Data Structure is a systematic way to organize data in order to use it efficiently. Following terms are the foundation terms of a data structure.
· Interface − Each data structure has an interface. Interface represents the set of operations that a data structure supports. An interface only provides the list of supported operations, type of parameters they can accept and return type of these operations.
· Implementation − Implementation provides the internal representation of a data structure. Implementation also provides the definition of the algorithms used inthe operations of the data structure.
Characteristics of a Data Structure· Correctness − Data structure implementation should implement its interface correctly.
· Time Complexity − Running time or the execution time of operations of data structure must be as small as possible.
· Space Complexity − Memory usage of a data structure operation should be as
little as possible.
Need for Data StructureAs applications are getting complex and data rich, there are three common problems that applications face now-a-days.
· Data Search − Consider an inventory of 1 million(106) items of a store. If the application is to search an item, it has to search an item in 1 million(106) items every time slowing down the search. As data grows, search will become slower.
· Processor Speed − Processor speed although being very high, falls limited if the data grows to billion records.
· Multiple Requests − As thousands of users can search data simultaneously on a webserver, even the fast server fails while searching the data.
To solve the above-mentioned problems, data structures come to rescue. Data can be organized ina data structure insuch a way that all items may not be required to be searched, and the required data can be searched almost instantly.
Execution Time CasesThere are three cases which are usually used to compare various data structure's execution time in a relative manner.
· Worst Case − This is the scenario where a particular data structure operation takes maximum time it can take. If an operation's worst case time is Æ’(n) then this operation will not take more than Æ’(n) time, where Æ’(n) represents function ofn.
· Average Case − This is the scenario depicting the average execution time of an operation of a data structure. If an operation takes Æ’(n) time in execution, then m operations will take mÆ’(n) time.
· Best Case − This is the scenario depicting the least possible execution time of an operation of a data structure. If an operation takes Æ’(n) time in execution, then theactual operation may take time as the random number which would be maximum as Æ’(n).
Basic Terminology· Data − Data are values or set of values.
· Data Item − Data item refers to single unit of values.
· Group Items − Data items that are divided into sub items are called as Group
Items.
· Elementary Items − Data items that cannot be divided are called as Elementary
Items.
· Attribute and Entity − An entity is that which contains certain attributes or properties, which may be assigned values.
· Entity Set − Entities of similar attributes form an entity set.
· Field − Field is a single elementary unit of information representing an attribute
of an entity.
· Record − Record is a collection of field values of a given entity.
· File − File is a collection of records of the entities in a given entity set.
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Try it Option OnlineYou really do not need to set up your own environment to start learning C programming language. Reason is very simple, we already have set up C Programming environment online, so that you can compile and execute all the available examples online at the same time when you are doing your theory work. This gives you confidence inwhat you are reading and to check the result with different options. Feel free to modify any example and execute it online.
Try the following example using the Try it option available at the top right corner of the
sample code box −
#include <stdio.h>
int main(){
/* My first program in C */
printf("Hello, World! \n");return 0;
}
For most of the examples given in this tutorial, you will find Try it option, so just make useof it and enjoy your learning.
Local Environment SetupIf you are still willing to set up your environment for C programming language, you need thefollowing two tools available on your computer, (a) Text Editor and (b) The C Compiler.
Text Editor
This will be used to type your program. Examples of few editors include Windows Notepad, OS Edit command, Brief, Epsilon, EMACS, and vim or vi.
The name and the version of the text editor can vary ondifferent operating systems. For example, Notepad will be used on Windows, and vim or vi can be used on Windows as well as Linux or UNIX.
The files you create with your editor are called source files and contain program source code. The source files for C programs are typically named with the extension ".c".
Before starting your programming, make sure you have one text editor in place andyou have enough experience towrite a computer program, save it in a file, compile it, and finally execute it.
The C Compiler
The source code written in the source file is the human readable source for your program. It needs to be "compiled", to turn into machine language so that your CPU can actually execute the program as per thegiven instructions.
This C programming language compiler will be used to compile your source code into a final executable program. We assume you have the basic knowledge about a programming language compiler.
Most frequently used and free available compiler is GNU C/C++ compiler. Otherwise, you can have compilers either from HP or Solaris if you have respective Operating Systems (OS).
The following section guides you on how to install GNU C/C++ compiler onvarious OS. Weare mentioning C/C++ together because GNU GCC compiler works for both C and C++ programming languages.
Installation on UNIX/LinuxIf you are using Linux orUNIX, then check whether GCC is installed on your system by
entering the following command from the command line −
$ gcc -vIf you have GNU compiler installed on your machine, then it should print a message such asthe following −
Using built-in specs. Target: i386-redhat-linux
Configured with: ../configure --prefix=/usr .......Thread model: posix
gcc version 4.1.2 20080704 (Red Hat 4.1.2-46)
If GCC is not installed, then you will have to install it yourself using the detailed instructions available at http://gcc.gnu.org/install/
This tutorial has been written based on Linux and all the given examples have been compiled on Cent OS flavor of Linux system.
