Section 8.5
Multi-Dimensional Arrays
ANY TYPE CAN BE USED AS THE BASE TYPE
FOR AN ARRAY. You can have an array of ints, an array of
Strings, an array of Objects, and so on.
In particular, since an
array type is a first-class Java type, you can
have an array of arrays. For example, an array of ints
has type int[]. This means that there is
automatically another type, int[][], which represents
an "array of arrays of ints". Such an
array is said to be a two-dimensional array.
Of course once you have the type int[][], there is nothing
to stop you from forming the type int[][][], which represents a
three-dimensional array -- and so on.
There is no limit on the number of dimensions that an array
type can have. However, arrays of dimension three or higher
are fairly uncommon, and I concentrate here mainly on two-dimensional
arrays. The type BaseType[][] is usually read
"two-dimensional array of BaseType" or "BaseType array
array".
The declaration statement "int[][] A;" declares a variable
named A of type int[][]. This variable can hold a reference
to an object of type int[][]. The assignment statement
"A = new int[3][4];" creates a new two-dimensional array
object and sets A to point to the newly created object.
As usual, the declaration and assignment could be combined in
a single declaration statement "int[][] A = new int[3][4];".
The newly created object is an array of arrays-of-ints.
The notation int[3][4]
indicates that there are 3 arrays-of-ints in the array A,
and that there are 4 ints in each array-of-ints. However,
trying to think in such terms can get a bit confusing -- as you might
have already noticed. So it is customary to think of a two-dimensional
array of items as a rectangular grid or
matrix of items. The notation
"new int[3][4]" can then be taken to describe a grid of ints
with 3 rows and 4 columns. The following picture might help:
For the most part, you can ignore the reality and keep the picture
of a grid in mind. Sometimes, though, you will need to remember
that each row in the grid is really an array in itself. These
arrays can be referred to as A[0], A[1], and
A[2]. Each row is in fact a value of type int[].
It could, for example, be passed to a subroutine that asks for
a parameter of type int[].
The notation A[1] refers to one of the rows of the
array A. Since A[1] is itself an array of ints,
you can another subscript to refer to one of the positions in that row.
For example, A[1][3]
refers to item number 3 in row number 1. Keep in mind, of
course, that both rows and columns are numbered starting from zero.
So, in the above example, A[1][3] is 5. More generally,
A[i][j] refers to the grid position in row number i
and column number j. The 12 items in A are
named as follows:
A[0][0] A[0][1] A[0][2] A[0][3]
A[1][0] A[1][1] A[1][2] A[1][3]
A[2][0] A[2][1] A[2][2] A[2][3]
A[i][j] is actually a variable of type int.
You can assign integer values to it or use it in any other
context where an integer variable is allowed.
It might be worth noting that A.length gives the
number of rows of A. To get the number of columns in
A, you have to ask how many ints there are in a row;
this number would be given by A[0].length, or equivalently
by A[1].length or A[2].length. (There is actually
no rule that says that all the rows of an array must have the same length,
and some advanced applications of arrays use varying-sized rows.
But if you use the new operator to create an array in the
manner described above, you'll always get an array with equal-sized rows.)
Three-dimensional arrays are treated similarly. For example, a
three-dimensional array of ints could be created with the declaration
statement "int[][][] B = new int[7][5][11];". It's possible
to visualize the value of B as a solid 7-by-5-by-11 block
of cells. Each cell holds an int and represents one
position in the three-dimensional array. Individual positions in
the array can be referred with variable names of the form
B[i][j][k]. Higher-dimensional arrays follow the same
pattern, although for dimensions greater than three, there is no
easy way to visualize the structure of the array.
It's possible to fill a multi-dimensional array with specified
items at the time
it is declared. Recall that when an ordinary one-dimensional
array variable is declared, it can be assigned an "array
initializer," which is just a list of values enclosed
between braces, { and }. Array initializers can also be used
when a multi-dimensional array is declared. An initializer
for a two-dimensional array consists of a
list of one-dimensional array initializers, one for each
row in the two-dimensional array. For example, the array A shown in
the picture above could be created with:
int[][] A = { { 1, 0, 12, -1 },
{ 7, -3, 2, 5 },
{ -5, -2, 2, 9 }
};
If no initializer is provided for an array, then when the array is
created it is automatically filled with the appropriate value:
zero for numbers, false for boolean, and null
for objects.
