CS.3.Arrays.pdf

C O M P U T E R S C I E N C E
S E D G E W I C K / W A Y N E
PART I: PROGRAM M IN G IN JAVA
http://introcs.cs.princeton.edu
R O B E R T S E D G E W I C K
K E V I N W A Y N E
Computer
Science
Computer
Science
An Interdisciplinary Approach
3. Arrays
1.4

3. Arrays
•Basic concepts
•Typical array-processing code
•Two-dimensional arrays
CS.3.A.Arrays.Basics

Basic building blocks for programming
3
any program you might want to write
objects
functions and modules
arrays
conditionals and loops
Math text I/O
assignment statements
primitive data types
graphics, sound, and image I/O
Ability to store and process
huge amounts of data
arrays

4
Your first data structure
Main purpose. Facilitate storage and manipulation of data.
Examples.
• 52 playing cards in a deck.
• 100 thousand students in an online class.
• 1 billion pixels in a digital image.
• 4 billion nucleotides in a DNA strand.
• 73 billion Google queries per year.
• 86 billion neurons in the brain.
• 50 trillion cells in the human body.
• 6.02 × 1023 particles in a mole.
A data structure is an arrangement of data that enables efficient processing by a program.
index value
0 2♥
1 6♠
2 A♦
3 A♥
...
49 3♣
50 K♣
51 4♠
An array is an indexed sequence of values of the same type.

5
Processing many values of the same type
double a0 = 0.0;
double a1 = 0.0;
double a2 = 0.0;
double a3 = 0.0;
double a4 = 0.0;
double a5 = 0.0;
double a6 = 0.0;
double a7 = 0.0;
double a8 = 0.0;
double a9 = 0.0;
...
a4 = 3.0;
...
a8 = 8.0;
...
double x = a4 + a8;
10 values, without arrays
tedious and error-prone code
double[] a;
a = new double[10];
...
a[4] = 3.0;
...
a[8] = 8.0;
...
double x = a[4] + a[8];
10 values, with an array
an easy alternative
double[] a;
a = new double[1000000];
...
a[234567] = 3.0;
...
a[876543] = 8.0;
...
double x = a[234567] + a[876543];
1 million values, with an array
scales to handle huge amounts of data

Memory representation of an array
6
A computer's memory is also an indexed sequence of memory locations.
• Each primitive type value occupies a fixed number of locations.
• Array values are stored in contiguous locations.
An array is an indexed sequence of values of the same type.
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9]
Critical concepts
• Indices start at 0.
• Given i, the operation of accessing the value a[i] is extremely efficient.
• The assignment b = a makes the names b and a refer to the same array.
stay tuned for many details
it does not copy the array,
as with primitive types
(stay tuned for details)
a for simplicity in this lecture, think of a as the memory address of the first location
the actual implementation in Java is just slightly more complicated.

Java language support for arrays
7
operation typical code
Declare an array double[] a;
Create an array of a given length a = new double[1000];
Refer to an array entry by index a[i] = b[j] + c[k];
Refer to the length of an array a.length;
Basic support
Default initialization to 0 for numeric types a = new double[1000];
Declare, create and initialize in one statement double[] a = new double[1000];
Initialize to literal values double[] x = { 0.3, 0.6, 0.1 };
Initialization options
BUT cost of creating an
array is proportional to
its length.
no need to use a loop like
for (int i = 0; i < 1000; i++)
a[i] = 0.0;

Copying an array
8
To copy an array, create a new array , then copy all the values.
a
0.3 0.6 0.99 0.01 0.5
b
0.3 0.6 0.99 0.01 0.5
double[] b = new double[a.length];
for (int i = 0; i < a.length; i++)
b[i] = a[i];
i i
Important note: The code b = a does not copy an array (it makes b and a refer to the same array).
a
0.3 0.6 0.99 0.01 0.5
b
double[] b = new double[a.length];
b = a;

