1. CSC 205 Programming II Lecture 18 The Eight Queens Problem

2. Recap: Backtracking The strategy guessing at a solution and backing up when an impasse is reached The solution template A general-purpose BackTrack class Application specific classes A class implementing the Application interface A Position class A class implementing the Java Iterator interface

3. The tryToSolve Method boolean success = false; Iterator itr = app.iterator (pos); while (!success && itr.hasNext()) { pos = (Position)itr.next(); if (app.valid (pos)) { app.record (pos); if (app.done (pos)) success = true; else { success = tryToSolve (pos); if (!success) app.undo (pos); } // not done } // a valid position } // while return success; 0-north 1-east3-west 2-south

4. The Eight Queens Problem A queen can attack pieces in her row, in her column, or in either of her diagonals The goal is to put eight queens on a board and none of them is under attack An efficient solution is needed There are 4,426,165,368 ways to arrange 8 queens on a chessboard of 64 squares The number is reduced to 40,320 after taking the fact that no two queens can be in the same row or column into account

5. Key Elements The start position The square in which the first queen is placed The finish position The first position on the last column which is not under attack The way to iterate Place the next queen in the column right to the current one, starting from the first row; check against the existing queen(s) Put the queen in the first row which is not under attack Backtrack if all eight rows have been tried in vain

6. Scenario I Let’s use a smaller board (4X4) and place the first queen in the first square on the board (in the upper-left corner) Q

7. Scenario II As a second example, let’s place the first queen in the second square in the first column Q

8. Define the Problem – the Position class public class Position { protected int row, column; public Position () { row = 0; column = 0; } // constructor public Position (int row, int column) { this.row = row; this.column = 0; } // constructor public int row () { return row; } public int column () { return column ; } }

9. Define the Problem – the QueensIterator class private class QueensIterator implements Iterator { int row, column; int count = 0; public QueensIterator (Position pos) { row = pos.row(); column = pos.column(); } // constructor public boolean hasNext() { return count < 8; } // method hasNext

10. Define the Problem – the next method // Precondition: 0 <= count <= 7. // Postcondition: the choice for the next Position has been //returned. public Object next() { Position nextPosition = new Position(); //add your code here: set row to the right value return nextPosition; } // method next public void remove() { throw new UnsupportedOperationException(); } // method next } // class QueensIterator

11. Define the Problem – the EightQueens class This class implements the Application interface (see the next two slides for details) Instance variables Methods to be implemented The valid method The done method The undo method The record method The toString method

12. The Application Interface import java.util.*; public interface Application { // Postcondition: true has been returned if pos could be on a //path to a goal position. Otherwise, false has been returned. public boolean valid (Position pos); // Postcondition: the position specified by pos has been //marked as being on a path to a goal position. public void record (Position pos); // Postcondition: true has been returned if pos is a goal //position. Otherwise, false has been returned. public boolean done (Position pos);

13. The Application Interface // Postcondition: the position specified by pos has been //marked as not being on a path to a goal position. public void undo (Position pos); // Postcondition: a string representation of this Application has //been returned. public String toString(); // Postcondition: an iterator over the positions directly //accessible from pos has been returned. public Iterator iterator (Position pos); } // interface Application

14. In The Heart of the Solution – When Should valid Return true ? false should be returned when Out of the valid range The attempted queen is under attack In the same row as an existing queen In the same column as an existing queen In the same descending diagonal as an existing queen In the same ascending diagonal as an existing queen Otherwise, true should be returned

15. Another Example Key West,FL:Miami,FL-70 Pensacola,FL:Tallahassee,FL-120 St Petersburg,FL:Tampa,FL-20;Naples,FL-100 Miami,FL:Fort Lauderdale,FL-15;Key West,FL-70 Tallahassee,FL:Pensacola,FL-120;Lake City,FL-70 Jacksonville,FL:Lake City,FL-50;Daytona Beach,FL-90 Naples,FL:Fort Lauderdale,FL-90;St Petersburg,FL-100 Orlando,FL:Daytona Beach,FL-40;Lake City,FL-120;Tampa,FL-60 Tampa,FL:Orlando,FL-60;St Petersburg,FL-20;Lake City,FL-125 Fort Pierce,FL:Daytona Beach,FL-110;Fort Lauderdale,FL-50 Fort Lauderdale,FL:Fort Pierce,FL-90;Naples,FL-90;Miami,FL-15 Daytona Beach,FL:Orlando,FL-40;Jacksonville,FL-90;Fort Pierce,FL-110 Lake City,FL:Tallahassee,FL-70;Orlando,FL-120;Jacksonville,FL-50;Tampa,FL-125

16. The Eight Queens Problem Q