Data Structures & AlgorithmsIT 0501

Algorithm Analysis I

Problem Solving: Main Steps

1. Problem definition2. Algorithm design / Algorithm

specification3. Algorithm analysis4. Implementation (coding/writing

program)5. Testing6. [Maintenance]

1. Problem Definition

What is the task to be accomplished?Calculate sum of first n natural numbers

What are the time / space / speed / performance requirements ?

2. Algorithm Design / Specifications

Algorithm: Finite set of instructions that, if followed, accomplishes a particular task.Describe: in natural language / pseudo-code / flow chart / etc. Criteria to follow:

Input: Zero or more quantities (externally produced)Output: One or more quantities Definiteness: Clarity, precision of each instructionFiniteness: The algorithm has to stop after a finite (may be very large) number of stepsEffectiveness: Each instruction has to be basic enough and feasible

Algorithm for finding sum of first n natural numbers

Sum_natural_nos (n)1. sum =0 2 for i=1 to n do3 sum= sum+i4 endfor5 return sum

4,5,6: Implementation, Testing, Maintainance

ImplementationDecide on the programming language to use

• C, C++, Lisp, Java, Perl, Prolog, assembly, etc. , etc.

Write clean, well documented code

Test, test, test

Integrate feedback from users, fix bugs, ensure compatibility across different versions Maintenance

3. Algorithm Analysis

Space complexityHow much space is required

Time complexityHow much time does it take to run the algorithm

Often, we deal with estimates!

Space Complexity

Space complexity = The amount of memory required by an algorithm to run to completion

[Core dumps = the most often encountered cause is “memory leaks” – the amount of memory required larger than the memory available on a given system]

Space Complexity (cont’d)

1. Fixed part: The size required to store certain data/variables, that is independent of the size of the problem:- e.g. name of the data collection

1. Variable part: Space needed by variables, whose size is dependent on the size of the problem:- e.g. actual text - load 2GB of text VS. load 1MB of text

Space Complexity (cont’d)

Find the space needed for the Sum_natural_nos algorithm

Time Complexity

Find time complexity for Sum_natural_nos algorithmOften more important than space complexity

space available (for computer programs!) tends to be larger and largertime is still a problem for all of us

3-4GHz processors on the market still … researchers estimate that the computation of various transformations for 1 single DNA chain for one single protein on 1 TerraHZ computer would take about 1 year to run to completion

Algorithms running time is an important issue

Running time

Input

1 ms

2 ms

3 ms

4 ms

5 ms

A B C D E F G

worst-case

best-case}average-case?

Suppose the program includes an if-then statement that may execute or not: variable running timeTypically algorithms are measured by their worst case

Experimental Approach

Write a program that implements the algorithmRun the program with data sets of varying size.Determine the actual running time using a system call to measure time (e.g. system (date) );

Experimental Approach

It is necessary to implement and test the algorithm in order to determine its running time. Experiments can be done only on a limited set of inputs, and may not be indicative of the running time for other inputs. The same hardware and software should be used in order to compare two algorithms. – condition very hard to achieve!

Use a Theoretical Approach

Based on high-level description of the algorithms, rather than language dependent implementationsMakes possible an evaluation of the algorithms that is independent of the hardware and software environments

Generality

Algorithm DescriptionHow to describe algorithms independent of a programming language Pseudo-Code = a description of an algorithm that is

more structured than usual prose but less formal than a programming language

(Or diagrams)Example: find the maximum element of an array.Algorithm arrayMax(A, n):

Input: An array A storing n integers.Output: The maximum element in A.currentMax A[0]for i 1 to n -1 do

if currentMax < A[i] then currentMax A[i]return currentMax

Pseudo CodeExpressions: use standard mathematical symbols

use for assignment ( ? in C/C++)use = for the equality relationship (? in C/C++)

Method Declarations: -Algorithm name(param1, param2) Programming Constructs:

decision structures: if ... then ... [else ..]while-loops while ... do repeat-loops: repeat ... until ... for-loop: for ... do array indexing: A[i]

Methodscalls: object method(args)returns: return value

Use commentsInstructions have to be basic enough and feasible!

Low Level Algorithm Analysis

Based on primitive operations (low-level computations independent from the programming language)E.g.:

Make an addition = 1 operationCalling a method or returning from a method = 1 operationIndex in an array = 1 operationComparison = 1 operation etc.

Method: Inspect the pseudo-code and count the number of primitive operations executed by the algorithm

ExampleAlgorithm Linear Search(A[1..n], n, key):

Input: An array A storing n integers.Output: position of the key ( i if found,

otherwise NULL)for i 1 to n doif A[i] == key, then

return iEndifEndfor Return NULL

How many operations ?

Problem

Fibonacci numbersF[0] = 0F[1] = 1F[i] = F[i-1] + F[i-2] for i 2

Pseudo-codeNumber of operations