Contents

## M/R Algorithms

### Basic Algorithms

#### Addition

#### Addition of multiple matrices

#### Multiplication

- Iterative Approach

For i = 0 step 1 until N -1 Job: Computes the ith row of C = Matrix-Vector multiplication Iterative job: - A map task receives a row n of B as a key, and vector of row as its value - Multiplying by all columns of ith row of A - Reduce task find and add the ith product 1st + + + + | a11 a12 a13 | | a11 a21 a31 | | ... ... ... | X | a12 a22 a32 | | ... ... ... | | a13 a23 a33 | + + + + 2nd + + + + | ... ... ... | | a11 a21 a31 | | a21 a22 a23 | X | a12 a22 a32 | | ... ... ... | | a13 a23 a33 | + + + + ....

- Blocking Algorithm Approach

To mutliply two dense matrices A and B, We collect the blocks to 'collectionTable' firstly using map/reduce. Rows are named as c(i, j) with sequential number ((N^2 * i) + ((j * N) + k) to avoid duplicated records.

CollectionTable: matrix A matrix B ------------------------+------------------------------- block(0, 0)-0 block(0, 0) block(0, 0) block(0, 0)-1 block(0, 1) block(1, 0) block(0, 0)-2 block(0, 2) block(2, 0) ... N ... block(N-1, n-1)-(N^3-1) block(N-1, N-1) block(N-1, N-1)

Each row has a two sub matrices of a(i, k) and b(k, j) so that minimized data movement and network cost.

Blocking jobs: Collect the blocks to 'collectionTable' from A and B. - A map task receives a row n as a key, and vector of each row as its value - emit (blockID, sub-vector) pairs - Reduce task merges block structures based on the information of blockID Multiplication job: - A map task receives a blockID n as a key, and two sub-matrices of A and B as its value - Multiply two sub-matrices: a[i][j] * b[j][k] - Reduce task computes sum of blocks - c[i][k] += multiplied blocks

#### Matrix Norm

- Find the maximum absolute row sum of matrix

Matrix.Norm.One is that find the maximum absolute row sum of matrix. Comparatively, it's a good fit with MapReduce model because doesn't need iterative jobs or table/file JOIN operations.

j=n The maximum absolute row sum = max ( sum | a_{i,j} | ) 1<=i<=n j=1 - A map task receives a row n as a key, and vector of each row as its value - emit (row, the sum of the absolute value of each entries) - Reduce task select the maximum one

NOTE: Matrix.infinity, Matrix.Maxvalue and Matrix.Frobenius are almost same with this.

#### Compute the transpose of matrix

The transpose of a matrix is another matrix in which the rows and columns have been reversed. The matrix must be square for this work.

+ + + + | a11 a12 a13 | | a11 a21 a31 | | a21 a22 a23 | => | a12 a22 a32 | | a31 a32 a33 | | a13 a23 a33 | + + + + - A map task receives a row n as a key, and vector of each row as its value - emit (Reversed index, the entry with the given index) - Reduce task sets the reversed values

#### Compute the determinant of square matrix

### Decomposition Algorithms

#### Cholesky Decomposition