Data StructuresArray Representation

Andres Mendez-Vazquez

May 6, 2015

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Introduction

ObservationWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

ExampleWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

We could use a name variable for each datadouble score0; double score1; double score2;double score3; double score4; double score5;

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Introduction

ObservationWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

ExampleWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

We could use a name variable for each datadouble score0; double score1; double score2;double score3; double score4; double score5;

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Introduction

ObservationWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

ExampleWhen a program manipulates many variables that contain “similar” formsof data, organizational problems quickly arise.

We could use a name variable for each datadouble score0; double score1; double score2;double score3; double score4; double score5;

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Making your life miserable

Because you can askWhich is the highest score?

Look at the code

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Making your life miserable

Because you can askWhich is the highest score?

Look at the code

doub l e h i gh_sco r e = s c o r e 0 ;i f ( s c o r e 1 > h igh_sco r e ) { h i gh_sco r e = s c o r e 1 ; }i f ( s c o r e 2 > h igh_sco r e ) { h i gh_sco r e = s c o r e 2 ; }i f ( s c o r e 3 > h igh_sco r e ) { h i gh_sco r e = s c o r e 3 ; }i f ( s c o r e 4 > h igh_sco r e ) { h i gh_sco r e = s c o r e 4 ; }i f ( s c o r e 5 > h igh_sco r e ) { h i gh_sco r e = s c o r e 5 ; }System . out . p r i n t l n ( h i gh_sco r e ) ;

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How do we solve this?

ThusJava, C and C++ use array variables to put together this collection ofequal elements.

Memory Layout

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How do we solve this?

ThusJava, C and C++ use array variables to put together this collection ofequal elements.

Memory Layout

a b c d

START

MEMORY LAYOUT

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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1D Array Representation In Java, C, and C++

NotesFor 1-dimensional array:

The elements are mapped into contiguous memory locationsThey are accessed through the use of location (x [i ]) = start + i

How much memory is used for representing an array?1 Space Overhead

1 Storing the start address: 4 bytes2 Storing x.length: 4 bytes

2 Space for the elements, for example 4 bytes for n elements

Total: 4 + 4 + 4n = 8 + 4n bytes

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Now, 2D Arrays

We declare 2-dimensional arrays as follow (Java, C)int[][] A = new int[3][4]

It can be shown as a tableA[0][0] A[0][1] A[0][2] A[0][3]A[1][0] A[1][1] A[1][2] A[1][3]A[2][0] A[2][1] A[2][2] A[2][3]

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Now, 2D Arrays

We declare 2-dimensional arrays as follow (Java, C)int[][] A = new int[3][4]

It can be shown as a tableA[0][0] A[0][1] A[0][2] A[0][3]A[1][0] A[1][1] A[1][2] A[1][3]A[2][0] A[2][1] A[2][2] A[2][3]

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Rows and Columns in a 2-D Array

RowsA[0][0] A[0][1] A[0][2] A[0][3]A[1][0] A[1][1] A[1][2] A[1][3]A[2][0] A[2][1] A[2][2] A[2][3]

Columns

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Rows and Columns in a 2-D Array

RowsA[0][0] A[0][1] A[0][2] A[0][3]A[1][0] A[1][1] A[1][2] A[1][3]A[2][0] A[2][1] A[2][2] A[2][3]

Columns

A[0][0] A[0][1] A[0][2] A[0][3]A[1][0] A[1][1] A[1][2] A[1][3]A[2][0] A[2][1] A[2][2] A[2][3]

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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2D Array Representation In Java, C, and C++

Given the following 2-dimensional array x

x =

a b c de f g hi j k l

This information is stored as 1D arrays with a reference to them

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2D Array Representation In Java, C, and C++Given the following 2-dimensional array x

x =

a b c de f g hi j k l

This information is stored as 1D arrays with a reference to them

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Some Properties

If we call the lengths of each of themx.length ⇒ 3x[0].length == x[1].length == x[2].length ⇒ 3

Space OverheadRemember, it is 8 bytes per 1D array

1 Thus, we have 4× 8 = 32 bytes2 Or we can see this as (number of rows + 1)× 8 bytes

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Some Properties

If we call the lengths of each of themx.length ⇒ 3x[0].length == x[1].length == x[2].length ⇒ 3

Space OverheadRemember, it is 8 bytes per 1D array

1 Thus, we have 4× 8 = 32 bytes2 Or we can see this as (number of rows + 1)× 8 bytes

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Some Properties

If we call the lengths of each of themx.length ⇒ 3x[0].length == x[1].length == x[2].length ⇒ 3

Space OverheadRemember, it is 8 bytes per 1D array

1 Thus, we have 4× 8 = 32 bytes2 Or we can see this as (number of rows + 1)× 8 bytes

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Some Properties

If we call the lengths of each of themx.length ⇒ 3x[0].length == x[1].length == x[2].length ⇒ 3

Space OverheadRemember, it is 8 bytes per 1D array

1 Thus, we have 4× 8 = 32 bytes2 Or we can see this as (number of rows + 1)× 8 bytes

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Some Properties

If we call the lengths of each of themx.length ⇒ 3x[0].length == x[1].length == x[2].length ⇒ 3

Space OverheadRemember, it is 8 bytes per 1D array

1 Thus, we have 4× 8 = 32 bytes2 Or we can see this as (number of rows + 1)× 8 bytes

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More Properties

FirstThis representation is called the array-of-arrays representation.

