Algorithm Design Paradigms

Techniques for the Design of Algorithms

The first part of this course is concerned with:

Algorithmic Paradigms

That is:

General approaches to the construction of efficient solutions to problems.

Such methods are of interest because:

They provide templates suited to solving a broad range of diverse problems.
They can be translated into common control and data structures provided by most high-level languages.
The temporal and spatial requirements of the algorithms which result can be precisely analysed.

Over the next few lectures we shall examine the following paradigms:

Divide and Conquer.
Dynamic Programming
Greedy Method
Backtracking.

Although more than one technique may be applicable to a specific problem, it is often the case that an algorithm constructed by one approach is clearly superior to equivalent solutions built using alternative techniques.

The choice of design paradigm is an important aspect of algorithm synthesis.

Basic Algorithm Analysis

Questions

How does one calculate the running time of an algorithm?
How can we compare two different algorithms?
How do we know if an algorithm is `optimal'?

1. Count the number of basic operations performed by the algorithm on the worst-case input

A basic operation could be:

An assignment
A comparison between two variables
An arithmetic operation between two variables. The worst-case input is that input assignment for which the most basic operations are performed.

Simple Example:

n := 5;
loop
    get(m);
    n := n -1;
until (m=0 or n=0)

Worst-case: 5 iterations

Usually we are not concerned with the number of steps for a single fixed case but wish to estimate the running time in terms of the `input size'.

get(n);
loop
    get(m);
    n := n -1;
until (m=0 or n=0)

Worst-case: n iterations

Examples of `input size':

Sorting:

n == The number of items to be sorted;

Basic operation: Comparison.

Multiplication (of x and y):

n == The number of digits in x plus the number of digits in y.

Basic operations: single digit arithmetic.

Graph `searching':

n == the number of nodes in the graph or the number of edges in the graph.

Counting the Number of Basic Operations

Sequence: P and Q are two algorithm sections:


Time( P ; Q )  =  Time( P ) + Time( Q )

Iteration:

while < condition > loop
     P;
end loop;

for i in 1..n loop
    P;
end loop


Time  =  Time( P ) * ( Worst-case number of iterations )

Conditional

if < condition > then
    P;
else
    Q;
end if;


Time  =  Time(P)   if  < condition > =true
         Time( Q ) if  < condition > =false

We shall consider recursive procedures later in the course.

Example:

for i in 1..n loop
    for j in 1..n loop
       if i < j then
          swop (a(i,j), a(j,i)); -- Basic operation
       end if;
    end loop;
end loop;


Time  <  n*n*1
       =  n^2

Asymptotic Performance

We are usually concerned with the growth in running time as the input size increases, e.g.

Suppose P, Q and R are 3 algorithms with the following worst-case run times:

If each is run on a machine that executes one million (10^6) operations per second

Thus,

The growth of run-time in terms of n (2^n; n^2; n) is more significant than the exact constant factors (1; 5; 100)

`O'-notation

Let f:N-> R and g:N-> R. Then: f( n ) = O( g(n) ) means that

There are values, n0 and c, such that f( n ) < = c * g( n ) whenever n > = n0.

Thus we can say that an algorithm has, for example,

Run-time O( n^2 )

Examples

There is an algorithm (mergesort) to sort n items which has run-time O( n log n ).
There is an algorithm to multiply 2 n-digit numbers which has run-time O( n^2 ).
There is an algorithm to compute the nth Fibonacci number which has run-time O( log n ).

OM-notation

To express the concept of an algorithm taking at least some number of steps

OM-notation

can be used.

Again let, f:N-> R and g:N-> R. Then: f( n ) = OM ( g(n) ) means that there are values, n0 and c, such that f( n ) > = c * g( n ) whenever n > = n0.

THETA-notation

Again let, f:N-> R and g:N-> R.

If f(n)=O(g(n)) and f(n)=OM(g(n))

Then we write, f( n ) = THETA ( g(n) )

In this case, f(n) is said to be asymptotically equal to g(n).

If f( n ) = THETA( g(n) ) then algorithms with running times f(n) and g(n) appear to perform similarly as n gets larger.

Manipulating O- and OM-notation

f(n)=O(g(n)) if and only if g(n)=OM(f(n))

If f(n) = O(g(n)) then

f(n)+g(n)=O(g(n)) ; f(n)+g(n)=OM(g(n))

f(n)*g(n)=O(g(n)^2) ; f(n)*g(n)=OM(f(n)^2)

Examples

Suppose f(n)=10n and g( n )=2n^2

f(n)=O(g(n))
10n+2n^2 = O(n^2)
10n+2n^2 = OM(n^2)
(10n)*(2n^2) = O(n^4)
(10n)*(2n^2) = OM(n^2)

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