Just as in the case of one-dimensional arrays, two-dimensional
arrays are often processed using for statements. To process
all the items in a two-dimensional array, you have to use one
for statement nested inside another. If the array A
is declared as
int[][] A = new int[3][4];
then you could store a zero into each location in A with:
for (int row = 0; row < 3; row++) {
for (int column = 0; column < 4; column++) {
A[row][column] = 0;
}
}
The first time the outer for loop executes (with row = 0),
the inner for loop fills in the four values in the first row
of A, namely A[0][0] = 0, A[0][1] = 0,
A[0][2] = 0, and A[0][3] = 0. The next execution
of the outer for loop fills in the second row of A.
And the third and final execution of the outer loop fills in the
final row of A.
Similarly, you could add up all the items in A with:
int sum = 0;
for (int i = 0; i < 3; i++)
for (int j = 0; j < 4; i++)
sum = sum + A[i][j];
To process a three-dimensional array, you would, of course, use triply nested
for loops.
A two-dimensional array can be used whenever the data being
represented can be naturally arranged into rows and columns.
Often, the grid is built into the problem. For example,
a chess board is a grid with 8 rows and 8 columns. If a class
named ChessPiece is available to represent individual
chess pieces, then the contents of a chess board could be
represented by a two-dimensional array:
ChessPiece[][] board = new ChessPiece[8][8];
Or consider the "mosaic" of colored rectangles used as
an example in Section 4.6. The
mosaic is implemented by a class named MosaicCanvas.
The data about the color of each of the rectangles in the mosaic is stored in
an instance variable named grid of type Color[][]. Each position
in this grid is occupied by a value of type Color. There is one position
in the grid for each colored rectangle in the mosaic. The actual two-dimensional
array is created by the statement:
grid = new Color[ROWS][COLUMNS];
where ROWS is the number of rows of rectangles in the mosaic
and COLUMNS is the number of columns. The value of the
Color variable grid[i][j] is the color of the
rectangle in row number i and column number j.
When the color of that rectangle is changed to some color value, c,
the value stored in grid[i][j] is changed with a statement of
the form "grid[i][j] = c;". When the mosaic is redrawn, the
values stored in the two-dimensional array are used to decide
what color to make each rectangle. Here is a simplified version
of the code from the MosaicCanvas class that draws all
the colored rectangles in the grid. You can see how it uses the array:
int rowHeight = getSize().height / ROWS;
int colWidth = getSize().width / COLUMNS;
for (int row = 0; row < ROWS; row++) {
for (int col = 0; col < COLUMNS; col++) {
g.setColor( grid[row][col] ); // Get color from array.
g.fillRect( col*colWidth, row*rowHeight,
colWidth, rowHeight );
}
}
Sometimes two-dimensional arrays are used in problems in which
the grid is not so visually obvious. Consider a
company that owns 25 stores. Suppose that the company has data
about the profit earned at each store for each month in the year 2000.
If the stores are numbered from 0 to 24, and if the twelve months from
January '00 through December '00 are numbered from 0 to 11, then
the profit data could be stored in an array, profit,
constructed as follows:
double[][] profit = new double[25][12];
profit[3][2] would be the amount of profit earned at
store number 3 in March, and more generally, profit[storeNum][monthNum]
would be the amount of profit earned in store number storeNum
in month number monthNum. In this example, the one-dimensional
array profit[storeNum] has a very useful meaning: It is just
the profit data for one particular store for the whole year.
Let's assume that the profit array has already been
filled with data. This data can be processed in a lot of
interesting ways. For example, the total profit for
the company -- for the whole year from all its stores -- can be
calculated by adding up all the entries in the array:
double totalProfit; // Company's total profit in 2000.
totalProfit = 0;
for (int store = 0; store < 25; store++) {
for (int month = 0; month < 12; month++)
totalProfit += profit[store][month];
}
Sometimes it is necessary
to process a single row or a single column of an array, not
the entire array. For
example, to compute the total profit earned by the company in
December, that is, in month number 11, you could use the loop:
double decemberProfit = 0.0;
for (storeNum = 0; storeNum < 25; storeNum++)
decemberProfit += profit[storeNum][11];
Let's extend this idea to create a one-dimensional array
that contains the total profit for each month of the year:
double[] monthlyProfit; // Holds profit for each month.
monthlyProfit = new double[12];
for (int month = 0; month < 12; month++) {
// compute the total profit from all stores in this month.
monthlyProfit[month] = 0.0;
for (int store = 0; store < 25; store++) {
// Add the profit from this store in this month
// into the total profit figure for the month.
monthlyProfit[month] += profit[store][month];
}
}
As a final example of processing the profit array,
suppose that we wanted to know which store generated the
most profit over the course of the year. To do this, we have
to add up the monthly profits for each store. In array terms,
this means that we want to find the sum of each row in the
array. As we do this, we need to keep track of which row
produces the largest total.
double maxProfit; // Maximum profit earned by a store.
int bestStore; // The number of the store with the
// maximum profit.
double total = 0.0; // Total profit for one store.