SecondIt requires contiguous memory of size 3 bytes, 4 bytes, 4 bytes, and 4bytes for the 4 1D-arrays.

ThirdOne memory block of size number of rows and number of rows blocksof size number of columns.

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More Properties

FirstThis representation is called the array-of-arrays representation.

SecondIt requires contiguous memory of size 3 bytes, 4 bytes, 4 bytes, and 4bytes for the 4 1D-arrays.

ThirdOne memory block of size number of rows and number of rows blocksof size number of columns.

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More Properties

FirstThis representation is called the array-of-arrays representation.

SecondIt requires contiguous memory of size 3 bytes, 4 bytes, 4 bytes, and 4bytes for the 4 1D-arrays.

ThirdOne memory block of size number of rows and number of rows blocksof size number of columns.

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Row-Major Mapping

Again

x =

a b c de f g hi j k l

For thisWe will convert it into 1D-array y by collecting elements by rows.

ThusWithin a row elements are collected from left to right.Rows are collected from top to bottom.

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Row-Major Mapping

Again

x =

a b c de f g hi j k l

For thisWe will convert it into 1D-array y by collecting elements by rows.

ThusWithin a row elements are collected from left to right.Rows are collected from top to bottom.

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Row-Major Mapping

Again

x =

a b c de f g hi j k l

For thisWe will convert it into 1D-array y by collecting elements by rows.

ThusWithin a row elements are collected from left to right.Rows are collected from top to bottom.

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We get

An arrayy[] = {a, b, c, d, e, f, g, h, i, j, k, l}

Memory Map

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We get

An arrayy[] = {a, b, c, d, e, f, g, h, i, j, k, l}

Memory Map

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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How do we locate an element?

Look At This

ThenAssume x has r rows and c columns.Each row has c elements.i rows to the left of row i.

ThusWe have ic elements to the left of x[i][0].Then, x[i][j] is mapped to position ic + j of 1D array.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Space Overhead

We have4 bytes for start of 1D array.4 bytes for length of 1D array.4 bytes for c (number of columns)

Total: 12 bytesNote: The number of rows = length/c

StillDisadvantage: Need contiguous memory of size rc.

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Column-Major Mapping

Similar SetupConvert into 1D array y by collecting elements by columns.Within a column elements are collected from top to bottom.Columns are collected from left to right.

Thus

x =

a b c de f g hi j k l

We get y = {a, e, i, b, f, j, c, g, k, d, h, l}

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Column-Major Mapping

Similar SetupConvert into 1D array y by collecting elements by columns.Within a column elements are collected from top to bottom.Columns are collected from left to right.

Thus

x =

a b c de f g hi j k l

We get y = {a, e, i, b, f, j, c, g, k, d, h, l}

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Column-Major Mapping

Similar SetupConvert into 1D array y by collecting elements by columns.Within a column elements are collected from top to bottom.Columns are collected from left to right.

Thus

x =

a b c de f g hi j k l

We get y = {a, e, i, b, f, j, c, g, k, d, h, l}

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Column-Major Mapping

Similar SetupConvert into 1D array y by collecting elements by columns.Within a column elements are collected from top to bottom.Columns are collected from left to right.

Thus

x =

a b c de f g hi j k l

We get y = {a, e, i, b, f, j, c, g, k, d, h, l}

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Matrix

Something NotableTable of values.It has rows and columns, but numbering begins at 1 rather than 0.

We use the following notationUse notation x(i,j) rather than x[i][j].

NoteIt may use a 2D array to represent a matrix.

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Matrix

Something NotableTable of values.It has rows and columns, but numbering begins at 1 rather than 0.

We use the following notationUse notation x(i,j) rather than x[i][j].

NoteIt may use a 2D array to represent a matrix.

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Matrix

Something NotableTable of values.It has rows and columns, but numbering begins at 1 rather than 0.

We use the following notationUse notation x(i,j) rather than x[i][j].

NoteIt may use a 2D array to represent a matrix.

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Drawbacks of using a 2D Array

FirstIndexes are off by 1.

SecondJava arrays do not support matrix operations such as add, transpose,multiply, and so on.

HoweverYou can develop a class Matrix for object-oriented support of all matrixoperations.

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Drawbacks of using a 2D Array

FirstIndexes are off by 1.

SecondJava arrays do not support matrix operations such as add, transpose,multiply, and so on.

HoweverYou can develop a class Matrix for object-oriented support of all matrixoperations.

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Drawbacks of using a 2D Array

FirstIndexes are off by 1.

SecondJava arrays do not support matrix operations such as add, transpose,multiply, and so on.

HoweverYou can develop a class Matrix for object-oriented support of all matrixoperations.

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Some Basic Matrices: Diagonal Matrix

Diagonal MatrixAn n × n matrix in which all nonzero terms are on the diagonal.

Propertiesx(i,j) is on diagonal iff i = j

Example

x =

a 0 0 00 b 0 00 0 c 00 0 0 d

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Some Basic Matrices: Diagonal Matrix

Diagonal MatrixAn n × n matrix in which all nonzero terms are on the diagonal.

