DATA STRUCTURE FOR MULTIDIMENSIONAL POLYNOMIALS

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✎: DATA STRUCTURE FOR MULTIDIMENSIONAL POLYNOMIALS

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Keywords: DATA STRUCTURE FOR MULTIDIMENSIONAL POLYNOMIALS

 

RESEARCH BODY

ABSTRACT

A polynomial is an algebraic expression that has a sum of terms where each term contains only variables with (non negative) integer exponents and real coefficients. A multivariate polynomial is a polynomial where each terms may have one or more variables of different names and the variable name in any given term need not be identical to the variable names of any other term in the polynomial.

The development in the application of multidimensional polynomial is binged on the use of computer for the manipulation of data structure which would determine the appropriate representation (Data Structure) that affords easy computation.

CHAPTER ONE

1.0 INTRODUCTION

1.1 DEFINITION OF DATA STRUCTURE

Data structure is the pattern in which data is organized. Data structure refers to the organisation of data in computer memory or the way in which data is efficiently stored, processed and retrieved.

Data structure can also be defined as a logical mode of organizing data in a specific way. The choice of particular data structure depends on the following consideration:

·        It should be able to represent the inherent relationship of the data in the real world.

·        Simplicity of data efficiently

For instance, in order to keep the records of students of a school using any type of data structure, the following are to be considered;

·        The individual characters

·        The group character

·        Individual records

1.2     CLASSES OF DATA STRUCTURE Data structures can be classified by the relationship that exist in them. The following are the classes of data structure · Simple data structure · Compound data structure · Linear data structure · Non linear data structure

Simple Data Structure:                                                                             Simple data structure can be constructed with the help of primitive data structure. A primitive data structure used to represent the standard data types of any one of the computer languages. Variables, arrays, pointers, structures, unions, etc. are examples of primitive data structures. Compound Data structure:                                                             Compound data structure can be constructed with the help of any one of the primitive data structure and it is having a specific functionality. It can be designed by user. It can be classified as 1) Linear data structure 2) Non-linear data structure Linear Data Structure :Linear data structures can be constructed as a continuous arrangement of data elements in the memory. It can be constructed by using array data type. In the linear Data Structures the relationship of adjacency is maintained between the Data elements. Operations Applied on Linear Data Structure : The following list of operations applied on linear data structures 1. Add an element 2. Delete an element 3. Traverse 4. Sort the list of elements 5. Search for a data element.

By applying one or more functionalities to create different types of data structures. For example Stack, Queue, Tables, List, and Linked Lists. Non-linear data structure: Non-linear data structure can be constructed as a collection of randomly distributed set of data item joined together by using a special pointer (tag). In non-linear Data structure the relationship of adjacency is not maintained between the Data items. Operations Applied on Non-linear Data Structures : The following list of operations applied on non-linear data structures. 1. Add elements 2. Delete elements 3. Display the elements 4. Sort the list of elements 5. Search for a data element

1.3     TYPES OF DATA STRUCTURES

There are various types in which data structure exists, some of the types of data structure are discussed below;

1.3.1  ARRAYS

An array is an aggregate data structure that is designed to store a group of objects of the same or different types and each identified by at least one array index or key. The simplest type of data structure is a linear array, also called one-dimensional array .The general form of a one-dimensional array is <type specifier> array name [size]

                     Fig 1.1 Arrays structure

Index; maps array value to a stored object. Index is a logical representation of address within the array.

Address; is the location where array is being stored in physical memory.

1.3.2  LINKED LISTS

A linked list is a collection of objects which are linked to each other in a linear pattern. Each of these objects is called a node, the first object is called the front node or the head. Each of the nodes store some data in them. One important feature of this type of list is that the data is not stored in contiguous locations. Every objects has two components- one is the data part and the other is the address of the node to which its pointing. the nodes are not contiguous and can lie in any part of the memory.

Fig 1.2: linked lists

1.3.2.1        SINGLY LINKED LIST

In singly linked list, each node in the list stores the contents of the node and a pointer or reference to the next node in the list. It does not store any pointer or reference to the previous node

Fig 1.3  Illustration of a singly-linked list

1.3.2.2        MULTI-LINKED LISTS

Multi-Linked Lists can be best described in the fig below

Fig1.4: Illustration multi-linked lists

1.3.2.3 CIRCULAR LINKED LIST

A circular linked list is one in which the last node is linked back to the first node instead of being linked to NULL.

Fig 1.5 Illustration of circular linked list

1.3.3  TREES

Trees are non linear structures, it’s a data structure which reflects the hierarchical relationship between its various elements for example, family trees, organisation chart e.t.c A tree is a finite set T of one or more disjoint nodes  such that

·        There is one specially designated node called the root of the tree and is denoted as root (T)

·        The node of a tree is the root of some other sub trees contained in the whole tree. The number of sub-trees of a node is the degree of that node. A node of degree zero is called TERMINAL NODE otherwise called a LEAF. A non-terminal node is called a branch node.