Programming with arrays: typical examples
9
double[] a = new double[N];
for (int i = 0; i < N; i++)
a[i] = Math.random();
Create an array with N random values
for (int i = 0; i < N; i++)
System.out.println(a[i]);
Print array values, one per line
double sum = 0.0;
for (int i = 0; i < N; i++)
sum += a[i];
double average = sum / N;
Compute the average of array values
double max = a[0];
for (int i = 1; i < N; i++)
if (a[i] > max) max = a[i];
Find the maximum of array values
double[] b = new double[N];
for (int i = 0; i < N; i++)
b[i] = a[i];
Copy to another array
For brevity, N is a.length and b.length in all this code.
int stake = Integer.parseInt(args[0]);
int goal = Integer.parseInt(args[1]);
int trials = Integer.parseInt(args[2]);
Access command-line args in system array

Pop quiz 1 on arrays
Q. What does the following code print?
10
public class PQarray1
{
public static void main(String[] args)
{
int[] a = new int[6];
int[] b = new int[a.length];
b = a;
for (int i = 1; i < b.length; i++)
b[i] = i;
System.out.print(a[i] + " ");
System.out.println();
System.out.print(b[i] + " ");
}
}

Q. What does the following code print?
11
public class PQarray1
{
{
int[] a = new int[6];
int[] b = new int[a.length];
b = a;
b[i] = i;
System.out.print(a[i] + " ");
System.out.print(b[i] + " ");
}
}
A. % java PQ4_1
0 1 2 3 4 5
0 1 2 3 4 5
After this, b and a refer to the same array

Programming with arrays: typical bugs
12
a = new double[10];
for (int i = 0; i < 10; i++)
What type of data does a refer to?
Undeclared variable
double[] a;
for (int i = 0; i < 9; i++)
Never created the array
Uninitialized array
double[] a = new double[10];
for (int i = 1; i <= 10; i++)
No a[10] (and a[0] unused)
Array index out of bounds

Image sources
http://commons.wikimedia.org/wiki/File:CERN_Server_03.jpg
CS.3.A.Arrays.Basics

3. Arrays
•Basic concepts
•Examples of array-processing code
CS.3.B.Arrays.Examples

15
Example of array use: create a deck of cards
Define three arrays
• Ranks.
• Suits.
• Full deck.
Use nested for loops to put all the cards in the deck.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
2♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 9♣ 10♣ J♣ Q♣ K♣ A♣ 2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ ...
String[] deck = new String[52];
deck
String[] suit = { "♣", "♦", "♥", "♠" };
0 1 2 3
♣ ♦ ♥ ♠
suit
String[] rank = {"2", "3", "4", "5", "6", "7", "8", "9", "10", "J", "Q", "K", "A" };
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 J Q K A
rank
for (int j = 0; j < 4; j++)
for (int i = 0; i < 13; i++)
deck[i + 13*j] = rank[i] + suit[j];
j
j
i
better style to use rank.length and suit.length
clearer in lecture to use 4 and 13

16
Example of array use: create a deck of cards
public class Deck
{
{
String[] rank = {"2", "3", "4", "5", "6", "7", "8", "9", "10", "J", "Q", "K", "A" };
String[] suit = { "♣", "♦", "♥", "♠" };
for (int j = 0; j < 4; j++)
for (int i = 0; i < 13; i++)
for (int i = 0; i < 52; i++)
System.out.print(deck[i] + " ");
}
}
% java Deck
2♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 9♣ 10♣ J♣ Q♣ K♣ A♣
2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ 10♦ J♦ Q♦ K♦ A♦
2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ 9♥ 10♥ J♥ Q♥ K♥ A♥
2♠ 3♠ 4♠ 5♠ 6♠ 7♠ 8♠ 9♠ 10♠ J♠ Q♠ K♠ A♠
%
no color in Unicode;
artistic license for lecture