Propertiesx(i,j) is on diagonal iff i = j

Example

x =

a 0 0 00 b 0 00 0 c 00 0 0 d

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Some Basic Matrices: Diagonal Matrix

Diagonal MatrixAn n × n matrix in which all nonzero terms are on the diagonal.

Propertiesx(i,j) is on diagonal iff i = j

Example

x =

a 0 0 00 b 0 00 0 c 00 0 0 d

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Some Basic Matrices: Diagonal Matrix

Propertiesx(i,j) is on diagonal iff i = j

DRAWBACKFor a n × n you are required to have:

For example using integers: 4n2 bytes...

What to do?Store diagonal only vs n2 whole

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Some Basic Matrices: Diagonal Matrix

Propertiesx(i,j) is on diagonal iff i = j

DRAWBACKFor a n × n you are required to have:

For example using integers: 4n2 bytes...

What to do?Store diagonal only vs n2 whole

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Some Basic Matrices: Diagonal Matrix

Propertiesx(i,j) is on diagonal iff i = j

DRAWBACKFor a n × n you are required to have:

For example using integers: 4n2 bytes...

What to do?Store diagonal only vs n2 whole

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Some Basic Matrices: Lower Triangular Matrix

DefinitionAn n × n matrix in which all nonzero terms are either on or below thediagonal.

Example

x =

1 0 0 02 3 0 04 5 6 07 8 9 10

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Some Basic Matrices: Lower Triangular Matrix

DefinitionAn n × n matrix in which all nonzero terms are either on or below thediagonal.

Example

x =

1 0 0 02 3 0 04 5 6 07 8 9 10

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Some Basic Matrices: Lower Triangular Matrix

Propertiesx(i,j) is part of lower triangle iff i >= j.

Number of elements in lower triangle is

1 + 2 + 3 + ... + n = n (n + 1)2 (1)

ThusYou can store only the lower triangle

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Some Basic Matrices: Lower Triangular Matrix

Propertiesx(i,j) is part of lower triangle iff i >= j.

Number of elements in lower triangle is

1 + 2 + 3 + ... + n = n (n + 1)2 (1)

ThusYou can store only the lower triangle

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Some Basic Matrices: Lower Triangular Matrix

Propertiesx(i,j) is part of lower triangle iff i >= j.

Number of elements in lower triangle is

1 + 2 + 3 + ... + n = n (n + 1)2 (1)

ThusYou can store only the lower triangle

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Array Of Arrays Representation

We can use this

Something NotableYou can use an irregular 2-D array ... length of rows is not required to bethe same.

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Array Of Arrays Representation

We can use this

Something NotableYou can use an irregular 2-D array ... length of rows is not required to bethe same.

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Code

Code for irregular array

// d e c l a r e a two−d i m e n s i o n a l a r r a y v a r i a b l e// and a l l o c a t e the d e s i r e d number o f rowsi n t [ ] [ ] i r r e g u l a r A r r a y = new i n t [ numberOfRows ] [ ] ;

// now a l l o c a t e space f o r the e l ement s i n each rowf o r ( i n t i = 0 ; i < numberOfRows ; i ++)

i r r e g u l a r A r r a y [ i ] = new i n t [ s i z e [ i ] ] ;

// use the a r r a y l i k e any r e g u l a r a r r a yi r r e g u l a r A r r a y [ 2 ] [ 3 ] = 5 ;i r r e g u l a r A r r a y [ 4 ] [ 6 ] = i r r e g u l a r A r r a y [ 2 ] [ 3 ] + 2 ;i r r e g u l a r A r r a y [ 1 ] [ 1 ] += 3 ;

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However, You can do better

Map Lower Triangular Array Into A 1D ArrayYou can use a row-major order, but omit terms that are not part of thelower triangle.

For example

x =

1 0 0 02 3 0 04 5 6 07 8 9 10

We get1, 2, 3, 4, 5, 6, 7, 8, 9, 10

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However, You can do better

Map Lower Triangular Array Into A 1D ArrayYou can use a row-major order, but omit terms that are not part of thelower triangle.

For example

x =

1 0 0 02 3 0 04 5 6 07 8 9 10

We get1, 2, 3, 4, 5, 6, 7, 8, 9, 10

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However, You can do better

Map Lower Triangular Array Into A 1D ArrayYou can use a row-major order, but omit terms that are not part of thelower triangle.

For example

x =

1 0 0 02 3 0 04 5 6 07 8 9 10

We get1, 2, 3, 4, 5, 6, 7, 8, 9, 10

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The final mapping

We want

We have that1 Order is: row 1, row 2, row 3, ...2 Row i is preceded by rows 1, 2, ..., i-13 Size of row i is i.

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The final mapping

We want

We have that1 Order is: row 1, row 2, row 3, ...2 Row i is preceded by rows 1, 2, ..., i-13 Size of row i is i.

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The final mapping

We want

We have that1 Order is: row 1, row 2, row 3, ...2 Row i is preceded by rows 1, 2, ..., i-13 Size of row i is i.

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The final mapping

We want

We have that1 Order is: row 1, row 2, row 3, ...2 Row i is preceded by rows 1, 2, ..., i-13 Size of row i is i.

