CMPS 101 Algorithms and Abstract Data Types Programming Assignment 3 solution

Description

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  AikB


1
.
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 with 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
( )
3  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
involve adding or multiplying by zero, hence are unnecessary.
The sum S, and difference D, of A and B are the
n n
matrices having
ij
th
entries:
Sij
 Aij  Bij
and Dij
 Aij Bij
The scalar product of a real number x with A is denoted
xA
, and has
ij
th
entry
ij Aij (xA)  x 
. The transpose
of A, denoted
T A , is the matrix whose
th ij
entry is the
th ji
entry of A:
ij ji
T
(A )  A
. In other words, the
rows of A are the columns of
T A
, and the columns of A are the rows of
T A . Each of these operations can
be computed in time
( )
2  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
n n
array of doubles. The advantage of this representation is
that all of the above matrix operations have a straight-forward implementation using nested loops. This
project will use a very different representation however. Here 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.












0.0 4.0 5.0
3.0 0.0 0.0
1.0 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. As you
will see though, the matrix operations are much more difficult 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 our List ADT
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. First you must 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 module 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












7.0 8.0 9.0
4.0 5.0 6.0
1.0 2.0 3.0
A and











1.0 1.0 1.0
0.0 0.0 0.0
1.0 0.0 1.0
B
are encoded by the following input file:
3 9 5
1 1 1.0
1 2 2.0
1 3 3.0
2 1 4.0
2 2 5.0
2 3 6.0
3 1 7.0
3 2 8.0
3 3 9.0
1 1 1.0
1 3 1.0
3 1 1.0
3 2 1.0
3 3 1.0
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, 𝐴
𝑇
, 𝐴𝐵 and 𝐵
2
. The output file format is illustrated by the following example, which
corresponds to the above input file.
A has 9 non-zero entries:
1: (1, 1.0) (2, 2.0) (3, 3.0)
2: (1, 4.0) (2, 5.0) (3, 6.0)
3: (1, 7.0) (2, 8.0) (3, 9.0)
B has 5 non-zero entries:
1: (1, 1.0) (3, 1.0)
3: (1, 1.0) (2, 1.0) (3, 1.0)
(1.5)*A =
1: (1, 1.5) (2, 3.0) (3, 4.5)
2: (1, 6.0) (2, 7.5) (3, 9.0)
3: (1, 10.5) (2, 12.0) (3, 13.5)
A+B =
1: (1, 2.0) (2, 2.0) (3, 4.0)
2: (1, 4.0) (2, 5.0) (3, 6.0)
3: (1, 8.0) (2, 9.0) (3, 10.0)
A+A =
1: (1, 2.0) (2, 4.0) (3, 6.0)
2: (1, 8.0) (2, 10.0) (3, 12.0)
3: (1, 14.0) (2, 16.0) (3, 18.0)
B-A =
1: (2, -2.0) (3, -2.0)
2: (1, -4.0) (2, -5.0) (3, -6.0)
3: (1, -6.0) (2, -7.0) (3, -8.0)
A-A =
Transpose(A) =
1: (1, 1.0) (2, 4.0) (3, 7.0)
2: (1, 2.0) (2, 5.0) (3, 8.0)
3: (1, 3.0) (2, 6.0) (3, 9.0)
A*B =
1: (1, 4.0) (2, 3.0) (3, 4.0)
2: (1, 10.0) (2, 6.0) (3, 10.0)
3: (1, 16.0) (2, 9.0) (3, 16.0)
B*B =
1: (1, 2.0) (2, 1.0) (3, 2.0)
3: (1, 2.0) (2, 1.0) (3, 2.0)
Notice that the rows are to be printed in column sorted order, and zero rows are skipped altogether. On the
other hand, the input file may give the matrix entries in any order.
Matrix ADT Specifications
In addition to the main program Sparse.java and the altered List.java from pa1, you will implement a Matrix
ADT in a file called Matrix.java, which defines the Matrix class. This class will contain a private inner
class (similar to Node in your List ADT) that encapsulates the column and value information corresponding
to a matrix entry. You may give this inner class any name you wish, but I will refer to it 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 required 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>=1
// 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 the worst case run times
of the above functions should have the following asymptotic growth rates.
A.changeEntry(): Θ(𝑎)
A.copy(): Θ(𝑛 + 𝑎)
A.scalarMult(x): Θ(𝑛 + 𝑎)
A.transpose(): Θ(𝑛 + 𝑎)
A.add(B): Θ(𝑛 + 𝑎 + 𝑏)
A.sub(B): Θ(𝑛 + 𝑎 + 𝑏)
A.mult(B): Θ(𝑛
2 + 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, and are highly recommended.
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. Students often ask what should be the contents of these test files. In each case,
include enough calls to ADT operations to convince the grader that you did in fact test your List and Matrix
ADT modules in isolation before using them in the larger project. The best way to do this is to actually use
them for this purpose. At minimum they should call every public function in their respective ADT modules
at least once.
Also submit a README file and a Makefile that creates an executable jar file called Sparse. Thus seven
files in all will be turned in:
Sparse.java
Matrix.java
List.java
MatrixTest.java
ListTest.java
Makefile
README
Submit these to the assignment name pa3 by the due date. As always, start early and ask questions.