Project 1: Sparse Matrix Class

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Project 1 Sparse Matrix Class

• A matrix is sparse if it has only a small number
  of non-zeros.
• t non-zeros mn entries
• t is very small comparing to mn
• Example - A 55 sparse matrix M with 11 non-zeros

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Todays Outline

• Understand Sparse Matrix representation
• Go over methods in Sparse Matrix class
• Algorithms used in set and removeRow methods
• Discuss experiments on matrix multiplication
• Note
• In this project, we only deal with integer
  matrices.

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Store Sparse Matrices

• Sparse Matrices in real applications are usually
  huge. Much space can be saved if we only store
  non-zeros.
• Space cost for all matrix entries O(mn)
• Space cost for non-zeros entries O(t)
• Problem If only non-zeros are stored, we also
  need to store their location in a matrix. So a
  good representation is required.

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Representation of Sparse Matrix

• Five protected variables are defined in Sparse
  Matrix class
• Height number of rows
• Width number of columns
• 3 Vector Objects to store non-zeros entries and
  their locations.

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Representation of Sparse Matrix (cont.)

• Three Vector variables in class SparseMatrix
• values
• A Vector whose size of non-zero entries
• Elements are stored by column.
• Exercise
• Write the values vector for matrix M.

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Representation of Sparse Matrix (cont.)

• columnPointers
• A vector whose size n of columns
• It contains the starting location for each column
  in vector values.
• columnPointersi c means the first non-zero
  entry in column i has a location c in values.
• If column i are all zeros, then columnPointersi
  -1.
• Exercise
• 1. Write the columnPointers vector for matrix M.
• 2. How to decide the number of non-zeros in
  column i?

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Representation of Sparse Matrix (cont.)

• rowIndices
• A vector whose size size of values of
  non-zero entries
• It contains the row indices for those non-zero
  entries in values.
• rowIndicesi r means the ith element of
  values is in row r of the sparse matrix.
• Exercise
• Write the rowIndices vector for matrix M.

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Methods in Sparse Matrix Class

• Sparse Matrix class support all the methods in
  Matrix class, plus 3 new methods.
• More details about removeRow and removeCol methods

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Methods in Sparse Matrix Class

• Three New methods
• public SparseMatrix ( Matrix x )
• //constructs a SparseMatrix represeanting the
  same matrix as x.
• public Matrix getRow (int r)
• //returns the row in the same way as removeRow,
  without removing it from the matrix.
• public Matrix getCol (int c)
• //Return the row in the same way as removeCol,
  without removing it from the matrix.

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Algorithm for set methods

• The set method has the same function as in class
  Matrix. That is, set a new value at specified
  location.
• Receive 3 parameters public void set (int r, int
  c, int value)
• Algorithm
• Step 1. If (value 0 Mrc0), return
• If (value ! 0), go to Step 2
• If (value 0 Mrc!0), go to step
  4.
• Step 2. If (column c has no non-zeros entry ) or
  (Mrc 0), go to step 3. Otherwise go to
  Step 5.
• Step 3. Find the location in values and add
  value. Update columnPointers and rowIndices
  accordingly.
• Step 4. Remove Mrc from values and rowIndices
  , update columnPointers .
• Step 5. Find the location l of Mrc in values.
  Set valuesl value.
• Q1 At most how many values in columnPointers
  need to be updated in step 3?
• Q2 Is it easy to find the location of Mrc?

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Algorithm for removeRow methods

• The removeRow method removes a row from matrix
  and returns a 2-row matrix.
• Receive 1 parameter public void removeRow (int
  r)
• To remove a row, we need to do the following
  work.
• Decrease Height by 1.
• Remove elements from vector values and rowIndices
  if the row has non-zeros.
• Update the vector columnPointers
• Construct the returning matrix object

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Algorithm for removeRow methods (cont.)

• Algorithm - public void removeRow (int r)
• Step 1. Search rowIndices for next non-zero value
  in row r. If a non-zero in row r is found, say v
  valuesl , go to step 2. Otherwise go to step
  3
• Step 2. Determine the column index c for v. Add v
  to row_val_vect, c to col_vect, l to indice_vect
  Go back to step 1
• Step 3. Construct returning Matrix based on
  row_val_vect and col_vect
• Step 4. Remove non-zeros in row r from values and
  rowIndices. Update columnPointers.

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Tests and Experiments - Matrix Multiplication

• Let C be the product of matrix A and B, in order
  to the multiplication, the dimensions of the
  matrics must satisfy that
• (mn) (np) (mp)
• A B C
• Entries in C is defined as the following
• C(i,j) A(i,0)B(0,j)A(i,1)B((1,j)A(i,n-1)
  B(n-1, j)
• i.e. C(i,j) (the ith row of A ) (the jth
  column of B)
• Example

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Tests and Experiments - Matrix Multiplication
(cont.)

• In our experiments, we only use square matrices.
• A ,B, and their product C are all NN matrices.
• Run the program for N1000, 2000, ..., 10,000.
• In the construction of A and B, each entry is set
  to be non-zero with probability p. (You can use
  Random object in Java do this generate a number
  uniformly distributed on 0.0, 1.0).

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Tests and Experiments - Timing

• For both SparseMatrix and Matrix objects, record
  the time for
• Matrix construction
• Matrix multiplication
• 4 groups of data are generated to do the
  plotting.
• Matrix construction
• SparseMatrix construction
• Matrix mulitplicaion
• SparseMatrix muliplicaion

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Plotting tools

• Matlab, Excel, Gnuplot,
• x-axis value of N (matrix row/col number)
• y-axis time
• Explain your plots with several paragraphs