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More

Find the total number of elements before row i

1 + 2 + 3 + ... + i − 1 = i (i − 1)2 (2)

Thus

So element (i,j) is at position i(i-1)/2 + j -1 of the 1D array.

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More

Find the total number of elements before row i

1 + 2 + 3 + ... + i − 1 = i (i − 1)2 (2)

Thus

So element (i,j) is at position i(i-1)/2 + j -1 of the 1D array.

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Now the Interface Matrix

You will need this for your homework

p u b l i c i n t e r f a c e Matr ix<Item >{p u b l i c I tem get ( i n t row , i n t column ) ;p u b l i c v o i d add ( i n t row , i n t column , Item myobj ) ;p u b l i c I tem remove ( i n t row , i n t column ) ;p u b l i c v o i d output ( ) ;

}

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Sparse Matrices

DefinitionA sparse array is simply an array most of whose entries are zero (or null, orsome other default value).

For ExampleSuppose you wanted a 2-dimensional array of course grades, whose rowsare students and whose columns are courses.

PropertiesThere are about 90,173 studentsThere are about 5000 coursesThis array would have about 450,865,000 entries

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Sparse Matrices

DefinitionA sparse array is simply an array most of whose entries are zero (or null, orsome other default value).

For ExampleSuppose you wanted a 2-dimensional array of course grades, whose rowsare students and whose columns are courses.

PropertiesThere are about 90,173 studentsThere are about 5000 coursesThis array would have about 450,865,000 entries

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Sparse Matrices

DefinitionA sparse array is simply an array most of whose entries are zero (or null, orsome other default value).

For ExampleSuppose you wanted a 2-dimensional array of course grades, whose rowsare students and whose columns are courses.

PropertiesThere are about 90,173 studentsThere are about 5000 coursesThis array would have about 450,865,000 entries

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Sparse Matrices

DefinitionA sparse array is simply an array most of whose entries are zero (or null, orsome other default value).

For ExampleSuppose you wanted a 2-dimensional array of course grades, whose rowsare students and whose columns are courses.

PropertiesThere are about 90,173 studentsThere are about 5000 coursesThis array would have about 450,865,000 entries

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Thus

Something NotableSince most students take fewer than 5000 courses, there will be a lot ofempty spaces in this array.

Something NotableThis is a big array, even by modern standards.

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Thus

Something NotableSince most students take fewer than 5000 courses, there will be a lot ofempty spaces in this array.

Something NotableThis is a big array, even by modern standards.

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Example: Airline Flights

ExampleAirline flight matrix.

PropertiesAirports are numbered 1 through nflight(i,j) = list of nonstop flights from airport i to airport j

Assume n = 1000 and you use an integer for representationn × n array of list references =⇒ 4 million bytes when all the

airports are connected

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Example: Airline Flights

ExampleAirline flight matrix.

PropertiesAirports are numbered 1 through nflight(i,j) = list of nonstop flights from airport i to airport j

Assume n = 1000 and you use an integer for representationn × n array of list references =⇒ 4 million bytes when all the

airports are connected

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Example: Airline Flights

ExampleAirline flight matrix.

PropertiesAirports are numbered 1 through nflight(i,j) = list of nonstop flights from airport i to airport j

Assume n = 1000 and you use an integer for representationn × n array of list references =⇒ 4 million bytes when all the

airports are connected

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Example: Airline Flights

ExampleAirline flight matrix.

PropertiesAirports are numbered 1 through nflight(i,j) = list of nonstop flights from airport i to airport j

Assume n = 1000 and you use an integer for representationn × n array of list references =⇒ 4 million bytes when all the

airports are connected

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Example: Airline Flights

HoweverThe total number of flights is way lesser =⇒ 20,000

Needed StorageWe need at most 20,000 list references =⇒ Thus, we need at most 80,000bytes

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Example: Airline Flights

HoweverThe total number of flights is way lesser =⇒ 20,000

Needed StorageWe need at most 20,000 list references =⇒ Thus, we need at most 80,000bytes

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Web Page Matrix

Web page matrixWeb pages are numbered 1 through n.web(i,j) = number of links from page i to page j.

Web AnalysisAuthority page ... page that has many links to it.Hub page ... links to many authority pages.

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Web Page Matrix

Web page matrixWeb pages are numbered 1 through n.web(i,j) = number of links from page i to page j.

Web AnalysisAuthority page ... page that has many links to it.Hub page ... links to many authority pages.

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Web Page Matrix

Web page matrixWeb pages are numbered 1 through n.web(i,j) = number of links from page i to page j.

Web AnalysisAuthority page ... page that has many links to it.Hub page ... links to many authority pages.

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Web Page Matrix

Web page matrixWeb pages are numbered 1 through n.web(i,j) = number of links from page i to page j.

Web AnalysisAuthority page ... page that has many links to it.Hub page ... links to many authority pages.

