is the and x A. The transpose of A, denoted

Transcription

1 CMPS 101 lgorims and bstract Data ypes Programming ssignment 3 In is assignment you will create a calculator for performing matrix operations at exploits e (expected) sparseness of it s matrix operands. n square matrix is said to be sparse if e number of non-zero entries (abbreviated NNZ) is small compared to e total number of entries, n 2. he result will be a Java program capable of performing fast matrix operations, even on very large matrices, provided ey are sparse. n n Given n n matrices and B, eir product C B is e n n matrix whose entry is given by n C k 1 ik B kj. hus e element in e i row and j column of C is e vector dot product of e i row of wi e j column of B. If we consider addition and multiplication of real numbers to be our basic operations, en e above formula can be computed in time, which is impractical for matrix sizes n of more an a few ousand. If it so happens at and B are sparse, en a great many of ese arimetic operations involve adding or multiplying by zero, hence are unnecessary. ( n 3 ) he sum S, and difference D, of and B are e n n matrices having entries: S B B and D he scalar product of a real number x wi is denoted x, and has entry ( x) x. he transpose of, denoted, is e matrix whose entry is e ji entry of : ( ) ji. In oer words, e rows of are e columns of, and e columns of are e rows of. Each of ese operations can be computed in time ( n 2 ), and just as for multiplication, eir cost can be improved upon significantly when and B are sparse. s one would expect, e cost of a matrix operation depends heavily on e choice of data structure used to represent e matrix operands. here are several ways to represent a square matrix wi real entries. he standard approach is to use a 2-dimensional array of doubles. he advantage of is representation is at all of e above matrix operations have a straight-forward implementation using nested loops. his 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 e Matrix, but only e non-zero entries will be stored. herefore List elements must store, not just e matrix entries, but e column index in which ose entries reside. For example, e matrix below would have e following representation as an array of Lists. n n M 3.0 rray of Lists: : 2 : 3: (1, 1.0) (1, 3.0) (2, 4.0) (3, 2.0) (3, 5.0) his meod obviously results in a substantial space savings when e Matrix is sparse. In addition, e standard matrix operations defined above can be performed more efficiently on sparse matrices. s you

2 will see ough, e matrix operations are much more difficult to implement using is representation. he trade-off en, is a gain in space and time efficiency for sparse matrices, at e expense of more complicated algorims for performing standard matrix operations. Designing ese algorims in terms of our List D operations will constitute e majority of e work you do on is assignment. It will be necessary to make some minor changes to your List D from pa1. First you must convert your List D from a List of ints to a List of Objects. his entails changing certain field types, declaration statements, meod parameters, and return types from int to Object. he Objects referred to by ese List elements will be defined in e Matrix D specified below. Second, it will be necessary to eliminate e List operations copy() and cat() (which was optional anyway.) ll oer List operations from pa1 will be retained. he equals() operation however will be altered slightly so as to override, raer an overload Object's built in equals() meod. his 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 e superclass Object. Indeed, all equals() meods in is project should carry is same signature. File Formats he top level client module for is project will be called Sparse.java. It will take two command line arguments giving e names of e input and output files, respectively. he input file will begin wi a single line containing ree integers n, a and b, separated by spaces. he second line will be blank, and e following a lines will specify e non-zero entries of an matrix. Each of ese lines will contain a space separated list of ree numbers: two integers and a double, giving e row, column, and value of e corresponding matrix entry. fter anoer blank line, will follow b lines specifying e non-zero entries of an matrix B. For example, e two matrices n n n n and B are encoded by e following input file: Your program will read an input file such as above, initialize and build e rray of Lists representation of e matrices and B, en calculate and print e following matrices to e output file:, B, (1.5), + B,

3 +, B,,, B and B 2. he output file format is illustrated by e following example, which corresponds to e above input file. 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)* = 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) +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, 1) + = 1: (1, 2.0) (2, 4.0) (3, 6.0) 2: (1, 8.0) (2, 1) (3, 12.0) 3: (1, 14.0) (2, 16.0) (3, 18.0) B- = 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) - = ranspose() = 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) *B = 1: (1, 4.0) (2, 3.0) (3, 4.0) 2: (1, 1) (2, 6.0) (3, 1) 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 at e rows are to be printed in column sorted order, and zero rows are skipped altogeer. On e oer hand, e input file may give e matrix entries in any order. Matrix D Specifications In addition to e main program Sparse.java and e altered List.java from pa1, you will implement a Matrix D in a file called Matrix.java, which defines e Matrix class. his class will contain a private inner

4 class (similar to Node in your List D) at encapsulates e column and value information corresponding to a matrix entry. You may give is inner class any name you wish, but I will refer to it here as Entry. hus Entry will have two fields at store types int and double respectively. Entry must also contain its own equals() and tostring() meods which override e corresponding meods in e Object superclass. Your Matrix class will represent a matrix as an array of Lists of Entry Objects. It is required at ese Lists be maintained in column sorted order. Your Matrix D will export e following operations. // Constructor Matrix(int n) // Makes a new n x n zero Matrix. pre: n>=1 // ccess functions int getsize() // Returns n, e number of rows and columns of is Matrix int getnnz() // Returns e number of non-zero entries in is Matrix public boolean equals(object x) // overrides Object's equals() meod // Manipulation procedures void makezero() // sets is Matrix to e zero state Matrix copy()// returns a new Matrix having e same entries as is Matrix void changeentry(int i, int j, double x) // changes i row, j column of is Matrix to x // pre: 1<=i<=getSize(), 1<=j<=getSize() Matrix scalarmult(double x) // returns a new Matrix at is e scalar product of is Matrix wi x Matrix add(matrix M) // returns a new Matrix at is e sum of is Matrix wi M Matrix sub(matrix M) // returns a new Matrix at is e difference of is Matrix wi M Matrix transpose() // returns a new Matrix at is e transpose of is Matrix Matrix mult(matrix M) // returns a new Matrix at is e product of is Matrix wi M // Oer functions public String tostring() // overrides Object's tostring() meod It is required at your program perform ese operations efficiently. Let n be e number of rows in, and let a and b denote e number of non-zero entries in and B respectively. hen e worst case run times of e above functions should have e following asymptotic grow rates..changeentry():.copy():.scalarmult(x):.transpose():.add(b):.sub(b):.mult(b): Θ(a) Θ(n + a + b) Θ(n + a + b) Θ(n 2 + a b) It will be helpful to include a private function wi signature

5 private static double dot(list P, List Q) at computes e vector dot product of two matrix rows represented by Lists P and Q. Use is function togeer wi function transpose() to help implement mult(). Similar helper functions for e operations add() and sub() will also be useful, and are highly recommended. What to urn In Your project will be structured in ree files: Sparse.java, Matrix.java, and List.java. he main program, Sparse, will handle e input and output files and is e client of Matrix, which is itself e client of List. Note at Sparse is not itself a direct client of List, since it need not call any List operations. You will also write separate client modules Listest.java and Matrixest.java to test e List and Matrix Ds in isolation. Students often ask what should be e contents of ese test files. In each case, include enough calls to D operations to convince e grader at you did in fact test your List and Matrix D modules in isolation before using em in e larger project. he best way to do is is to actually use em for is purpose. t minimum ey should call every public function in eir respective D modules at least once. lso submit a REDME file and a Makefile at creates an executable jar file called Sparse. hus seven files in all will be turned in: Sparse.java Matrix.java List.java Matrixest.java Listest.java Makefile REDME Submit ese to e assignment name pa3 by e due date. s always, start early and ask questions.