Data Structures & Algorithm

What is a Data Structure?

Data Structure is a systematic way to organize data in order to use it efficiently. Following terms are 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 in the operations of the data structure.

Characteristics of a Data Structure

• Correctness − Data Structure implementation should implement its interface correctly.
• Time Complexity − Running time or 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 Structure

As applications are getting complex and data rich, there are three common problems applications face now-a-days.

• Data Search − Consider an inventory of 1 million(10⁶) items of a store. If application is to search an item. It has to search 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 data grows to billion records.
• Multiple requests − As thousands of users can search data simultaneously on a web server,even very fast server fails while searching the data.

To solve above problems, data structures come to rescue. Data can be organized in a data structure in such a way that all items may not be required to be search and required data can be searched almost instantly.

Execution Time Cases

There are three cases which are usual used to compare various data structure's execution time in relative manner.

• Worst Case − This is the scenario where a particular data structure operation takes maximum time it can take. If a operation's worst case time is ƒ(n) then this operation will not take time more than ƒ(n) time where ƒ(n) represents function of n.
• Average Case − This is the scenario depicting the average execution time of an operation of a data structure. If a operation takes ƒ(n) time in execution then m operations will take mƒ(n) time.
• Best Case − This is the scenario depicting the least possible execution time of an operation of a data structure. If a operation takes ƒ(n) time in execution then actual operation may take time as random number which would be maximum as ƒ(n).

Basic Terminology

• Data − Data are values or set of values.
• Data Item − Data item refers to single unit of values.
• Group Items − Data item that are divided into sub items are called as Group Items.
• Elementary Items − Data item 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.
• Local Environment Setup
• If you are still willing to set up your environment for C programming language, you need the following two softwares 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.
  Name and version of text editor can vary on different 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 and you have enough experience to write a computer program, save it in a file, compile it and finally execute it.

  The C Compiler

  The source code written in 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 instructions given.
  This C programming language compiler will be used to compile your source code into final executable program. I assume you have 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.
  Following section guides you on how to install GNU C/C++ compiler on various OS. I'm mentioning C/C++ together because GNU gcc compiler works for both C and C++ programming languages.

  Installation on UNIX/Linux

  If you are using Linux or UNIX, then check whether GCC is installed on your system by entering the following command from the command line −

  $ gcc -v
  

  If you have GNU compiler installed on your machine, then it should print a message something as follows −

  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 OS

  If you use Mac OS X, the easiest way to obtain GCC is to download the Xcode development environment from Apple's web site and follow the simple installation instructions. Once you have Xcode setup, you will be able to use GNU compiler for C/C++.
  Xcode is currently available at developer.apple.com/technologies/tools/.

  Installation on Windows

  To install GCC at 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 the latest 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 the MinGW 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

• Data Structures - Algorithms Basics

• Algorithm is a step by step procedure, which defines a set of instructions to be executed in 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 data structure point of view, following are some important categories of algorithms −

  • Search − Algorithm to search an item in a datastructure.
  • Sort − Algorithm to sort items in certain order
  • Insert − Algorithm to insert item in a datastructure
  • 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 Algorithm

  Not all procedures can be called an algorithm. An algorithm should have the below mentioned characteristics −

  • Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their input/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 be independent 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 constructscan be used to write an algorithm.

  We write algorithms in 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 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 & b
  step 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 c
  step 5 − STOP
  

  In design and analysis of algorithms, usually the second method is used to describe an algorithm. It makes it easy of 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 solution of a given problem. A problem can be solved in more than one ways.

  
  

  Hence, many solution algorithms can be derived for a given problem. Next step is to analyze those proposed solution algorithms and implement the best suitable.

  Algorithm Analysis

  Efficiency of an algorithm can be analyzed at two different stages, before implementation and after implementation, as mentioned below −

  • A priori analysis − This is theoretical analysis of an algorithm. Efficiency of algorithm is measured by assuming that all other factors e.g. processor speed, are constant and have no effect on implementation.
  • A posterior analysis − This is 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 here a priori algorithm analysis. Algorithm analysis deals with the execution or running time of various operations involved. Running time of an operation can be defined as no. of computer instructions executed per operation.

  Algorithm Complexity

  Suppose 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 − The time is measured by counting the number of key operations such as comparisons in sorting algorithm
  • Space Factor − The space is measured by counting the maximum memory space required by the algorithm.

  The complexity of an algorithm f(n) gives the running time and / or storage space required by the algorithm in terms of n as the size of input data.

  Space Complexity

  Space complexity of an algorithm represents the amount of memory space required by the algorithm in its life cycle. Space required by an algorithm is equal to the sum of the following two components −

  • A fixed part that is a space required to store certain data and variables, that are independent of the size of the problem. For example simple variables & constant used, program size etc.
  • A variable part is a space required by variables, whose size depends on the size of the 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 and S(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)
  Step 1 -  START
  Step 2 -  C ← A + B + 10
  Step 3 -  Stop
  

  Here we have three variables A, B and C and one constant. Hence S(P)=1+3. Now space depends on data types of given variables and constant types and it will be multiplied accordingly.

  Time Complexity

  Time 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, the total computational time is T(n)= c*n, where c is the time taken for addition of two bits. Here, we observe that T(n) grows linearly as input size increases.

• Data Structures - Asymptotic Analysis

• 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 an algorithm.

  Asymptotic analysis are input bound i.e., if there's no input to the algorithm it is concluded to work 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, running time of one operation is computed as f(n) and may be for another operation it is computed as g(n²). Which means first operation running time will increase linearly with the increase in n and running time of second operation will increase exponentially when n increases. Similarly the running time of both operations will be nearly same if n is significantly small.

  Usually, 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 Notations

  Following are commonly used asymptotic notations used in calculating running time complexity of an algorithm.

  • Ο Notation
  • Ω Notation
  • θ Notation

  Big Oh Notation, Ο

  The 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 longest amount of time an algorithm can possibly take to complete. For example, for a functionf(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 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 best amount of time an algorithm can possibly 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 θ(n) is the formal way to express both the lower bound and upper bound of an algorithm's running time. It is represented as following.

  θ(f(n)) = { g(n) if and only if g(n) =  Ο(f(n)) and g(n) = Ω(f(n)) for all n > n0. }

  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)