudo code • Accessing an attribute of an object is expressed by the attribute name followed by the object in brackets (length[a]) • More on pseudo code style in the text book

What is the running time of these methods ? Proc 2(A[1. . n]) i 1; j 1; s 0 repeat if A[i] < A[j] s s + 1 if j=n j=i + 1; i i+1 else j j+1 until i+j > 2 n Proc 1(A[1. . n]) s 0 for i = 1. . n do for j = (i +1). . n do if A[i] < A[j] s s + 1

Analysis of Bubble sort • Bubble sort (A) – 1. n length[A] – 2. for j n-1 to 1 – 3. for i 0 to j – 1 – 4. if A[i] > A[i + 1] – 5. Swap (A[i] , A[i +1])

Time Analysis • Best case – the array is already sorted and therefore no swap operations are required

Time Analysis • Worst case – the array is sorted in descending order and therefore all swap operations will be executed • For both inputs the solution requires time

Asymptotic Notation • Considering two algorithms, A and B, and the running time for each algorithm for a problem of size n is TA(n) and TB(n) respectively • It should be a fairly simple matter to compare the two functions TA(n) and TB(n) and determine which algorithm is the best!

Asymptotic Notation • Suppose the problem size is n 0 and that TA(n 0) < TB(n 0 ) Then clearly algorithm A is better than algorithm B for problem size n 0 • What happens if we do not know the problem size in advance ? If we can show that TA(n) < TB(n) regardless of n then algorithm A is better then algorithm B regardless of the problem size • Since we don’t know size in advance we tend to compare the asymptotic behavior of the two.

Asymptotic Upper Bound - O • O(g(n)) is the group of all functions f(n) which are non-negative for all integers, if there exists an integer n 0 and a constant c>0 such that for all integers

Big O notation cg(n) f(n)

Example

Asymptotic lower Bound - • is the group of all functions f(n) which are non-negative for all integers and if there exists an integer n 0 and a constant c>0 such that for all integers

Asymptotic lower Bound - f(n) cg(n)

Example

Asymptotic tight bound - • is the group of all functions f(n) which are non-negative for all integers and if there exists an integer n 0 and two constants such that for all integers

Asymptotic tight Bound dg(n) f(n) cg(n)

Example •

Asymptotic Notation • When we use the term f = O(n) we mean that the function f O(n) • When we write we mean that the aside from the function the sum includes an additional function from O(n) which we have no interest of stating explicitly

Example • Show that the function f(n)=8 n+128 =O(n 2) – lets choose c = 1, then

Conventions for using Asymptotic notation • drop all but the most significant terms – Instead of we write • drop constant coefficients – Instead of we use

Back to improved bubble sort • We improve the sorting algorithm so that if in a complete iteration over the array no swap operations were performed, the execution stops • Best case – • Worst case – • Average case –

Comparing functions

Example • Compare the functions

Example • Compare the functions • The logarithmic base does not change the order of magnitude

Example • Compare the functions

Properties of asymptotic notation • Transitivity:

Properties of asymptotic notation • Symmetry

Properties of asymptotic notation • Reflexivity:

Example • Give a proof or a counter example to the following statements: •

Example •

Example •

Example •

Example • Show that – 1. – 2.

Example • Is it true that for any two functions f, g either f=O(g) or g=O(f) ?