// First compute the profit from store number 0.
for (int month = 0; month < 12; month++)
total += profit[0][month];
bestStore = 0; // Start by assuming that the best
maxProfit = total; // store is store number 0.
// Now, go through the other stores, and whenever we
// find one with a bigger profit than maxProfit, revise
// the assumptions about bestStore and maxProfit.
for (store = 1; store < 25; store++) {
// Compute this store's profit for the year.
total = 0.0;
for (month = 0; month < 12; month++)
total += profit[store][month];
// Compare this store's profits with the highest
// profit we have seen among the preceding stores.
if (total > maxProfit) {
maxProfit = total; // Best profit seen so far!
bestStore = store; // It came from this store.
}
} // end for
// At this point, maxProfit is the best profit of any
// of the 25 stores, and bestStore is a store that
// generated that profit. (Note that there could also be
// other stores that generated exactly the same profit.)
For the rest of this section, we'll look at a more substantial example.
Here is an applet that lets two users play checkers against each other.
A player moves by clicking on the piece to be moved and then on the
empty square to which it is to be moved. The squares that the
current player can legally click are hilited. A piece that has
been selected to be moved is surrounded by a white border.
Other pieces that can legally be moved are surrounded by a cyan-colored
border. If a piece has been selected, each empty square that it can
legally move to is hilited with a green border. The game enforces
the rule that if the current player can jump one of the opponent's
pieces, then the player must jump. When a player's piece becomes
a king, by reaching the opposite end of the board, a big white
"K" is drawn on the piece.
I will only cover a part of the programming of this applet. I encourage you
to read the complete source code, Checkers.java.
At over 700 lines, this is a more substantial example than anything you've
seen before in this course, but it's an excellent example of state-based,
event-driven programming. The source file defines four classes. The logic
of the game is implemented in a class named CheckersCanvas.
The data about the pieces on the board are stored in a two-dimensional
array. Because of the complexity of the program, I wanted to divide it into
several classes. One of these classes is CheckersData, which handles
the data for the board. It is mainly this class that I want to talk about.
The CheckersData class has an instance variable named board
of type int[][]. The value of board is set to "new int[8][8]",
an 8-by-8 grid of integers. The values stored in the grid are defined as
constants representing the possible contents of a square on a checkerboard:
public static final int
EMPTY = 0, // Value representing an empty square.
RED = 1, // A regular red piece.
RED_KING = 2, // A red king.
BLACK = 3, // A regular black piece.
BLACK_KING = 4; // A black king.
The constants RED and BLACK are also used in my program
(or, perhaps, misused) to represent the two players in the game. When a game
is started, the values in the variable, board, are set to represent the
initial state of the board. The grid of values looks like
0 1 2 3 4 5 6 6
0 BLACK EMPTY BLACK EMPTY BLACK EMPTY BLACK EMPTY
1 EMPTY BLACK EMPTY BLACK EMPTY BLACK EMPTY BLACK
2 BLACK EMPTY BLACK EMPTY BLACK EMPTY BLACK EMPTY
3 EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY
4 EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY EMPTY
5 EMPTY RED EMPTY RED EMPTY RED EMPTY RED
6 RED EMPTY RED EMPTY RED EMPTY RED EMPTY
7 EMPTY RED EMPTY RED EMPTY RED EMPTY RED
A black piece can only move "down" the grid. That is, the row number of
the square it moves to must be greater than the row number of the square it comes
from. A red piece can only move up the grid. Kings of either color, of course,
can move in both directions.
One function of the CheckersData class is to take care of all the details
of making moves on the board. An instance method named makeMove() is
provided to do this. When a player moves a piece from one square to another,
the values stored at two positions in the array are changed. But that's not
all. If the move is a jump, then the piece that was jumped is removed from
the board. (The method checks whether the move
is a jump by checking if the square to which the piece is moving is
two rows away from the square where it starts.)