Q. What happens if the order of the for loops in Deck is switched?
17
for (int j = 0; j < 4; j++)
for (int i = 0; i < 13; i++)
for (int i = 0; i < 13; i++)
for (int j = 0; j < 4; j++)

Q. What happens if the order of the for loops in Deck is switched?
18
for (int j = 0; j < 4; j++)
for (int i = 0; i < 13; i++)
for (int i = 0; i < 13; i++)
for (int j = 0; j < 4; j++)
A. The array is filled in a different order, but the output is the same.
0 1 2 ... 12 13 14 15 ... 25 26 27 28 ... 38 39 40 41 ... 51
i
j
0 1 2 3
♣ ♦ ♥ ♠
suit
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 J Q K A
rank
deck 2♣ 3♣ 4♣ ... A♣ 2♦ 3♦ 4♦ ... A♦ 2♥ 3♥ 4♥ ... A♥ 2♠ 3♠ 4♠ ... A♠

Q. Change Deck to put the cards in rank order in the array.
19
% java Deck
2♣ 2♦ 2♥ 2♠ 3♣ 3♦ 3♥ 3♠ 4♣ 4♦ 4♥ 4♠ 5♣ 5♦ 5♥ 5♠ 6♣ 6♦ 6♥ 6♠ 7♣ 7♦ 7♥ 7♠ 8♣ 8♦ 8♥ 8♠ 9♣ 9♦ 9♥ 9♠
10♣ 10♦ 10♥ 10♠ J♣ J♦ J♥ J♠ Q♣ Q♦ Q♥ Q♠ K♣ K♦ K♥ K♠ A♣ A♦ A♥ A♠
%

Q. Change Deck to put the cards in rank order in the array.
20
for (int i = 0; i < 13; i++)
for (int j = 0; j < 4; j++)
deck[4*i + j] = rank[i] + suit[j];
A.
% java Deck
2♣ 2♦ 2♥ 2♠ 3♣ 3♦ 3♥ 3♠ 4♣ 4♦ 4♥ 4♠ 5♣ 5♦ 5♥ 5♠ 6♣ 6♦ 6♥ 6♠ 7♣ 7♦ 7♥ 7♠ 8♣ 8♦ 8♥ 8♠ 9♣ 9♦ 9♥ 9♠
10♣ 10♦ 10♥ 10♠ J♣ J♦ J♥ J♠ Q♣ Q♦ Q♥ Q♠ K♣ K♦ K♥ K♠ A♣ A♦ A♥ A♠
%
0 1 2 3 4 5 6 7 8 9 10 11 12 ...
2♣ 2♦ 2♥ 2♠ 3♣ 3♦ 3♥ 3♠ 4♣ 4♦ 4♥ 4♠ 5♣ ...
0 1 2 3
♣ ♦ ♥ ♠
suit
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 J Q K A
rank
i
j
deck

22
Array application: take a card, any card
Problem: Print a random sequence of N cards.
for (int i = 0; i < N; i++)
{
int r = (int) (Math.random() * 52);
System.out.println(deck[r]);
}
Algorithm
Take N from the command line and do the following N times
• Calculate a random index r between 0 and 51.
• Print deck[r].
Implementation: Add this code instead of printing deck in Deck.
Note: Same method is effective for printing a random sequence from any data collection.
each value between 0 and 51 equally likely

23
Array application: random sequence of cards
public class DrawCards
{
{
int N = Integer.parseInt(args[0]);
String[] rank = {"2", "3", "4", "5", "6", "7", "8", "9","10", "J", "Q", "K", "A" };
String[] suit = { "♣", "♦", "♥", "♠" };
for (int i = 0; i < 13; i++)
for (int j = 0; j < 4; j++)
for (int i = 0; i < N; i++)
{
int r = (int) (Math.random() * 52);
System.out.print(deck[r] + " ");
}
}
}
% java DrawCards 10
6♥ K♦ 10♠ 8♦ 9♦ 9♥ 6♦ 10♠ 3♣ 5♦
% java DrawCards 10
2♦ A♠ 5♣ A♣ 10♣ Q♦ K♣ K♠ A♣ A♦
% java DrawCards 10
6♠ 10♦ 4♥ A♣ K♥ Q♠ K♠ 7♣ 5♦ Q♠
% java DrawCards 10
A♣ J♣ 5♥ K♥ Q♣ 5♥ 9♦ 9♣ 6♠ K♥
Note: Sample is with replacement (same card can appear multiple times). appears twice