Installation on Mac OSIf you use Mac OS X, the easiest way to obtain GCC is to download the Xcode development environment from Apple's website and follow the simple installation instructions. Once you haveXcode setup, you will be able to use GNU compiler for C/C++.
Xcode is currently available at developer.apple.com/technologies/tools/
Installation on WindowsTo install GCC on Windows, you need to install MinGW. To install MinGW, go to the MinGW homepage, www.mingw.org, and follow the link to the MinGW download page. Download thelatest version of the MinGW installation program, which should be named MinGW-
<version>.exe.
While installing MinWG, at a minimum, you must install gcc-core, gcc-g++, binutils, and theMinGW runtime, but you may wish to install more.
Add the bin subdirectory of your MinGW installation to your PATH environment variable, so that you can specify these tools on the command line by their simple names.
When the installation is complete, you will be able to run gcc, g++, ar, ranlib, dlltool, and several other GNU tools from the Windows command line.
Algorithm
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Algorithm is a step-by-step procedure, which defines a set of instructions to be executed ina certain order to get the desired output. Algorithms are generally created independent of underlying languages, i.e. an algorithm can be implemented in more than one programming language.
From the data structure point of view, following are some important categories of
algorithms −
· Search − Algorithm to search an item in a data structure.
· Sort − Algorithm to sort items in a certain order.
· Insert − Algorithm to insert item in a data structure.
· Update − Algorithm to update an existing item in a data structure.
· Delete − Algorithm to delete an existing item from a data structure.
Characteristics of an AlgorithmNot all procedures can be called an algorithm. An algorithm should have the following
characteristics −
· Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their inputs/outputs should be clear and must lead to only one meaning.
· Input − An algorithm should have 0 or more well-defined inputs.
· Output − An algorithm should have 1 or more well-defined outputs, and should match the desired output.
· Finiteness − Algorithms must terminate after a finite number of steps.
· Feasibility − Should be feasible with the available resources.
· Independent − An algorithm should have step-by-step directions, which should beindependent of any programming code.
How to Write an Algorithm?There are no well-defined standards for writing algorithms. Rather, it is problem and resource dependent. Algorithms are never written to support a particular programming code.
As we know that all programming languages share basic code constructs like loops (do, for, while), flow-control (if-else), etc. These common constructs can be used to write an algorithm.
We write algorithms in a step-by-step manner, but it is not always the case. Algorithm writing is a process and is executed after the problem domain is well-defined. That is, we should know the problem domain, for which we are designing a solution.
Example
Let's try to learn algorithm-writing by using an example.
Problem − Design an algorithm to add two numbers and display the result.
step 1 − START
step 2 − declare three integers a, b & c step 3 −define values of a & b
step 4 − add values of a & bstep 5 − store output of step 4 to c step 6 − print c
step 7 −STOP
Algorithms tell the programmers how to code the program. Alternatively, the algorithm
can be written as −
step 1 − START ADD
step 2 − get values of a & b step 3 − c ← a + b
step 4 − display cstep 5 − STOP
In design and analysis of algorithms, usually the second method is used to describe an algorithm. It makes it easy for the analyst to analyze the algorithm ignoring all unwanted definitions. He can observe what operations are being used and how the process is flowing.
Writing step numbers, is optional.
We design an algorithm to get a solution of a given problem. A problem can be solved in morethan one ways.
Hence, many solution algorithms can be derived for a given problem. The next step is to analyze those proposed solution algorithms and implement the best suitable solution.
Algorithm AnalysisEfficiency of an algorithm can be analyzed at two different stages, before implementation andafter implementation. They are the following −
· A Priori Analysis − This is a theoretical analysis of an algorithm. Efficiency of an algorithm is measured by assuming that all other factors, for example, processor speed, are constant and have no effect on the implementation.
· A Posteriori Analysis − This is an empirical analysis of an algorithm. The selected algorithm is implemented using programming language. This is then executed on target computer machine. In this analysis, actual statistics like running time and space required, are collected.
We shall learn about a priori algorithm analysis. Algorithm analysis deals with the execution or running time of various operations involved. The running time of an operation canbe defined as the number of computer instructions executed per operation.
Algorithm ComplexitySuppose X is an algorithm and n is the size of input data, the time and space used by the algorithm X are the two main factors, which decide the efficiency of X.
· Time Factor – Time is measured by counting the number of key operations such as comparisons in thesorting algorithm.
· Space Factor − Space is measured by counting the maximum memory space required by thealgorithm.
The complexity of an algorithm f(n) gives the running time and/or the storage space required by the algorithm in terms of n as thesize of input data.
Space ComplexitySpace complexity of an algorithm represents the amount of memory space required by the algorithm in its life cycle. The space required by an algorithm is equal to the sum of thefollowing two components −
· A fixed part that is a space required to store certain data and variables, that are independent ofthe size of the problem. For example, simple variables and constants used, program size, etc.
· A variable part is a space required by variables, whose size depends on the size ofthe problem. For example, dynamic memory allocation, recursion stack space, etc.