Furthermore, a RED piece that moves to row 0 or
a BLACK piece that moves to row 7 becomes a king. This is good programming:
the rest of the program doesn't have to worry about any of these details. It just
calls this makeMove() method:
public void makeMove(int fromRow, int fromCol, int toRow, int toCol) {
// Make the move from (fromRow,fromCol) to (toRow,toCol). It is
// ASSUMED that this move is legal! If the move is a jump, the
// jumped piece is removed from the board. If a piece moves
// to the last row on the opponent's side of the board, the
// piece becomes a king.
board[toRow][toCol] = board[fromRow][fromCol]; // Move the piece.
board[fromRow][fromCol] = EMPTY;
if (fromRow - toRow == 2 || fromRow - toRow == -2) {
// The move is a jump. Remove the jumped piece from the board.
int jumpRow = (fromRow + toRow) / 2; // Row of the jumped piece.
int jumpCol = (fromCol + toCol) / 2; // Column of the jumped piece.
board[jumpRow][jumpCol] = EMPTY;
}
if (toRow == 0 && board[toRow][toCol] == RED)
board[toRow][toCol] = RED_KING; // Red piece becomes a king.
if (toRow == 7 && board[toRow][toCol] == BLACK)
board[toRow][toCol] = BLACK_KING; // Black piece becomes a king.
} // end makeMove()
An even more important function of the CheckersData class is to find legal
moves on the board. In my program, a move in a Checkers game is represented by
an object belonging to the following class:
class CheckersMove {
// A CheckersMove object represents a move in the game of
// Checkers. It holds the row and column of the piece that is
// to be moved and the row and column of the square to which
// it is to be moved. (This class makes no guarantee that
// the move is legal.)
int fromRow, fromCol; // Position of piece to be moved.
int toRow, toCol; // Square it is to move to.
CheckersMove(int r1, int c1, int r2, int c2) {
// Constructor. Set the values of the instance variables.
fromRow = r1;
fromCol = c1;
toRow = r2;
toCol = c2;
}
boolean isJump() {
// Test whether this move is a jump. It is assumed that
// the move is legal. In a jump, the piece moves two
// rows. (In a regular move, it only moves one row.)
return (fromRow - toRow == 2 || fromRow - toRow == -2);
}
} // end class CheckersMove.
The CheckersData class has an instance method which finds all
the legal moves that are currently available for a specified player. This method
is a function that returns an array of type CheckersMove[]. The
array contains all the legal moves, represented as CheckersMove
objects. The specification for this method reads
public CheckersMove[] getLegalMoves(int player)
// Return an array containing all the legal CheckersMoves
// for the specified player on the current board. If the player
// has no legal moves, null is returned. The value of player
// should be one of the constants RED or BLACK; if not, null
// is returned. If the returned value is non-null, it consists
// entirely of jump moves or entirely of regular moves, since
// if the player can jump, only jumps are legal moves.
A brief pseudocode algorithm for the method is
Start with an empty list of moves
Find any legal jumps and add them to the list
if there are no jumps:
Find any other legal moves and add them to the list
if the list is empty:
return null
else:
return the list
Now, what is this "list"? We have to return the legal moves in an array.
But since an array has a fixed size, we can't create the array until we know
how many moves there are, and we don't know that until near the end of the method,
after we've already made the list!
A neat solution is to use a ArrayList instead of an array
to hold the moves as we find them. As we add moves to the list,
it will grow just as large as necessary. At the end of the method,
we can create the array that we really want and copy the data into it:
Let "moves" be an empty ArrayList
Find any legal jumps and add them to moves
if moves.size() is 0:
Find any other legal moves and add them to moves
if moves.size() is 0:
return null
else:
Let moveArray be an array of CheckersMoves of length moves.size()
Copy the contents of moves into moveArray
return moveArray
Now, how do we find the legal jumps or the legal moves?
The information we need is in the board array, but it
takes some work to extract it. We have to look through all
the positions in the array and find the pieces that belong
to the current player. For each piece, we have to check each
square that it could conceivably move to, and check whether
that would be a legal move. There are four squares to consider.
For a jump, we want to look at squares that are two rows
and two columns away from the piece. Thus,
the line in the algorithm
that says "Find any legal jumps and add them to moves"
expands to:
For each row of the board:
For each column of the board:
if one of the player's pieces is at this location:
if it is legal to jump to row + 2, column + 2
add this move to moves
if it is legal to jump to row - 2, column + 2
add this move to moves
if it is legal to jump to row + 2, column - 2
add this move to moves
if it is legal to jump to row - 2, column - 2
add this move to moves
The line that says "Find any other legal moves and add them to moves"
expands to something similar, except that we have to look at
the four squares that are one column and one row away from the piece.
Testing whether a player can legally move from one given
square to another given square is itself non-trivial. The square the
player is moving to must actually be on the board, and it must
be empty. Furthermore, regular red and black pieces can only move in one
direction. I wrote the following utility method to check whether
a player can make a given non-jump move:
private boolean canMove(int player, int r1, int c1, int r2, int c2) {
// This is called by the getLegalMoves() method to determine
// whether the player can legally move from (r1,c1) to (r2,c2).
