January 16 Do not hand in a listing of the file InvalidRowCol.java.

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1 0 Changes CSE 2011 Fundamentals of Data Structures Report 1: Sparse Matrices Due: Thursday, January 31, 1pm Where: In class If the class has begun your report is late January 16 Do not hand in a listing of the file InvalidRowCol.java. 1 Main points Be sure to read and follow all the guidelines from the links on reports and academic honesty from the WWW home page for the course. The specification is the union of this document plus the program text you are given. 1.1 Learning objectives Review Java programming with arrays and singly linked lists Review abstract data types, classes and objects Specification of interfaces Program assertions pre- and post-conditions, class invariants and loop invariants Big O analysis for running time and space requirements Testing 1.2 To hand in Hand in the following items as a package in order given in the following. 1. Cover page printed from the course web pages 2. Design document Section 4 3. A listing of the file IntMatrixElement.java Section A listing of the file IntSparseMatrix.java Section A listing of the file MinOutTest.java Section Electronic submission Before the deadline, submit a directory called report1 that should contain all java files for the system. No other files should be submitted. To submit, use the following command on Prism: submit 2011 r1 report1. Files cannot be deleted the submit command can only add or replace files so be very careful to clean up your directory before any submission. While you can develop your system on your personal computer, be sure your system will compile and execute on Prism. 1.4 To get started Copy the file called report1.tar.gz from the course directory /cs/course/2011 to a local directory. When you untar with the command tar xzf report1.tar.gz, you will get a directory called report1 that contains the following files. The method main in the class MinOutTest is the starting point for the execution of the system. 1. IntMatrixElement you need to complete the class definition. If you complete this correctly, then the system will compile and execute with no changes to the other files. 2. IntSparseMatrix you need to complete the class definition

2 3. MinOutTest.java you need to complete the class definition. 4. MatrixElement do not change the file. 5. SparseMatrix do not change the file. 2 Sparse matrix representation A sparse matrix is a matrix with elements with a predominance of zero or null values. For example, Figure 1 shows a standard representation for a 9 x 10 matrix that contains only four non-zero matrix elements the values 3, -1, 4 and 7 shown in magenta. If the standard two-dimensional array storage were used then the array would use 90 integers of space 360 bytes of storage, plus the array descriptor overhead. Figure 1: The standard representation of a 9x10 matrix shown. All zero elements take up space. Rather than use storage space to store the zero values, in a sparse matrix representation only non-zero entries are stored. There are many ways a sparse matrix can be implemented. Figure 2 shows a sparse matrix representation for the matrix represented in Figure 1 that is used for the system you are to complete for Report 1. Note that the 9x10 array is embedded in a representation that can hold a 12x12 array. The example shown permits storage of arrays of size P x Q, where P and Q can range over the values 1 through 12 inclusive. In addition to the user data consisting of the values stored in the matrix, there is a need to store metadata data about the user data. For the sparse matrix representation chosen for this report, the metadata consists of the following. 1. As there is one singly linked list for each row and for each column, there are two one-dimensional arrays rowheader and columnheader that store pointers to the row and column lists. 2. MaxRowCol that records the size of the row and column header arrays. This number represents the maximum number of rows and the maximum number of columns for matrices that can be stored. 3. actualrows and actualcols that represent the actual number of rows and the actual number of columns for the current matrix. 4. maxnonzerorow and maxnonzerocol that represent the maximum row with a nonzero element and the maximum column with a nonzero element

3 Figure 2: Example of a 9 x10 sparse matrix with four non-zero elements at [3,2], [3,10], [8,4] and [8,10] stored in a representation that can store up to a 12 x 12 matrix. Each matrix element contains, in addition to the user data, metadata consisting of the row and column position for the element, which must be explicitly stored, as the physical location cannot be used. For example, the value 1 is physically the second element in row 3 but logically it is in column 10. Pointers to the next row element and the next column element are also needed to maintain the singly linked lists for the rows and columns. As a consequence, each array element has four fields, of 4 bytes each, as metadata; each element requires, in the example, 20 bytes of storage, compared to 4 bytes in the standard representation. Figure 3: Shows the fields for a sparse matrix element

