Module 9: Recursion, Part II

public class MazeMaker {

    // These variables will be used across methods:
    static int desiredPathLength;    
    static Maze maze;


    public static void main (String[] argv)
    {
        generateMaze (5, 5);
    }


    public static void generateMaze (int numRows, int numCols)
    {
        maze = new Maze (numRows, numCols);

        // We want to generate a path of this length:
	desiredPathLength = numRows * numCols;

	// Initially, we'll start with the top left cell and mark that as visited.
        Coord start = new Coord (0, 0);
	maze.markVisited (start);

        // Generate the maze path recursively.
	boolean found = recursiveGenerate (start, 1);

        if (! found) {
            System.out.println ("Could not create the whole maze");
        }
        maze.display();
    }


    static boolean recursiveGenerate (Coord c, int pathLength)
    {
        // Bottom out condition 1:
	if (pathLength == desiredPathLength) {
	    // Done. We've create a maze path of the desired length.
	    return true;
	}

        // Bottom out condition 2: see if there's a neighbor to visit.
	Coord[] validNeighbors = maze.getUnvisitedClosedNeighbors (c);
	if ((validNeighbors == null) || (validNeighbors.length == 0)) {
            // There's no neighbor whose wall we can break. So quit trying.
	    return false;
	}

        // Otherwise, pick a random neighbor and proceed.
        int i = UniformRandom.uniform (0, validNeighbors.length-1);

        // Break the wall and mark the neighbor as visited.
        maze.breakWall (c, validNeighbors[i]);
        maze.markVisited (validNeighbors[i]);
            
        // Recurse.
        return recursiveGenerate (validNeighbors[i], pathLength+1);
    }

}

public class MazeMaker2 {

    static int desiredPathLength;
    static Maze maze;


    public static void main (String[] argv)
    {
        generateMaze (5, 5);
    }


    public static void generateMaze (int numRows, int numCols)
    {
        maze = new Maze (numRows, numCols);

	desiredPathLength = numRows * numCols;

	// Initially, we'll start with the top left cell.
        Coord start = new Coord (0, 0);
	maze.markVisited (start);

        // Generate the maze path recursively.
	boolean found = recursiveGenerate (start, 1);

        if (! found) {
            System.out.println ("Could not create the whole maze");
        }
        maze.display();
    }


    static boolean recursiveGenerate (Coord c, int pathLength)
    {
        // Bottom out condition 1:
	if (pathLength == desiredPathLength) {
	    return true;
	}

        // Bottom out condition 1: see if we're stuck.
	Coord[] validNeighbors = maze.getUnvisitedClosedNeighbors (c);
	if ((validNeighbors == null) || (validNeighbors.length == 0)) {
	    return false;
	}


	// Otherwise, we have some neighbors to explore.
	// Permute the directions randomly.
	permute (validNeighbors);

	for (int i=0; i < validNeighbors.length; i++) {

            // Try neighbor i.
            maze.breakWall (c, validNeighbors[i]);
            maze.markVisited (validNeighbors[i]);
            
	    boolean ok = recursiveGenerate (validNeighbors[i], pathLength+1);
	    if (ok) {
                // If neighbor i worked out, great.
		return true;
	    }

	    // Otherwise, undo assignment to i.
	    maze.fixWall (c, validNeighbors[i]);
	    maze.markUnvisited (validNeighbors[i]);

	} // end-for

	// Couldn't make it work.
	return false;
    }


    static void permute (Coord[] coords)
    {
	for (int i=0; i < coords.length; i++) {
	    // Find a random element to place into i-th place.
	    int k = (int) UniformRandom.uniform (i, coords.length-1);
	    Coord temp = coords[i];
	    coords[i] = coords[k];
	    coords[k] = temp;
	}
    }

}

public class MazeMaker3 {

    // ... variables ...

    public static void main (String[] argv)
    {
        generateMaze (5, 5);
    }


    public static void generateMaze (int numRows, int numCols)
    {
        // ... create and generate single-path maze ...