24
Array application: shuffle and deal from a deck of cards
Problem: Print N random cards from a deck.
Algorithm: Shuffle the deck, then deal.
• Consider each card index i from 0 to 51.
• Calculate a random index r between i and 51.
• Exchange deck[i] with deck[r]
• Print the first N cards in the deck.
for (int i = 0; i < 52; i++)
{
int r = i + (int) (Math.random() * (52-i));
String t = deck[r];
deck[r] = deck[i];
deck[i] = t;
}
for (int i = 0; i < N; i++)
System.out.print(deck[i]);
Implementation
each value
between i and 51
equally likely

25
Array application: shuffle a deck of 10 cards (trace)
for (int i = 0; i < 10; i++)
{
String t = deck[r];
deck[r] = deck[i];
deck[i] = t;
}
i r
deck
0 1 2 3 4 5 6 7 8 9
2♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 9♣ 10♣ J♣
0 7 9♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 2♣ 10♣ J♣
1 3 9♣ 5♣ 4♣ 3♣ 6♣ 7♣ 8♣ 2♣ 10♣ J♣
2 9 9♣ 5♣ J♣ 3♣ 6♣ 7♣ 8♣ 2♣ 10♣ 4♣
3 9 9♣ 5♣ J♣ 4♣ 6♣ 7♣ 8♣ 2♣ 10♣ 3♣
4 6 9♣ 5♣ J♣ 4♣ 8♣ 7♣ 6♣ 2♣ 10♣ 3♣
5 9 9♣ 5♣ J♣ 4♣ 8♣ 3♣ 6♣ 2♣ 10♣ 7♣
6 8 9♣ 5♣ J♣ 4♣ 8♣ 3♣ 10♣ 2♣ 6♣ 7♣
7 9 9♣ 5♣ J♣ 4♣ 8♣ 3♣ 10♣ 7♣ 6♣ 2♣
8 8 9♣ 5♣ J♣ 4♣ 8♣ 3♣ 10♣ 7♣ 6♣ 2♣
9 9 9♣ 5♣ J♣ 4♣ 8♣ 3♣ 10♣ 7♣ 6♣ 2♣
Q. Why does this method work?
• Uses only exchanges, so the deck after
the shuffle has the same cards as before.
• N −i equally likely values for deck[i].
• Therefore N ×(N −1)×(N −1)... ×2×1 = N !
equally likely values for deck[].
Initial order is immaterial.
Note: Same method is effective for randomly rearranging any type of data.

26
Array application: shuffle and deal from a deck of cards
public class DealCards
{
{
String[] rank = {"2", "3", "4", "5", "6", "7", "8", "9", "10", "J", "Q", "K", "A" };
String[] suit = { "♣", "♦", "♥", "♠" };
for (int i = 0; i < 13; i++)
for (int j = 0; j < 4; j++)
for (int i = 0; i < 52; i++)
{
String t = deck[r];
deck[r] = deck[i];
deck[i] = t;
}
for (int i = 0; i < N; i++)
System.out.print(deck[i]);
}
}
% java DealCards 5
9♣ Q♥ 6♥ 4♦ 2♠
random poker hand
% java DealCards 13
3♠ 4♥ 10♦ 6♥ 6♦ 2♠ 9♣ 8♠ A♠ 3♥ 9♠ 5♠ Q♥
random bridge hand