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Web Page Matrix

Something Notablen= 2,000,000,000 (and growing by 1 million a day)

If we used integers for representationn × n array of integers =⇒16 ∗ 1018 bytes (16 ∗ 109 GB)

PropertiesEach page links to 10 (say) other pages on average.On average there are 10 nonzero entries per row.Space needed for non-zero elements is approximately 20,000,000,000x 4 bytes = 80,000,000,000 bytes (80 GB)

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Web Page Matrix

Something Notablen= 2,000,000,000 (and growing by 1 million a day)

If we used integers for representationn × n array of integers =⇒16 ∗ 1018 bytes (16 ∗ 109 GB)

PropertiesEach page links to 10 (say) other pages on average.On average there are 10 nonzero entries per row.Space needed for non-zero elements is approximately 20,000,000,000x 4 bytes = 80,000,000,000 bytes (80 GB)

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Web Page Matrix

Something Notablen= 2,000,000,000 (and growing by 1 million a day)

If we used integers for representationn × n array of integers =⇒16 ∗ 1018 bytes (16 ∗ 109 GB)

PropertiesEach page links to 10 (say) other pages on average.On average there are 10 nonzero entries per row.Space needed for non-zero elements is approximately 20,000,000,000x 4 bytes = 80,000,000,000 bytes (80 GB)

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Web Page Matrix

Something Notablen= 2,000,000,000 (and growing by 1 million a day)

If we used integers for representationn × n array of integers =⇒16 ∗ 1018 bytes (16 ∗ 109 GB)

PropertiesEach page links to 10 (say) other pages on average.On average there are 10 nonzero entries per row.Space needed for non-zero elements is approximately 20,000,000,000x 4 bytes = 80,000,000,000 bytes (80 GB)

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Web Page Matrix

Something Notablen= 2,000,000,000 (and growing by 1 million a day)

If we used integers for representationn × n array of integers =⇒16 ∗ 1018 bytes (16 ∗ 109 GB)

PropertiesEach page links to 10 (say) other pages on average.On average there are 10 nonzero entries per row.Space needed for non-zero elements is approximately 20,000,000,000x 4 bytes = 80,000,000,000 bytes (80 GB)

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What we need...

We need...There are ways to represent sparse arrays efficiently

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Sparse Matrices

DefinitionSparse ... many elements are zeroDense ... few elements are zero

Examples of structured sparse matricesDiagonalTridiagonalLower triangular (Actually in the Middle of Sparse and Dense)

ActuallyThey may be mapped into a 1D array so that a mapping function can beused to locate an element.

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Representation Of Unstructured Sparse Matrices

We can use aSingle linear list in row-major order.

ThusWe scan the non-zero elements of the sparse matrix in row-majororder.Each nonzero element is represented by a triple (row, column,value).The list of triples may be an array list or a linked list (chain).

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Representation Of Unstructured Sparse Matrices

We can use aSingle linear list in row-major order.

ThusWe scan the non-zero elements of the sparse matrix in row-majororder.Each nonzero element is represented by a triple (row, column,value).The list of triples may be an array list or a linked list (chain).

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Representation Of Unstructured Sparse Matrices

We can use aSingle linear list in row-major order.

ThusWe scan the non-zero elements of the sparse matrix in row-majororder.Each nonzero element is represented by a triple (row, column,value).The list of triples may be an array list or a linked list (chain).

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Representation Of Unstructured Sparse Matrices

We can use aSingle linear list in row-major order.

ThusWe scan the non-zero elements of the sparse matrix in row-majororder.Each nonzero element is represented by a triple (row, column,value).The list of triples may be an array list or a linked list (chain).

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Outline1 Introduction

1D Arrays2D Arrays

2 Representation2D Array Representation

3 Improving the representationRow-Major Mapping

4 MatrixDefinitionTypes of Matrices

5 Sparse MatricesSparse MatricesRepresentation Of Unstructured Sparse MatricesSparse Arrays

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Example with an Sparse Array

FirstWe will start with sparse one-dimensional arrays, which are simpler

Example

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Example with an Sparse Array

FirstWe will start with sparse one-dimensional arrays, which are simpler

Example

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Representation

we can have the following representation

Where1 The front element is the index.2 The second element is the value at cell index.3 A pointer to the next element

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Representation

we can have the following representation

Where1 The front element is the index.2 The second element is the value at cell index.3 A pointer to the next element

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Representation

we can have the following representation

Where1 The front element is the index.2 The second element is the value at cell index.3 A pointer to the next element

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Representation

we can have the following representation

Where1 The front element is the index.2 The second element is the value at cell index.3 A pointer to the next element

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Sparse Array ADT

For a one-dimensional array of ItemA constructor:

I SparseArray(int length)

A way to get values from the array:I Object get(int index)

A way to store values in the array:I void add(int index, Item value)

In addition, it is OK to ask for a value from an empty array position1 For number, it should return ZERO.2 For Item, it should return NULL.

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Other Operations

What about?1 int length():

I return the size of the original array2 int elementCount() :

I how many non-null values are in the array

QuestionDo we forget something?

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Other Operations

What about?1 int length():

I return the size of the original array2 int elementCount() :

I how many non-null values are in the array

QuestionDo we forget something?

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Other Operations

What about?1 int length():

I return the size of the original array2 int elementCount() :

I how many non-null values are in the array

QuestionDo we forget something?

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Other Operations

What about?1 int length():

I return the size of the original array2 int elementCount() :

I how many non-null values are in the array

QuestionDo we forget something?

