1. Multidimensional Arrays

2. Two Dimensional Arrays
Table of data for each experiment and outcome.
Table of grades for each student and assignments.
Table of grayscale values for each pixel in a 2D image.
Mathematical abstraction. Matrix.
Java abstraction. 2D array.
"Microarrays allow us to see if a particular gene is on or off in a particular tissue. The colors on this picture correspond to whether or not the gene is expressed. So each row is a gene, and each column is a particular experiment, for example a particular type of tissue. If you see a red spot for some gene for breast tumors, that means this gene is expressed in breast cancer. But that same gene is not expressed in the brain, for example."
Gene 1
Gene n
gene expressed
Reference: Botstein & Brown group
gene not expressed

3. Two Dimensional Arrays in Java
Array access. Use a[i][j] to access element in row i and column j.
Zero-based indexing. Row and column indices start at 0.
double[][] a = new double[10][3];
for (int i = 0; i < 10; i++) {
for (int j = 0; j < 3; j++) {
a[i][j] = 0.0; }
Declaring and Initializing a 10-by-3 Array

4. Setting 2D Array Values at Compile Time
Initialize 2D array by listing values.
double[][] p = {
{ .02, .92, .02, .02, .02 },
{ .02, .02, .32, .32, .32 },
{ .02, .02, .02, .92, .02 },
{ .92, .02, .02, .02, .02 },
{ .47, .02, .47, .02, .02 }, };

5. Memory Layout 2-D Arrays
Row

6. Row lengths can be non-uniform Can use .length
Ragged Arrays
Row lengths can be non-uniform
Can use .length
int[][] a = ...;
for (int rows=0; rows < a.length; rows++) {
for (int cols=0; cols < a[rows].length; cols++) System.out.print(" " + a[rows][cols]);
System.out.println(""); }
Number of rows
Number of columns for a given row

7. Matrix Addition
Matrix addition. Given two N-by-N matrices a and b, define c to be the N-by-N matrix where c[i][j] is the sum a[i][j] + b[i][j].
double[][] c = new double[N][N];
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
c[i][j] = a[i][j] + b[i][j];

8. Matrix Multiplication
Matrix multiplication. Given two N-by-N matrices a and b, define c to be the N-by-N matrix where c[i][j] is the dot product of the ith row of a and the jth column of b.
all values initialized to 0
double[][] c = new double[N][N];
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
// dot-product of row i and column j
for (int k = 0; k < N; k++)
c[i][j] += a[i][k] * b[k][j];

9. Self-Avoiding Random Walk

10. Self-Avoiding Random Walk
Model.
N-by-N lattice.
Start in the middle.
Randomly move to a neighboring intersection, avoiding all previous intersections.
Applications. Polymers, statistical mechanics, etc.
A1. Not hard to convince yourself that you will always escape in a 5-by-5 grid

11. Made up of long chains of “monomers”
Example: Polymers
Made up of long chains of “monomers”
DNA: 108 nucleotides
Excluded Volume: No two monomers can occupy the same place
Polymer chains cannot self-intersect
Form “coils”
Polymer chains can be modeled as random walks
DNA

12. Self-Avoiding Random Walk
Model.
N-by-N lattice.
Start in the middle.
Randomly move to a neighboring intersection, avoiding all previous intersections.
Applications. Polymers, statistical mechanics, etc.
Q. What fraction of time will you escape the lattice?
Q. What is the average path length? …
A1. Not hard to convince yourself that you will always escape in a 5-by-5 grid

13. Simulating Self-Avoiding Random Walks
Represent the 2-D grid as a 2-D array
Array element[x][y] denotes whether the given intersection has been visited (boolean array)
Initialize all the elements of the array to false
Set element [x][y] to true when you visit that intersection

14. Self-Avoiding Walk: Implementation Dr. Java Demo
public class SelfAvoidingWalk {
public static void main(String[] args) {
int N = Integer.parseInt(args[0]); // lattice size
int T = Integer.parseInt(args[1]); // number of trials
int deadEnds = 0; // trials resulting in dead end
for (int t = 0; t < T; t++) {
boolean[][] a = new boolean[N][N]; // intersections visited
int x = N/2, y = N/2; // current position - center
while (x > 0 && x < N-1 && y > 0 && y < N-1) {
if (a[x-1][y] && a[x+1][y] && a[x][y-1] && a[x][y+1]) {
deadEnds++;
break; }
a[x][y] = true; // mark as visited
double r = Math.random();
if (r < 0.25 && !a[x+1][y]) x++;
else if (r < 0.50 && !a[x-1][y]) x--;
else if (r < 0.75 && !a[x][y+1]) y++;
else if (r < 1.00 && !a[x][y-1]) y--;
System.out.println(100*deadEnds/T + "% dead ends");

15. Self-Avoiding Random Walks in a 21-by-21 Grid

16. Summary Arrays. Organized way to store huge quantities of data.
Almost as easy to use as primitive types.
Can directly access an element given its index.