4 3 Tasks 3.1 The interfaces & exceptions MatrixElement interface The interface MatrixElement specifies the methods associated with a matrix element in the sparse matrix representation to be used for this report. You are to use the interface but you are not to change it. The following is a list of the methods. getrow setrow(row) getcolumn setcolumn(col) getdata setdata(data) getnextrow setnextrow(nextrowelem) getnextcolumn setnextcolumn(nextcolelem) SparseMatrix interface The interface file SparseMatrix.java specifies the three operations that are described in the following paragraphs. insertelement(row, col, data) InsertElement modifies the sparse matrix structure. If there is no element at the location [row, col], then a new matrix element is inserted in the matrix. If there already exists a matrix element at [row, col] then the data replaces the existing value. Matrix elements with a zero data value are not stored, as a consequence replacing an existing value with zero is equivalent to deleting the element from the matrix. By definition, the maximum non-zero row and column for a particular matrix contain nonzero data values and all rows and columns above the maximum non-zero row or column contains only zero values, as a consequence inserting a data value may increase or decrease the maximum non-zero row and/or column metadata for the matrix. display For a matrix with no rows or columns, print the message The matrix is empty. For a non-empty matrix, the procedure is to print the matrix in the style shown in the following example display for a 3 x 4 matrix. For the purpose of the report, positive integers are to be in the range and negative integers are to be in the range ; that way each data value takes up 3 columns in the output. The character represents a space character. sum(sparsematrix) ** ** 2 ** 1-99 ** 3 ** ** ** 6 Sum is a function to be used as in the following expression a new matrix is created, neither matrix1 nor matrix2 are to be modified newmatrix matrix1.sum(matrix2) It is the equivalent of the following mathematical expression. newmatrix = matrix1 + matrix2 The basis of the sum algorithm is the merging of the elements from both matrices. Think of the elements of each matrix as being logically a single sequence by considering the rows as being appended to each other (alternately the columns could be appended). Then do a merge operation using the row and column indices. If the locations differ, then a copy of the earlier element is put into the sum. If the locations are the same, then the data values are summed and the element put into the sum. Use the insertelement operation to add the new elements to the sum matrix

5 3.1.3 Invalid Row or Column exception Create and document a run time exception class InvalidRowCol.java for out of range row or column indices. Your minimum output test must produce two lines of output showing that the exception is caught for both row and column exceptions. 3.3 Integer sparse matrix Complete and document, with appropriate comments, require clauses (pre-conditions), ensure clauses (post-conditions), class invariants, and loop invariants, the following two classes. 1. IntMatrixElement.java implements MatrixElement.java using integers. 2. IntSparseMatrix.java implements SparseMatrix.java using IntMatrixElement. You must program your own list operations. While a singly linked list class could be used for either the rows or the columns, it cannot easily be used for both, thus leading to an asymmetric implementation. It is better to treat rows and columns symmetrically in the same way. You will need private support features to simplify the implementation. 3.4 Test implementation Complete and document a minimal output test program, MinOutTest.java, to test IntSparseMatrix. The Java file should contain appropriate comments to help the reader follow and understand what you are doing. Add, by hand, appropriate diagrams. 4 Design document Your design document is a maximum of 10 pages. It should have the following sections. Aside from diagrams, which can be hand drawn, documentation should be written using a word processor or text editor. 4.1 System overview This section presents an overview and description of the system, the classes and their relationship. It should include appropriate diagrams. 4.2 Discussion of space overhead This section presents a discussion of the space overhead of a sparse matrix. Develop expressions that give the amount of overhead for the following cases as a function of the maximum number of rows and columns of a matrix i.e. the metadata variable MaxRowCol. Assume that the actual size of matrices is the maximum for the representation i.e. actualrows = actualcols = MaxRowCol. From your expressions describe the big O analysis of the overhead space requirement. 1. Assume a matrix contains only one non-zero element. 2. Assume a matrix contains all non-zero elements. Compare the space requirements of the sparse matrix representation used in this report with the space requirements for standard matrix storage. Describe the cutoff where the sparse matrix representation begins to save space. 4.3 Discussion of running time Give a discussion of the big O running time for the insert and display algorithms based on the assumptions in the first paragraph of Section

6 4.4 Multi-dimensional sparse matrices Describe how the representation used in this specification for two-dimensional sparse matrices can be extended to an arbitrary number of dimensions. 5 Grading Scheme The grade for the report is partitioned into the following 5 parts. Presentation and general considerations 10% Design document 30% Programming 20% Program text documentation comments and assertions 20% Testing 20% - 6 -