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Other Operations

What about?1 int length():

I return the size of the original array2 int elementCount() :

I how many non-null values are in the array

QuestionDo we forget something?

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Example of Get

Code

p u b l i c I tem get ( i n t i n d e x ) {ChainNode c u r r e n t = t h i s . f i r s t N o d e ;

do {i f ( i n d e x == c u r r e n t . i n d e x ) {

// found c o r r e c t l o c a t i o nr e t u r n c u r r e n t . v a l u e ;

}c u r r e n t = c u r r e n t . nex t ;

} w h i l e ( i n d e x < c u r r e n t . i n d e x && next != n u l l ) ;r e t u r n n u l l ;

}

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Time Analysis

FirstWe must search a linked list for a given indexWe can keep the elements in order by index

Expected Time for both get and addIf we have n elements, we have the following complexity O(n).

For the other methodslength, elementCount =⇒ O(1)

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Time Analysis

FirstWe must search a linked list for a given indexWe can keep the elements in order by index

Expected Time for both get and addIf we have n elements, we have the following complexity O(n).

For the other methodslength, elementCount =⇒ O(1)

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Time Analysis

FirstWe must search a linked list for a given indexWe can keep the elements in order by index

Expected Time for both get and addIf we have n elements, we have the following complexity O(n).

For the other methodslength, elementCount =⇒ O(1)

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Problem with this analysis

True factIn an ordinary array, indexing to find an element is the only operation wereally need!!!

True factIn a sparse array, we can do indexing reasonably quickly

False ConclusionIn a sparse array, indexing to find an element is the only operation wereally need

The problem is that in designing the ADT, we did not think enoughabout how it would be used !!!

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Problem with this analysis

True factIn an ordinary array, indexing to find an element is the only operation wereally need!!!

True factIn a sparse array, we can do indexing reasonably quickly

False ConclusionIn a sparse array, indexing to find an element is the only operation wereally need

The problem is that in designing the ADT, we did not think enoughabout how it would be used !!!

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Problem with this analysis

True factIn an ordinary array, indexing to find an element is the only operation wereally need!!!

True factIn a sparse array, we can do indexing reasonably quickly

False ConclusionIn a sparse array, indexing to find an element is the only operation wereally need

The problem is that in designing the ADT, we did not think enoughabout how it would be used !!!

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Look at this code

To find the maximum element in a normal array:

doub l e max = a r r a y [ 0 ] ;f o r ( i n t i = 0 ; i < a r r a y . l e n g t h ; i ++){

i f ( a r r a y [ i ] > max) max = a r r a y [ i ] ;}

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Look at this code

To find the maximum element in a sparse array:

Double max = ( Double ) a r r a y . ge t ( 0 ) ;f o r ( i n t i = 0 ; i < a r r a y . l e n g t h ( ) ; i ++){

Double temp = ( Double ) a r r a y . ge t ( i ) ;i f ( temp . compareTo (max) > 0) {

max = temp ;}

}

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What is the problem?

FirstA lot of wrapping and casting because using generics.

SecondMore importantly, in a normal array, every element is relevant

ThirdIf a sparse array is 1% full, 99% of its elements will be zero.

I This is 100 times as many elements as we should need to examine

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What is the problem?

FirstA lot of wrapping and casting because using generics.

SecondMore importantly, in a normal array, every element is relevant

ThirdIf a sparse array is 1% full, 99% of its elements will be zero.

I This is 100 times as many elements as we should need to examine

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What is the problem?

FirstA lot of wrapping and casting because using generics.

SecondMore importantly, in a normal array, every element is relevant

ThirdIf a sparse array is 1% full, 99% of its elements will be zero.

I This is 100 times as many elements as we should need to examine

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What is the problem?

FourthOur search time is based on the size of the sparse array, not on thenumber of elements that are actually in it

QuestionWhat to do?

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What is the problem?

FourthOur search time is based on the size of the sparse array, not on thenumber of elements that are actually in it

QuestionWhat to do?

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Fixing the ADT

Something NotableAlthough “stepping through an array” is not a fundamental operation onan array, it is one we do frequently.

HoweverThis is a very expensive thing to do with a sparse arrayThis should not be so expensive:

I We have a list, and all we need to do is step through it

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Fixing the ADT

Something NotableAlthough “stepping through an array” is not a fundamental operation onan array, it is one we do frequently.

HoweverThis is a very expensive thing to do with a sparse arrayThis should not be so expensive:

I We have a list, and all we need to do is step through it

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Fixing the ADT

Something NotableAlthough “stepping through an array” is not a fundamental operation onan array, it is one we do frequently.

HoweverThis is a very expensive thing to do with a sparse arrayThis should not be so expensive:

I We have a list, and all we need to do is step through it

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Fixing the ADT

Something NotableAlthough “stepping through an array” is not a fundamental operation onan array, it is one we do frequently.

HoweverThis is a very expensive thing to do with a sparse arrayThis should not be so expensive:

I We have a list, and all we need to do is step through it

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Fixing the ADT

Poor SolutionLet the user step through the list.The user should not need to know anything about implementation.We cannot trust the user not to screw up the sparse array.These arguments are valid even if the user is also the implementer!

Correct SolutionUse an Inner iterator!!!

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Fixing the ADT

Poor SolutionLet the user step through the list.The user should not need to know anything about implementation.We cannot trust the user not to screw up the sparse array.These arguments are valid even if the user is also the implementer!

Correct SolutionUse an Inner iterator!!!

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Fixing the ADT

Poor SolutionLet the user step through the list.The user should not need to know anything about implementation.We cannot trust the user not to screw up the sparse array.These arguments are valid even if the user is also the implementer!

Correct SolutionUse an Inner iterator!!!

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Fixing the ADT

Poor SolutionLet the user step through the list.The user should not need to know anything about implementation.We cannot trust the user not to screw up the sparse array.These arguments are valid even if the user is also the implementer!

Correct SolutionUse an Inner iterator!!!

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Fixing the ADT

Poor SolutionLet the user step through the list.The user should not need to know anything about implementation.We cannot trust the user not to screw up the sparse array.These arguments are valid even if the user is also the implementer!

Correct SolutionUse an Inner iterator!!!

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Example

Codepublic class SparseArrayIterator<Item> implements Iterator<Item> {

// any data you need to keep track of goes hereSparseArrayIterator() { ...You do need one... }

public boolean hasNext ( ) { ...you write this code... }

public Item next ( ) { ...you write this code... }

public Item remove ( ) { ...you write this code... }}

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So, we have that

Code for Max

S p a r s e A r r a y I t e r a t o r i t e r a t o r = new S p a r s e A r r a y I t e r a t o r ( a r r a y ) ;Double max = ( Double ) a r r a y . ge t ( 0 ) ;

w h i l e ( i t e r a t o r . hasNext ( ) ) {temp = ( Double ) i t e r a t o r . nex t ( ) ;i f ( temp . compareTo (max) > 0) {

max = temp ;}

}

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HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

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HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

62 / 81

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HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

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HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

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HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

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Images/cinvestav-1.jpg

HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

62 / 81

Images/cinvestav-1.jpg

HoweverProblem

Our SparseArrayIterator is fine for stepping through the elements ofan array, but...

I It does not tell us what index they were at.I For some problems, we may need this information

First SolutionRevise our iterator to tell us, not the value in each list cell, but the indexin each list cell

ProblemSomewhat more awkward to use, since we would needarray.get(iterator.next()) instead of just iterator.next().But it’s worse than that, because next is defined to return an Item, sowe would have to wrap the index.We could deal with this by overloading get to take an Item argument.

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Instead of that use an IndexIterator Too

Code

p u b l i c c l a s s I n d e x I t e r a t o r{

p r i v a t e ChainNode c u r r e n t ;I n d e x I t e r a t o r ( ) { // c o n s t r u c t o r

c u r r e n t = f i r s t N o d e ;}

p u b l i c boo l ean hasNext ( ){ // j u s t l i k e b e f o r e }

p u b l i c i n t nex t ( ) {i n t i n d e x = c u r r e n t . i n d e x ;c u r r e n t = c u r r e n t . nex t ;r e t u r n i n d e x ;

}}

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Now, Sparse Matrices

FirstSomething Simple!!

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Using Array Linear List for Matrices

Example 0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

Thus

list =row 1 1 2 2 4 4

column 3 5 3 4 2 3value 3 4 5 6 2 6

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Using Array Linear List for Matrices

Example 0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

Thus

list =row 1 1 2 2 4 4

column 3 5 3 4 2 3value 3 4 5 6 2 6

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Now, How do we represent this list using arrays?

Why?We want compact representations...

Use your friend element from arrays

Thus you declare an array element with the Node informationNode element[] = new Node[1000];

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Now, How do we represent this list using arrays?Why?We want compact representations...

Use your friend element from arrays

package d a t a S t r u c t u r e s ;// Us ing Gene r i c i n Javap u b l i c c l a s s Node<Item>{

// e l ement sp r i v a t e i n t row ;p r i v a t e i n t columnp r i v a t e Item va l u e ;// c o n s t r u c t o r s come he r e

}

Thus you declare an array element with the Node informationNode element[] = new Node[1000];

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Now, How do we represent this list using arrays?Why?We want compact representations...

Use your friend element from arrays

package d a t a S t r u c t u r e s ;// Us ing Gene r i c i n Javap u b l i c c l a s s Node<Item>{

// e l ement sp r i v a t e i n t row ;p r i v a t e i n t columnp r i v a t e Item va l u e ;// c o n s t r u c t o r s come he r e

}

Thus you declare an array element with the Node informationNode element[] = new Node[1000];

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What about other representations?

We can use chains tooWe need to modify our node

To something like this

Graphically

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What about other representations?

We can use chains tooWe need to modify our node

To something like thispackage d a t a S t r u c t u r e s ;// Us ing G e n e r i c i n Javap u b l i c c l a s s Node<Item >{

// e l ement sp r i v a t e i n t row ;p r i v a t e i n t column ;p r i v a t e Item v a l u e ;p r i v a t e Node next ;// c o n s t r u c t o r s come he r e

}

Graphically

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What about other representations?We can use chains tooWe need to modify our node

To something like thispackage d a t a S t r u c t u r e s ;// Us ing G e n e r i c i n Javap u b l i c c l a s s Node<Item >{

// e l ement sp r i v a t e i n t row ;p r i v a t e i n t column ;p r i v a t e Item v a l u e ;p r i v a t e Node next ;// c o n s t r u c t o r s come he r e

}

Graphically

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We have then

Example

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One Linear List Per RowExample

0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

Thus

row1 = [(3,3) (5,4)]row2 = [(3,5) (4,7)]row3 = []row4 = [(2,2) (3,6)]

Node Structure

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One Linear List Per RowExample

0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

Thus

row1 = [(3,3) (5,4)]row2 = [(3,5) (4,7)]row3 = []row4 = [(2,2) (3,6)]

Node Structure

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One Linear List Per RowExample

0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

Thus

row1 = [(3,3) (5,4)]row2 = [(3,5) (4,7)]row3 = []row4 = [(2,2) (3,6)]

Node Structure

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Array Of Row Chains

Example

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Compress the row

Use a sparse array!!!

null

null

null

null

null

null

0

1

2

3

4

5

6

7

8

9

0

2

5

8

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We have a problem here

With 99% of the matrix as zeros of nullsWe have the problem that many row pointers are null...

What can you do?Ideas

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We have a problem here

With 99% of the matrix as zeros of nullsWe have the problem that many row pointers are null...

What can you do?Ideas

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However

Problems, problems Will Robinson!!!Ideas?

Yes, access!!We have a problem there!!

It isNot so fast!!!! SOLUTIONS???

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However

Problems, problems Will Robinson!!!Ideas?

Yes, access!!We have a problem there!!

It isNot so fast!!!! SOLUTIONS???

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However

Problems, problems Will Robinson!!!Ideas?

Yes, access!!We have a problem there!!

It isNot so fast!!!! SOLUTIONS???

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Orthogonal List Representation

More complexityBoth row and column lists.

Node Structure

Example 0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

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Orthogonal List Representation

More complexityBoth row and column lists.

Node Structure

Example 0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

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Orthogonal List Representation

More complexityBoth row and column lists.

Node Structure

Example 0 0 3 0 40 0 5 7 00 0 0 0 00 2 6 0 0

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Row Lists

Row direction

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Column Lists

Column Direction

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Orthogonal Lists

All Together

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Go One Step Further

QuestionWe have another problem the sparse arrays in the column and rowpointers!!!

ThusHow it will look like for this (You can try!!!)

0 1 0 0 20 0 0 0 00 0 2 0 00 0 0 0 00 0 1 0 1

(3)

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Go One Step Further

QuestionWe have another problem the sparse arrays in the column and rowpointers!!!

ThusHow it will look like for this (You can try!!!)

0 1 0 0 20 0 0 0 00 0 2 0 00 0 0 0 00 0 1 0 1

(3)

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Do not worry!!!

You have MANY VARIATIONS!!!

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Approximate Memory Requirements

Example500 x 500 matrix with 1994 nonzero elements

2D array 500 x 500 x 4 = 1 million bytesSingle Array List 3 x 1994 x 4 = 23,928 bytesOne Chain Per Row 23928 + 500 x 4 = 25,928What about the sparse version even in row and column pointer?

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Approximate Memory Requirements

Example500 x 500 matrix with 1994 nonzero elements

2D array 500 x 500 x 4 = 1 million bytesSingle Array List 3 x 1994 x 4 = 23,928 bytesOne Chain Per Row 23928 + 500 x 4 = 25,928What about the sparse version even in row and column pointer?

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Approximate Memory Requirements

Example500 x 500 matrix with 1994 nonzero elements

2D array 500 x 500 x 4 = 1 million bytesSingle Array List 3 x 1994 x 4 = 23,928 bytesOne Chain Per Row 23928 + 500 x 4 = 25,928What about the sparse version even in row and column pointer?

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Approximate Memory Requirements

Example500 x 500 matrix with 1994 nonzero elements

2D array 500 x 500 x 4 = 1 million bytesSingle Array List 3 x 1994 x 4 = 23,928 bytesOne Chain Per Row 23928 + 500 x 4 = 25,928What about the sparse version even in row and column pointer?

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Approximate Memory Requirements

Example500 x 500 matrix with 1994 nonzero elements

2D array 500 x 500 x 4 = 1 million bytesSingle Array List 3 x 1994 x 4 = 23,928 bytesOne Chain Per Row 23928 + 500 x 4 = 25,928What about the sparse version even in row and column pointer?

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Runtime PerformanceExample Matrix Transpose500 x 500 matrix with 1994 nonzero elements:

Method Time2D Array 210 ms

Single Array List 6 msOne Chain per Row 12 ms

Example Matrix Addition500 x 500 matrix with 1994 nonzero elements:

Method Time2D Array 880 ms

Single Array List 18 msOne Chain per Row 29 ms

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Runtime PerformanceExample Matrix Transpose500 x 500 matrix with 1994 nonzero elements:

Method Time2D Array 210 ms

Single Array List 6 msOne Chain per Row 12 ms

Example Matrix Addition500 x 500 matrix with 1994 nonzero elements:

Method Time2D Array 880 ms

Single Array List 18 msOne Chain per Row 29 ms

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