Coupon collector
27
9♣ 5♠ 8♥ 10♦ 2♠ A♠ 10♥ Q♦ 3♠ 9♥ 5♦ 9♣ 7♦ 2♦ 8♣ 6♣ Q♥ K♣ 10♥ A♦ 4♦ J♥
2♠ 3♠ 4♦ 5♠ 6♣ 7♦ 8♥ 9♣ 10♦ J♥ Q♦ K♣ A♠
10♥
9♥
5♦
9♣
2♦ 8♣ Q♥
10♥
A♦
2 3 4 5 6 7 8 9 10 J Q K A
Example: Collect all ranks in a random sequence of cards (M =
Collection
Sequence
22 cards needed
to complete
collection
Coupon collector problem
• M different types of coupons.
• Collector acquires random coupons, one at a time, each type equally likely.
Q. What is the expected number of coupons needed to acquire a full collection?

Array application: coupon collector
28
public class Coupon
{
{
int M = Integer.parseInt(args[0]);
int cards = 0; // number of cards collected
int distinct = 0; // number of distinct cards
boolean[] found = new boolean[M];
while (distinct < M)
{
int r = (int) (Math.random() * M);
cards++;
if (!found[r])
{
distinct++;
found[r] = true;
}
}
System.out.println(cards);
}
}
Key to the implementation
• Create a boolean array of length M.
(Initially all false by default.)
• When r generated, check the rth
value in the array.
• If true, ignore it (not new).
• If false, count it as new distinct
value (and set rth entry to true)
% java Coupon 13
46
% java Coupon 13
22
% java Coupon 13
54
% java Coupon 13
27
Coupon collector simulation
• Generate random int values
between 0 and M −1.
• Count number used to generate
each value at least once.

29
Array application: coupon collector (trace for M = 6)
boolean[] found = new boolean[M];
{
cards++;
if (!found[r])
{
distinct++;
found[r] = true;
}
}
r
found
distinct cards
0 1 2 3 4 5
F F F F F F 0 0
2 F F T F F F 1 1
0 T F T F F F 2 2
4 T F T F T F 3 3
0 T F T F T F 3 4
1 T T T F T F 4 5
2 T T T F T F 4 6
5 T T T F T T 5 7
0 T T T F T T 5 8
1 T T T F T T 5 9
3 T T T T T T 6 10

30
Simulation, randomness, and analysis (revisited)
Pierre-Simon Laplace
1749-1827
Remarks
• Computer simulation can help validate mathematical analysis.
• Computer simulation can also validate software behavior.
% java Coupon 4
11
% java Coupon 13
38
% java Coupon 1200
8789
% java Coupon 12534
125671
type M expected wait
playing card suits 4 8
playing card ranks 13 41
baseball cards 1200 9201
Magic™ cards 12534 125508
Example: Is Math.random()
simulating randomness?
A. (known via mathematical analysis for centuries) About M ln M + .57721M .

31
public class Gambler
{
{
int stake = Integer.parseInt(args[0]);
int goal = Integer.parseInt(args[1]);
int wins = 0;
for (int i = 0; i < trials; i++)
{
int t = stake;
while (t > 0 && t < goal)
{
if (Math.random() < 0.5) t++;
else t--;
}
if (t == goal) wins++;
}
System.out.println(wins + " wins of " + trials);
}
}
Gambler's ruin simulation, previous lecture
public class CouponCollector
{
{
int M = Integer.parseInt(args[0]);
int cards = 0;
boolean[] found;
for (int i = 0; i < trials; i++)
{
int distinct = 0;
found = new boolean[M];
{
cards++;
if (!found[r])
{
distinct++;
found[r] = true;
}
}
}
System.out.println(cards/trials);
}
}
Analogous code for coupon collector, this lecture
Once simulation is debugged, experimental evidence is easy to obtain.