Fig 1.6.    Illustration of Trees

1.3.3.1           BINARY TREES

A binary tree is made of nodes, where each node contains a “left” reference, a “right” reference, and a data element. The topmost node in the tree is called the root. Every node (excluding a root) in a tree is connected by a directed edge from exactly one other node. This node is called a parent.

                             Fig 1.7        Binary Trees

The binary tree above contains 7 nodes represented by letters A through G. The root of the above tree is A, while B and C are the left and right successor of node A in that order. The left sub tree consists of B, D, E and the right sub tree of A consists of C,F,G. Any node in a binary tree has either 0,1,2 successors. Nodes with no successor is called terminal nodes.

1.3.4  STACK

A stack is a basic data structure that can be logically thought as linear structure represented by a real physical stack or pile, a structure where insertion and deletion of items takes place at one end called top of the stack. Stack is a LIFO structure. (Last in First out). push() function is used to insert new elements into the Stack and pop() is used to delete an element from the stack. Both insertion and deletion are allowed at only one end of Stack called Top.

Fig 1.8  Representation using stack

1.3.5  DEQUEUE

Queue is a linear list opened at both end. One end is always used to insert data (enqueue) and the other is used to remove data (dequeue). Queue follows First-In-First-Out methodology, i.e., the data item stored first will be accessed first.

Fig 1.9  Representation using dequeue

1.3.6  GRAPH

A graph data structure consists of a finite (and possibly mutable) set of vertices or nodes or points, together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph.

Fig 1.10 Illustration using graph

A graph may be defined as a finite set V of vertices and a set E of edges. The  notation used is as follows;

Graph G= (V, E). Consider the above graph;

The set of vertices for the graph is V= {A,B,C,D,E,F}

The set of edges for the graph is E={(A,C), (A,B), (C,B), (B,E), (E,D), (D,F)}

A graph can be used by airlines for maintaining the flight information such as the various flight, their routes and the distance between the places.

1.4     BASIC OPERATIONS OF DATA STRUCTURE

Data are processed by means of certain operations which appearing in the data structure. Data has situation on depends largely on the frequency with which specific operations are performed. This section introduces the reader to some of the most frequently used of these operations.

1.4.1  TRAVERSING

Accessing each records exactly once so that certain items in the record may be processed. It is mainly used for hierarchical data structure (Trees)

1.4.2  SEARCHING

Finding the location of a particular record with a given key value, or finding the location of all records which satisfy one or more conditions.

1.4.3  INSERTING

Adding a new record to the structure. 1.4.4  DELETING

Removing the record from the structure. 1.4.5  SORTING

Managing the data or record in some logical order(Ascending or descending order). 1.4.6  MERGING:

Combining the record in two different sorted files into a single sorted file.

1.5     USES OF DATA STRUCTURE

Different kinds of data structures are meant  for different kinds of applications, and some are highly specialized to specific tasks.

Data structures are important for the following reasons:

1. Data structures are used in almost every program or software system.

2. Specific data structures are essential ingredients of many efficient algorithms, and make possible the management of huge amounts of data, such as large integrated collection of databases.

3. Some programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design.

1.6     REAL LIFE APPLICATIONS OF DATA STRUCTURES.

a) To store a set of programs which are to be given access to a hard disk according to their priority. b) For representing a city region telephone network. c) To store a set of fixed key words which are referenced very frequently. d) To represent an image in the form of a bitmap. e) To implement back functionality in the internet browser. f) To store dynamically growing data which is accessed very frequently, based upon a key value. g) To implement printer spooler so that jobs can be printed in the order of their arrival. h) To record the sequence of all the pages browsed in one session. i) To implement the undo function in a text editor. j) To store information about the directories and files in a system.

1.7     ADVANTAGE OF DATA STRUCTURE

  • Data structures allow information storage on hard disks.
  • provides means for management of large dataset such as databases or internet indexing services.
  • Are necessary for design of efficient algorithms.
  • allows safe storage of information on a computer. The information is then available for later use and can be used by multiple programs. Additionally, the information is secures and cannot be lost (especially if it is stored on magnetic tapes).
  • allows the data use and processing on a software system.
  • Allows easier processing of data.
  • Using internet, we can access the data anytime from any connected machine (computer, laptop, tablet, phone, etc.)

1.8     SUBJECT OF STUDY

This project is therefore aimed at studying polynomials, basically with emphasis on multidimensional polynomials since a polynomial is a one dimensional array and thus, finding the means of evaluation and selection of appropriate data structure for implementing multidimensional polynomials. The appropriateness of such data structure will also be put into test by a set of routines, a multidimensional polynomial system that will perform a series of operation like sum, difference, product and division.

 

Keywords: DATA STRUCTURE FOR MULTIDIMENSIONAL POLYNOMIALS

 


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Keywords: DATA STRUCTURE FOR MULTIDIMENSIONAL POLYNOMIALS

 


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