// It is ASSUMED that (r1,c1) contains one of the player's
// pieces and that (r2,c2) is a neighboring square.
if (r2 < 0 || r2 >= 8 || c2 < 0 || c2 >= 8)
return false; // (r2,c2) is off the board.
if (board[r2][c2] != EMPTY)
return false; // (r2,c2) already contains a piece.
if (player == RED) {
if (board[r1][c1] == RED && r2 > r1)
return false; // Regular red piece can only move down.
return true; // The move is legal.
}
else {
if (board[r1][c1] == BLACK && r2 < r1)
return false; // Regular black piece can only move up.
return true; // The move is legal.
}
} // end canMove()
This method is called by my getLegalMoves() method to check whether
one of the possible moves that it has found is actually legal. I have a similar
method that is called to check whether a jump is legal. In this case,
I pass to the method the square containing the player's piece, the square that
the player might move to, and the square between those two, which the player
would be jumping over. The square that is being jumped must contain one of the
opponent's pieces. This method has the specification:
private boolean canJump(int player, int r1, int c1,
int r2, int c2, int r3, int c3) {
// This is called by other methods to check whether
// the player can legally jump from (r1,c1) to (r3,c3).
// It is assumed that the player has a piece at (r1,c1), that
// (r3,c3) is a position that is 2 rows and 2 columns distant
// from (r1,c1) and that (r2,c2) is the square between (r1,c1)
// and (r3,c3).
Given all this, you should be in a position to understand the complete
getLegalMoves() method. It's a nice way to finish off this
chapter, since it combines several
topics that we've looked at: one-dimensional arrays,
ArrayLists, and two-dimensional arrays:
public CheckersMove[] getLegalMoves(int player) {
if (player != RED && player != BLACK)
return null;
int playerKing; // The constant for a King belonging to the player.
if (player == RED)
playerKing = RED_KING;
else
playerKing = BLACK_KING;
ArrayList moves = new ArrayList();
// Moves will be stored in this list.
/* First, check for any possible jumps. Look at each square on
the board. If that square contains one of the player's pieces,
look at a possible jump in each of the four directions from that
square. If there is a legal jump in that direction, put it in
the moves ArrayList.
*/
for (int row = 0; row < 8; row++) {
for (int col = 0; col < 8; col++) {
if (board[row][col] == player || board[row][col] == playerKing) {
if (canJump(player, row, col, row+1, col+1, row+2, col+2))
moves.add(new CheckersMove(row, col, row+2, col+2));
if (canJump(player, row, col, row-1, col+1, row-2, col+2))
moves.add(new CheckersMove(row, col, row-2, col+2));
if (canJump(player, row, col, row+1, col-1, row+2, col-2))
moves.add(new CheckersMove(row, col, row+2, col-2));
if (canJump(player, row, col, row-1, col-1, row-2, col-2))
moves.add(new CheckersMove(row, col, row-2, col-2));
}
}
}
/* If any jump moves were found, then the user must jump, so we
don't add any regular moves. However, if no jumps were found,
check for any legal regular moves. Look at each square on
the board. If that square contains one of the player's pieces,
look at a possible move in each of the four directions from
that square. If there is a legal move in that direction,
put it in the moves ArrayList.
*/
if (moves.size() == 0) {
for (int row = 0; row < 8; row++) {
for (int col = 0; col < 8; col++) {
if (board[row][col] == player
|| board[row][col] == playerKing) {
if (canMove(player,row,col,row+1,col+1))
moves.add(new CheckersMove(row,col,row+1,col+1));
if (canMove(player,row,col,row-1,col+1))
moves.add(new CheckersMove(row,col,row-1,col+1));
if (canMove(player,row,col,row+1,col-1))
moves.add(new CheckersMove(row,col,row+1,col-1));
if (canMove(player,row,col,row-1,col-1))
moves.add(new CheckersMove(row,col,row-1,col-1));
}
}
}
}
/* If no legal moves have been found, return null. Otherwise, create
an array just big enough to hold all the legal moves, copy the
legal moves from the ArrayList into the array, and return the array.
*/
if (moves.size() == 0)
return null;
else {
CheckersMove[] moveArray = new CheckersMove[moves.size()];
for (int i = 0; i < moves.size(); i++)
moveArray[i] = (CheckersMove)moves.get(i);
return moveArray;
}
} // end getLegalMoves
End of Chapter 8