32
% java CouponCollector 4 1000000
8
41
236
9176
125920
type M M ln M + .57721M
playing card suits 4 8
playing card ranks 13 41
playing cards 52 236
baseball cards 1200 9201
magic cards 12534 125508
Predicted by mathematical analysis Observed by computer simulation
and Math.random() simulates randomness.
Hypothesis. Centuries-old analysis is correct

Image sources
http://www.vis.gr.jp/~nazoya/cgi-bin/catalog/img/CARDSBIC809_red.jpg
http://www.alegriphotos.com/Shuffling_cards_in_casino-photo-deae1081e5ebc6631d6871f8b320b808.html
http://iveypoker.com/wp-content/uploads/2013/09/Dealing.jpg
http://upload.wikimedia.org/wikipedia/commons/b/bf/Pierre-Simon,_marquis_de_Laplace_(1745-1827)_-_Guérin.jpg
CS.3.B.Arrays.Examples

3. Arrays
•Basic concepts
•Examples of array-processing code
CS.3.C.Arrays.2D

35
Two-dimensional arrays
Main purpose. Facilitate storage and manipulation of data.
Examples
• Matrices in math calculations.
• Grades for students in an online class.
• Outcomes of scientific experiments.
• Transactions for bank customers.
• Pixels in a digital image.
• Geographic data
• ...
0 1 2 3 4 5 ...
0 A A C B A C
1 B B B B A A
2 C D D B C A
3 A A A A A A
4 C C B C B B
5 A A A B A A
...
grade
student
ID
x-coordinate
y-coordinate
A two-dimensional array is a doubly-indexed sequence of values of the same type.

Java language support for two-dimensional arrays (basic support)
36
Declare a two-dimensional array double[][] a;
Create a two-dimensional array of a given length a = new double[1000][1000];
Refer to an array entry by index a[i][j] = b[i][j] * c[j][k];
Refer to the number of rows a.length;
Refer to the number of columns a[i].length;
Refer to row i a[i]
can be different
for each row
a[][]
a[1]
a[0][0] a[0][1] a[0][2] a[0][3] a[0][4] a[0][5] a[0][6] a[0][7] a[0][8] a[0][9]
a[1][0] a[1][1] a[1][2] a[1][3] a[1][4] a[1][5] a[1][6] a[1][7] a[1][8] a[1][9]
a[2][0] a[2][1] a[2][2] a[2][3] a[2][4] a[2][5] a[2][6] a[2][7] a[2][8] a[2][9]
no way to refer
to column j
a 3-by-10 array

Java language support for two-dimensional arrays (initialization)
37
Default initialization to 0
for numeric types
a = new double[1000][1000];
Declare, create and initialize
in a single statement
double[][] a = new double[1000][1000];
Initialize to literal values
double[][] p =
{
{ .92, .02, .02, .02, .02 },
{ .02, .92, .32, .32, .32 },
{ .02, .02, .02, .92, .02 },
{ .92, .02, .02, .02, .02 },
{ .47, .02, .47, .02, .02 },
};
BUT cost of creating an
array is proportional to
its size.
no need to use nested loops like
for (int i = 0; i < 1000; i++)
for (int j = 0; j < 1000; j++)
a[i][j] = 0.0;

Application of arrays: vector and matrix calculations
38
Mathematical abstraction: matrix
Java implementation: 2D array
Vector addition
double[] c = new double[N];
for (int i = 0; i < N; i++)
c[i] = a[i] + b[i];
Matrix addition
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];
.70 .20 .10
.30 .60 .10
.50 .10 .40
.80 .30 .50
.10 .40 .10
.10 .30 .40
1.5 .50 .60
.40 1.0 .20
.60 .40 .80
+ =
.30 .60 .10 + =
.50 .10 .40 .80 .70 .50
Mathematical abstraction: vector