Space complexity S(P) of any algorithm P is S(P) = C + SP(I), where C is the fixed part andS(I) is the variable part of the algorithm, which depends on instance characteristic I. Following is a simple example that tries to explain the concept −
Algorithm: SUM(A, B) Step1 - START
Step 2 - C ← A + B + 10Step 3 - Stop
Here we have three variables A, B, and C and one constant. Hence S(P) = 1+3. Now, spacedepends on data types of given variables and constant types and it will be multiplied accordingly.
Time ComplexityTime complexity of an algorithm represents the amount of time required by the algorithm to run to completion. Time requirements can be defined as a numerical function T(n), where T(n) can be measured as the number of steps, provided each step consumes constant time.
For example, addition of two n-bit integers takes n steps. Consequently, thetotal computational timeis T(n) = c*n, where c is the time taken for the addition of two bits. Here, we observe that T(n) grows linearly as the input size increases.
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Asymptotic analysis of an algorithm refers to defining the mathematical boundation/framing of its run-time performance. Using asymptotic analysis, we can very well conclude the best case, average case, and worst case scenario of analgorithm.
Asymptotic analysis is input bound i.e., if there's no input to the algorithm, it is concluded towork in a constant time. Other than the "input" all other factors are considered constant.
Asymptotic analysis refers to computing the running time of any operation in mathematical units of computation. For example, the running time of one operation is computed as f(n) and may be for another operation it is computed as g(n2). This means the first operation running time will increase linearly with the increase in n and the running time of the second operation will increase exponentially when n increases. Similarly, the running time of both operations will be nearly the same if n is significantly small.
Usually, the time required by an algorithm falls under three types −
· Best Case − Minimum time required for program execution.
· Average Case − Average time required for program execution.
· Worst Case − Maximum time required for program execution.
Asymptotic NotationsFollowing are the commonly used asymptotic notations to calculate the running time complexity of an algorithm.
· Ο Notation
· Ω Notation
· θ Notation
Big Oh Notation, Ο
The notation Ο(n) is the formal way to express the upper bound of an algorithm's running time. It measures the worst case time complexity or the longest amount of time an algorithm can possibly take to complete.
For example, for a function f(n)
Ο(f(n)) = { g(n) : there exists c > 0 and n0 such that g(n) ≤ c.f(n) for all n
> n0. }Omega Notation, Ω
The notation Ω(n) is the formal way to express the lower bound of an algorithm's running time. It measures the best case time complexity or the best amount of time an algorithm canpossibly take to complete.
For example, for a function f(n)
Ω(f(n)) ≥ { g(n) : there exists c > 0 and n0 such that g(n) ≤ c.f(n) for all n
> n0. }Theta Notation, θ
The notation θ(n) is the formal way to express both the lower bound and theupper bound of an algorithm's running time. It is represented as follows −
θ(f(n)) = { g(n) if and only if g(n) = Ο(f(n)) and g(n) = Ω(f(n)) for all n >
n0. }Common Asymptotic NotationsFollowing is a list of some common asymptotic notations:
| constant | − | Ο(1) |
| logarithmic | − | Ο(log n) |
| linear | − | Ο(n) |
| n log n | − | Ο(n log n) |
| quadratic | − | Ο(n2) |
| cubic | − | Ο(n3) |
| polynomial | − | nΟ(1) |
| exponential | − | 2Ο(n) |
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An algorithm isdesigned to achieve optimum solution for a given problem. In greedy algorithm approach, decisions are made from the given solution domain. As being greedy, theclosest solution that seems to provide an optimum solution is chosen.
Greedy algorithms try to find a localized optimum solution, which may eventually lead to globally optimized solutions. However, generally greedy algorithms do not provide globally optimized solutions.
Counting CoinsThis problem is to count to a desired value by choosing the least possible coins and the greedy approach forces the algorithm to pick the largest possible coin. If we are provided coins of € 1, 2, 5 and 10 and we are asked to count € 18 then the greedy procedure will be −
· 1 − Select one € 10 coin, the remaining count is 8
· 2 − Then select one € 5 coin, the remaining count is 3
· 3 − Then select one € 2 coin, the remaining count is 1
· 3 − And finally, the selection of one € 1 coins solves the problem
Though, it seems to be working fine, for this count we need to pick only 4 coins. But if we slightly change the problem then the same approachmay not be able to produce the same optimum result.
For the currency system, where we have coins of 1, 7, 10 value, counting coins for value
18 will be absolutely optimum but for count like 15, it may use more coins than necessary. For example, the greedy approach will use 10 + 1 + 1 + 1 + 1 + 1, total 6 coins. Whereas thesame problem could be solved by using only 3 coins (7 + 7 + 1)
Hence, we may conclude that the greedy approachpicks an immediate optimized solution and may fail where global optimization is a major concern.
Data Structures & Algorithms
Examples
Most networking algorithms use thegreedy approach. Here is a list of few of them −
· Travelling Salesman Problem
· Prim's Minimal Spanning Tree Algorithm
· Kruskal's Minimal Spanning Tree Algorithm
· Dijkstra's Minimal Spanning Tree Algorithm
· Graph - Map Coloring
· Graph - Vertex Cover
· Knapsack Problem
· Job Scheduling Problem
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