CS100J Lecture 18

1
CS100J Lecture 18

• Previous Lecture
• Programming concepts
• Two-dimensional arrays
• Java Constructs
• Constructors for two-dimensional arrays
• Initializers for arrays
• final
• Reading
• Lewis Loftus Section 6.3
• Savitch, Section 6.5
• This Lecture
• Two dimensional arrays.
• Reasonable size problem (a past assignment).
• Stepwise refinement.
• Use of comments as high-level specifications
• as high-level commands,
• as representation invariants.
• Incremental development and testing.
• Use of sentinels.

2
Knights Tour

• Chess is played on an 8-by-8 board.
• A knight can move 2 in one direction (horizontal
  or vertical) and 1 in the other direction
  (vertical or horizontal). For example, Kn can
  move to each square marked X.
• Problem. Write a program that tries to find a
  "knight's tour", e.g., starting in the upper left
  corner, the knight visits each of the 64 squares
  exactly once.

3
Possible Solution
4
Representation

• Rule of Thumb. Avoid using literal constants in
  the code define and use symbolic constants.
• Static class declarations visible to all methods.
  
• class Knight
• / Chess board B is N-by-N int array, for N
  8. /
• final static int N 8
• static int B new int NN
• / Unvisited squares are Blank. /
• static final int Blank 0
• / Board subscripts range from lo to hi. /
• static final int lo 0
• static final int hi 7
• / UNDEFINED, an illegal coordinate. /
• static final int UNDEFINED -1

5
Main

• / Try to find a Knight's Tour. /
• static void main(String args)
• / Set B to all Blank. /
• Initialize()
• / Set B to arrangement of the integers 1,2,3,...
  representing consecutive positions of the knight
  during the tour. Squares that the knight cannot
  reach remain Blank. /
• Solve()
• / Print B in an 8-by-8 grid. /
• Display()

6
Initialize

• / Set B to all Blank. /
• static void Initialize()
• for (int r lo r lt hi r)
• for (int c lo c lt hi c)
• Brc Blank

7
Solve

• / Set B to arrangement of the integers 1,2,3,...
  representing consecutive positions of the knight
  during the tour. Squares that the knight cannot
  reach remain Blank. /
• static void Solve()
• / This procedure "stub" permits us to test the
  main program, Initialize, and Display routines.
  /

8
Display

• / Print B in an 8-by-8 grid. /
• static void Display()
• for (int r lo r lt hi r)
• for (int c lo c lt hi c)
• System.out.print(Brc )
• System.out.println()
• Rule of Thumb. Develop your program using small
  procedures. Test it incrementally.

9
Sample Intermediate State

• while ( ________________________)

10
Pattern

• / Start at the beginning /
• . . .
• while ( / not past the end / )
• / Process the current place /
• . . .
• / Go on to the next place (or to past the
  end). /
• . . .

11
Solve

• / Initial configuration. /
• int move 0 // moves so far.
• int r lo // current row of Kn.
• int c lo // current column of Kn.
• while (r ! UNDEFINED)
• / Visit ltr,cgt. /
• move
• Brc move
• / Let ltr,cgt be coordinates of an unvisited
  "neighbor" of current ltr,cgt, or
  ltUNDEFINED,UNDEFINEDgt if current square has no
  unvisited neighbors. /
• . . .

12
Choosing a Neighbor to Visit

• / Let ltr,cgt be coordinates of an unvisited
  "neighbor" of current ltr,cgt, or
  ltUNDEFINED,UNDEFINEDgt if current square has no
  unvisited neighbors. /
• int unvisitedR UNDEFINED
• int unvisitedC UNDEFINED
• for (int k 0 k lt 8 k)
• int Nrow _____________________________
• int Ncol _____________________________
• if ( BNrowNcol Blank )
• unvisitedR Nrow
• unvisitedC Ncol
• r unvisitedR
• c unvisitedC

13
Neighbors

• // 0 1 2 3 4 5 6 7
• int deltaR -1, -2, -2, -1, 1, 2, 2, 1
• int deltaC 2, 1, -1, -2, -2, -1, 1, 2

14
Boundary Conditions

• where X can be any non-zero value

0 1 2 3 4 5 6 7 8 9
10 11
0 1 2 3 4 5 6 7 8 9 10 11
x x x x x x x x x x x x
x x x x x x x x x x x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x x x x x x x x x x x
x x x x x x x x x x x x
15
Representation Revisited

• Static class declarations visible to all methods.
  
• / Chess board B is N-by-N int array, for N
  12. /
• final int N 12
• int B new int NN
• / Unvisited squares are Blank. /
• final int Blank 0
• / Board subscripts range from lo to hi. /
• final int lo 2
• final int hi 9
• / UNDEFINED, an illegal coordinate. /
• final int UNDEFINED -1
• / ltdeltaRk,deltaCkgt is ltrow,colgt offset to
  neighbor k. /
• final static int deltaR
• -1,-2,-2,-1, 1, 2,2,1
• final static int deltaC

16
Improvement Using Heuristic

• Go where the options are running out, i.e., go to
  the unvisited neighbor with the fewest unvisited
  neighbors.

• / Let ltr,cgt be coordinates of an unvisited
  "neighbor" of current ltr,cgt, or
  ltUNDEFINED,UNDEFINEDgt if current square has no
  unvisited neighbors. /
• int fewest 9 // min unvisited neighbors
• int neighborR UNDEFINED
• int neighborC UNDEFINED
• for (int k 0 k lt 8 k)
• int Nrow r deltaRk
• int Ncol c deltaCk
• if ( BNrowNcol Blank )
• int n unvisited(Nrow, Ncol)
• if (n lt fewest )
• neighborR Nrow neighborC Ncol
• fewest n
• r neighborR

17
Unvisited

• / the number of neighbors of square ltr,cgt
  that are unvisited. /
• static int unvisited(int r, int c)
• int count 0 // unvisited neighs.
• for (int k 0 k lt 8 k)
• int Nrow r deltaRk
• int Ncol c deltaCk
• if ( BNrowNcol Blank ) count
• return count