Application of arrays: vector and matrix calculations
39
Mathematical abstraction: matrix
.70 .20 .10
.30 .60 .10
.50 .10 .40
.80 .30 .50
.10 .40 .10
.10 .30 .40
.59 .32 .41
.31 .36 .25
.45 .31 .42
* =
.30 .60 .10 .25
· =
.50 .10 .40
Vector dot product
double sum = 0.0;
for (int i = 0; i < N; i++)
sum += a[i]*b[i];
Matrix multiplication
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
for (int k = 0; k < N; k++)
c[i][j] += a[i][k] * b[k][j];
i x[i] y[i] x[i]*y[i] sum
0 0.3 0.5 0.15 0.15
1 0.6 0.1 0.06 0.21
2 0.1 0.4 0.04 0.25
end-of-loop trace
Mathematical abstraction: vector

Q. How many multiplications to multiply two N-by-N matrices?
40
1. N
2. N2
3. N3
4. N4
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
for (int k = 0; k < N; k++)
c[i][j] += a[i][k] * b[k][j];

1. N
2. N2
3. N3
4. N4
Q. How many multiplications to multiply two N-by-N matrices?
41
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
for (int k = 0; k < N; k++)
c[i][j] += a[i][k] * b[k][j];
Nested for loops: N × N × N

42
Self-avoiding random walks
Approach: Use Monte Carlo simulation, recording visited positions in an N-by-N array.
Q. What are the chances of reaching a dead end? dead end
escape
Model: a random process in an N-by-N lattice
• Start in the middle.
• Move to a random neighboring intersection
but do not revisit any intersection.
• Outcome 1 (escape): reach edge of lattice.
• Outcome 2 (dead end): no unvisited neighbors.
A dog walks around at
random in a city, never
revisiting any intersection.
Q. Does the dog escape?

Application of 2D arrays: self-avoiding random walks
44
public class SelfAvoidingWalker
{
{
int deadEnds = 0;
for (int t = 0; t < trials; t++)
{
boolean[][] a = new boolean[N][N];
int x = N/2, y = N/2;
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;
double r = Math.random();
if (r < 0.25) { if (!a[x+1][y]) x++; }
else if (r < 0.50) { if (!a[x-1][y]) x--; }
else if (r < 0.75) { if (!a[x][y+1]) y++; }
else if (r < 1.00) { if (!a[x][y-1]) y--; }
}
}
System.out.println(100*deadEnds/trials + "% dead ends");
}
}
% java SelfAvoidingWalker 10 100000
5% dead ends
32% dead ends
58% dead ends
77% dead ends
87% dead ends
93% dead ends
96% dead ends
98% dead ends
99% dead ends
99% dead ends
0%
25%
50%
75%
100%
10 20 30 40 50 60 70 80 90 100

45
Simulation, randomness, and analysis (revisited again)
Remark: Computer simulation is often the only effective way to study a scientific phenomenon.
Self-avoiding walk in an N-by-N lattice
• Start in the middle.
• Move to a random neighboring intersection (do not revisit any intersection).
A. 99+% for N >100 (clear from simulations). YOU can!
Applications
• Model the behavior of solvents and polymers.
• Model the physics of magnetic materials.
• (many other physical phenomena)
Computational models play
an essential role in modern
scientific research.
Paul Flory
1910-1985
Nobel Prize 1974
Q. What is the probability of reaching a dead end?
A. Nobody knows (despite decades of study).
Mathematicians and
physics researchers
cannot solve the problem.

Your first data structure
46
Some applications in this course:
LFSR
^
digital audio
digital images
Arrays: A basic building block in programming
• They enable storage of large amounts of data (values all of the same type).
• With an index, a program can instantly access a given value.
• Efficiency derives from low-level computer hardware organization (stay tuned).
N-body simulation

Image sources
http://en.wikipedia.org/wiki/Airedale_Terrier#mediaviewer/File:Airedale_Terrier.jpg
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1974/flory_postcard.jpg
CS.3.C.Arrays.2D

CS.3.Arrays.pdf

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