Sparse matrix, Data Structure & Algorithms

Assignment Help:

Q. Define a sparse matrix. Explain different types of sparse matrices? Show how a triangular array is stored in memory. Evaluate the method to calculate address of any element ajk of a matrix stored in memory.                                                                                                                             

Ans.

Sparse Matrix

A m x n matrix A is said to be sparse if MOST of its elements are zero. A matrix that is not sparse is called dense matrix.

Types of Sparse matrix

1)            Diagonal Matrix

488_Diagonal matrix.png 

This is the square matrix where the non zero elements are only where row = col ie at

diagonal.

2)  Tridiagonal Matrix

897_tridiagonal matrix.png

In  this  square  matrix  all  elements  other  than  those  on  and  on  the  diagonals immediately above and below this one are zero.

Triangular Matrices

Tiangular Matrices is of 2 types:

a)  Lower triangular b)  Upper triangular

720_Diagonal matrix1.png

In an n*n lower triangular matrix A, row 1 has one non zero element, row 2 has 2,

....., and row n has n. whereas, in an n*n upper triangular matrix A, row 1 has n non zero elements, row 2 has n-1 ,.... , and row n has 1. In both the cases, the total number of non-zero elements is n(n+1)/2.

Both of these matrices can be represented using an one dimensional array la of size n(n+1)/2.

Consider lower triangular matrix L. the elements can be mapped by rows or by columns.

In a row-wise mapping, the element L[i,j], i>=j, is preceded by ∑k  for k=1 to i-1, elements that are in row 1 through i-1, and j-1 such elements from row i. the total number of elements that precede it in a row-wise mapping is

1620_Diagonal matrix2.png

 

This expression also gives the position l[i,j] in la.

Method  to  calculate  address  of  any  element  ajk   of  a  matrix     stored  in memory.

Let us consider 2 dimensional array a of size m*n further consider that the lower bound for the row index is lbr and for column index is lbc.

Like linear array, system keeps track of the first element only i.e. , the base address of the array.

Using this base address, the computer computes the address of the element in the ith row and jth column i.e. loc(a[i][j]), using the following formulae:

Column major order:-

Loc (a[i][j]) = base (a) + w [m (j - lbc) + ( i - lbr)] in general

Row major order:-

Loc (a[i][j]) = base (a) + w [n(i - lbr) + ( j - lbc)]            in general

where w is number of bytes per storage location for any one element of the array.


Related Discussions:- Sparse matrix

Sorting algorithm is best if the list is already sorted, Which sorting algo...

Which sorting algorithm is best if the list is already sorted? Why? Insertion sort as there is no movement of data if the list is already sorted and complexity is of the order

Flowchart, conversion of centrigral to frahenhit

conversion of centrigral to frahenhit

A difference between linear and non linear, state difference between linear...

state difference between linear and non linear data structure. give one example scenario of each

C++, #What is the pointer

#What is the pointer

Complexity, Complexity : How do the resource needs of a program or algorith...

Complexity : How do the resource needs of a program or algorithm scale (the growth of resource requirements as a function of input). In other words, what happens with the performan

State phong shading, Phong Shading Phong shading too is based on interp...

Phong Shading Phong shading too is based on interpolation, but instead of interpolating the colour value, it is the normal vector, which is interpolated for each point and a co

Elaborate the symbols of abstract data type, Elaborate the symbols of abstr...

Elaborate the symbols of abstract data type length(a)-returns the number of characters in symbol a. capitalize(a)-returns the symbol generated from a by making its first cha

B-tree, Draw a B-tree of order 3 for the following sequence of keys: 2,4,9,...

Draw a B-tree of order 3 for the following sequence of keys: 2,4,9,8,7,6,3,1,5,10.and delete 8 and 10

Explain the different types of traversal on binary tree, Question 1 What i...

Question 1 What is a data structure? Discuss briefly on types of data structures Question 2 Explain the insertion and deletion operation of linked list in detail Qu

Two - way merge sort, Merge sort is also one of the 'divide & conquer' clas...

Merge sort is also one of the 'divide & conquer' classes of algorithms. The fundamental idea in it is to split the list in a number of sublists, sort each of these sublists & merge

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd