Matrix Algorithms and Sparse Representation Assignment for CMPS 101 | Assignments Computer Science

CMPS 101

Algorithms and Abstract Data Types

Summer 2009

Programming Assignment 3

Due Saturday July 25, 10:00 pm

In this assignment you will create a calculator for performing matrix operations that exploits the

(expected) sparseness of it’s matrix operands. An

nn

square matrix is said to be sparse if the number

of non-zero entries (abbreviated NNZ) is small compared to the total number of entries

. The result

will be a Java program capable of performing fast matrix operations, even on very large matrices,

provided they are sparse.

Given

nn

matrices A and B, their product

BAC 

is the

nn

matrix whose

th entry is given by

ikij

BAC 



Thus the element in the ith row and jth column of C is the vector dot product of the ith row of A and the jth

column of B. If we consider addition and multiplication of real numbers to be our basic operations, then

the above formula can be computed in time

)(

n

, which is impractical for matrix sizes n of more than a

few thousand. If it so happens that A and B are sparse, then a great many of these arithmetic operations

are unnecessary. The sum S, and difference D, of A and B are the

nn

matrices having

th entries:

ijijij

BAS 

and

ijijij

BAD 

The scalar product of a real number x with A is denoted

, and has

th entry

ijij

AxxA )(

. The

transpose of A, denoted

, is the matrix T whose

entry is the

entry of A:

jiij

AT 

. In other

words, transposition “flips” the rows and columns of A. Each of these operations can be computed in

time

)(

n

, and just as for multiplication, their cost can be improved upon significantly when A and B

are sparse.

As one would expect, the cost of a matrix operation depends heavily on the choice of data structure used

to represent the matrix operands. There are several ways to represent a matrix with real entries. The

standard approach is to use a 2-dimensional

nn

array of doubles. The advantage of this representation

is that all of the above matrix operations have a straight-forward implementation. In this project you will

represent a matrix as a 1-dimensional array of Lists. Each List will represent one row of the Matrix, but

only the non-zero entries will be stored. Therefore List elements must store not just the matrix entries, but

the columns in which those entries reside. For example, the matrix below would have the following

representation as an array of Lists.















0.50.40.0

0.00.00.3

0.20.00.1

Array of Lists:







)0.5,3()0.4,2(:3

)0.3,1(:2

)0.2,3()0.1,1(:1

This method obviously results in a substantial space savings when the Matrix is sparse. In addition, the

standard matrix operations defined above can be performed more efficiently on sparse matrices. You will

find however that the matrix operations are much less straight-forward to implement using this

representation. The trade-off then, is a gain in space and time efficiency for sparse matrices, at the

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CMPS 101

Algorithms and Abstract Data Types Summer 2009

Programming Assignment 3

Due Saturday July 25, 10:00 pm

In this assignment you will create a calculator for performing matrix operations that exploits the

(expected) sparseness of it’s matrix operands. An n^  n^ square matrix is said to be sparse if the number

of non-zero entries (abbreviated NNZ) is small compared to the total number of entries (^) n^2. The result will be a Java program capable of performing fast matrix operations, even on very large matrices, provided they are sparse.

Given n^  n^ matrices A and B , their product C  A  B is the n^  n^ matrix whose ij^ th^ entry is given by

kj n k

Cij  A ikB



Thus the element in the i th^ row and j th^ column of C is the vector dot product of the i th^ row of A and the j th column of B. If we consider addition and multiplication of real numbers to be our basic operations, then the above formula can be computed in time ( n^3 ), which is impractical for matrix sizes n of more than a few thousand. If it so happens that A and B are sparse, then a great many of these arithmetic operations

are unnecessary. The sum S , and difference D , of A and B are the n^  n^ matrices having ij^ th^ entries:

S (^) ij  A (^) ij  B ij and Dij  Aij  Bij The scalar product of a real number x with A is denoted xA^ , and has ij^ th^ entry (^ xA^ ) ij  x  Aij. The transpose of A , denoted (^) AT , is the matrix T whose ij th entry is the ji th entry of A : Tij^  A^ ji. In other words, transposition “flips” the rows and columns of A. Each of these operations can be computed in time ( n^2 ), and just as for multiplication, their cost can be improved upon significantly when A and B are sparse. As one would expect, the cost of a matrix operation depends heavily on the choice of data structure used to represent the matrix operands. There are several ways to represent a matrix with real entries. The

standard approach is to use a 2-dimensional n^  n^ array of doubles. The advantage of this representation

is that all of the above matrix operations have a straight-forward implementation. In this project you will represent a matrix as a 1-dimensional array of Lists. Each List will represent one row of the Matrix, but only the non-zero entries will be stored. Therefore List elements must store not just the matrix entries, but the columns in which those entries reside. For example, the matrix below would have the following representation as an array of Lists.           

0 4. 0 5. 0
0 0. 0 0. 0
0 0. 0 2. 0 M (^) Array of Lists:      3 : ( 2 , 4. 0 ) ( 3 , 5. 0 ) 2 : ( 1 , 3. 0 ) 1 : ( 1 , 1. 0 ) ( 3 , 2. 0 ) This method obviously results in a substantial space savings when the Matrix is sparse. In addition, the standard matrix operations defined above can be performed more efficiently on sparse matrices. You will find however that the matrix operations are much less straight-forward to implement using this representation. The trade-off then, is a gain in space and time efficiency for sparse matrices, at the

expense of more complicated algorithms for performing standard matrix operations. Designing these algorithms in terms of List operations will constitute the majority of the work you do on this assignment. It will be necessary to make some minor changes to your List ADT from pa1. You must first convert your List ADT from a List of ints to a List of Objects. This entails changing certain field types, declaration statements, method parameters, and return types from int to Object. The Objects referred to by these List elements will be defined in the Matrix ADT specified below. Second, it will be necessary to eliminate the List operations copy() and cat() (which was optional anyway.) All other List operations from pa1 will be retained. The equals() operation however will be altered slightly so as to override, rather than overload, Object's built in equals() method. This is done by changing it's signature from boolean equals(List L), as in pa1 to public boolean equals(Object x), which is it's signature in the superclass Object. Indeed, all equals() methods in this project should carry this same signature. File Formats The top level client for this project will be called Sparse.java. It will take two command line arguments giving the names of the input and output files, respectively. The input file will begin with a single line containing three integers n , a , and b , separated by spaces. The second line will be blank, and the

following a lines will specify the non-zero entries of an n^  n^ matrix A. Each of these lines will contain a

space separated list of three numbers: two integers and a double, giving the row, column, and value of the corresponding matrix entry. After another blank line, will follow b lines specifying the non-zero entries

of an n^  n^ matrix B. For example, the two matrices

          

0 8. 0 9. 0
0 5. 0 6. 0
0 2. 0 3. 0 A (^) and           
0 1. 0 1. 0
0 0. 0 0. 0
0 0. 0 1. 0 B are encoded by the following input file: 3 9 5 1 1 1. 1 2 2. 1 3 3. 2 1 4. 2 2 5. 2 3 6. 3 1 7. 3 2 8. 3 3 9. 1 1 1. 1 3 1. 3 1 1. 3 2 1. 3 3 1. Your program will read an input file such as above, initialize and build the Array of Lists representation of the matrices A and B , then calculate and print the following matrices to the output file: A , B , (^1.^5 ) A , A  B , A  A , B  A , A  A , (^) AT , AB , and (^) B^2. The output file format is illustrated by the following example, which corresponds to the above input file.

here as Entry. Thus Entry will have two fields that store types int and double respectively. Entry must also contain its own equals() and toString() methods which override the corresponding methods in the Object superclass. Your Matrix class will represent a matrix as an array of Lists of Entry Objects. It is highly recommended that these Lists be maintained in column sorted order. Your Matrix ADT will export the following operations. // Constructor //////////////////////// Matrix(int n) // Makes a new n x n zero Matrix. pre: n>= // Access functions /////////////////// int getSize() // Returns n, the number of rows and columns of this Matrix int getNNZ() // Returns the number of non-zero entries in this Matrix public boolean equals(Object x) // overrides Object's equals() method // Manipulation procedures //////////// void makeZero() // sets this Matrix to the zero state Matrix copy()// returns a new Matrix having the same entries as this Matrix void changeEntry(int i, int j, double x) // changes ith row, jth column of this Matrix to x // pre: 1<=i<=getSize(), 1<=j<=getSize() Matrix scalarMult(double x) // returns a new Matrix that is the scalar product of this Matrix with x Matrix add(Matrix M) // returns a new Matrix that is the sum of this Matrix with M // pre: getSize()==M.getSize() Matrix sub(Matrix M) // returns a new Matrix that is the difference of this Matrix with M // pre: getSize()==M.getSize() Matrix transpose() // returns a new Matrix that is the transpose of this Matrix Matrix mult(Matrix M) // returns a new Matrix that is the product of this Matrix with M // pre: getSize()==M.getSize() // Other functions //////////////////// public String toString() // overrides Object's toString() method It is required that your program perform these operations efficiently. Let n be the number of rows in A , and let a and b denote the number of non-zero entries in A and B respectively. Then functions copy(), changeEntry(), scalarMult(), and transpose() should have cost (^ n^  a ) when applied to A , in worst case. Functions add() and sub() can be implemented to run in time (^ n^  a  b ), and mult() should run in time  ( n  a  b ). It will be helpful to include a private function with signature private static double dot(List P, List Q) that computes the vector dot product of two matrix rows represented by Lists P and Q. Use this function together with function transpose() to help implement mult(). Similar helper functions for the operations add() and sub() will also be useful. What to Turn In Your project will be structured in three files: Sparse.java, Matrix.java, and List.java. The main program, Sparse, will handle the input and output files and is the client of Matrix, which is itself the client of List.

Note that Sparse is not itself a direct client of List, since it need not call any List operations. You will also write separate client modules ListTest.java and MatrixTest.java to test the List and Matrix ADTs in isolation. Also submit a README and a Makefile that creates an executable jar file called Sparse. Thus seven files in all will be turned in: Sparse.java, Matrix.java, MatrixTest.java, List.java, ListTest.java, Makefile, and README. Submit these to the assignment name pa3 by the due date. As always start early and ask questions.

Matrix Algorithms and Sparse Representation Assignment for CMPS 101, Assignments of Computer Science

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Partial preview of the text

Download Matrix Algorithms and Sparse Representation Assignment for CMPS 101 and more Assignments Computer Science in PDF only on Docsity!

CMPS 101

Programming Assignment 3

Due Saturday July 25, 10:00 pm

(expected) sparseness of it’s matrix operands. An n^  n^ square matrix is said to be sparse if the number

Given n^  n^ matrices A and B , their product C  A  B is the n^  n^ matrix whose ij^ th^ entry is given by

Cij  A ikB

are unnecessary. The sum S , and difference D , of A and B are the n^  n^ matrices having ij^ th^ entries:

standard approach is to use a 2-dimensional n^  n^ array of doubles. The advantage of this representation

following a lines will specify the non-zero entries of an n^  n^ matrix A. Each of these lines will contain a

of an n^  n^ matrix B. For example, the two matrices