        // Break a few more walls, randomly.
        breakRandomWalls (maze, 10);

        maze.display();
    }


    static boolean recursiveGenerate (Coord c, int pathLength)
    {
        // ... same as before ...
    }

 
    static void breakRandomWalls (Maze maze, int numWalls)
    {
	for (int k=0; k < numWalls; k++) {
            // Create random coordinates, i.e., identify a random cell.
	    int x = UniformRandom.uniform (0, maze.numRows-1);
	    int y = UniformRandom.uniform (0, maze.numCols-1);
	    Coord c = new Coord (x,y);

            // Get its neighbors that are separated by a wall.
	    Coord[] validNeighbors = maze.getClosedNeighbors (c);
	    if (validNeighbors != null) {
                // Pick one and break the wall.
                int m = UniformRandom.uniform (0, validNeighbors.length-1);
		maze.breakWall (c, validNeighbors[m]);
	    }
	}
    }

}

import java.util.*;

public class MazeMaker4 {

    static int desiredPathLength;
    static Maze maze;


    public static void main (String[] argv)
    {
        Maze maze = generateMaze (5, 5);
	if (maze != null) {
            // We will seek a path from the top-left to the bottom-right corner.
	    Coord start = new Coord (0,0);
	    Coord end = new Coord (4,4);
	    solveMaze (maze, start, end);
	    maze.display ();
	}
	else {
	    System.out.println ("Maze creation did not work");
	}
    }


    // A path is a list of cells, i.e., a list of Coord instances.
    static LinkedList<Coord> solutionPath;


    static void solveMaze (Maze maze, Coord start, Coord end)
    {
        // We'll mark visited cells as we go along our path. Initially:
	maze.markAllUnvisited ();

        // Mark the start cell as visited.
	maze.markVisited (start);

        // Create the list.
	solutionPath = new LinkedList<Coord> ();

        // Recursively find the path and fill in coord's into the list.
	recursivelyFindPath (maze, start, end);

        // Pass the path into the GUI.
	maze.setSolutionPath (solutionPath);
    }


    static boolean recursivelyFindPath (Maze maze, Coord c, Coord end)
    {
        // First add the current cell into the path.
	solutionPath.add (c);

        // If we've reached the end, we're done.
	if ( (c.row == end.row) && (c.col == end.col) ) {
	    return true;
	}

        // Otherwise, let's find a neighbor to explore.
	Coord[] validNeighbors = maze.getUnvisitedOpenNeighbors (c);

	if (validNeighbors == null) {
            // If there aren't any, we're done, but couldn't find a path.
	    return false;
	}

	// Take the first one and explore from there.
	maze.markVisited (validNeighbors[0]);

	return recursivelyFindPath (maze, validNeighbors[0], end);
    }


    // ...  Maze generation code ... same as before ...

}

import java.util.*;

public class MazeMaker5 {

    // ... variables ... same as before (for generation) ...


    public static void main (String[] argv)
    {
        // ... same ...
    }


    // A path is a list of cells, i.e., a list of Coord instances.
    static LinkedList<Coord> solutionPath;


    static void solveMaze (Maze maze, Coord start, Coord end)
    {
        // We'll mark visited cells as we go along our path. Initially:
	maze.markAllUnvisited ();

        // Mark the start cell as visited.
	maze.markVisited (start);

        // Create the list.
	solutionPath = new LinkedList<Coord> ();

        // Recursively find the path and fill in coord's into the list.
	recursivelyFindPath (maze, start, end);

        // The start node gets added last. Why?
        solutionPath.addFirst (start);

        // Pass the path into the GUI.
	maze.setSolutionPath (solutionPath);
    }


    static boolean recursivelyFindPath (Maze maze, Coord c, Coord end)
    {
        // If we've reached the end, we're done.
	if ( (c.row == end.row) && (c.col == end.col) ) {
	    return true;
	}

        // Otherwise, let's find a neighbor to explore.
	Coord[] validNeighbors = maze.getUnvisitedOpenNeighbors (c);
	if (validNeighbors == null) {
            // If we couldn't find any neighbors to explore, we're stuck.
	    return false;
	}

        // Try each neighbor, as many as needed.

        for (int i=0; i < validNeighbors.length; i++) {

            // Try neighbor i.
	    maze.markVisited (validNeighbors[i]);
	    boolean found = recursivelyFindPath (maze, validNeighbors[i], end);

	    if (found) {
                // Notice that we add this to the front of the list, and only
                // after the solution has been found up to here.
		solutionPath.addFirst (validNeighbors[i]);
		return true;
	    }
            
            // If neighbor i didn't work out, un-do the visit and continue.
	    maze.markUnvisited (validNeighbors[i]);
	}

        // If we reached here, then none of the neighbors worked out.
	return false;
    }

}

We will of course solve this recursively:

• This is another example of recursion with backtracking.

• Key ideas:
  • We'll proceed columm-by-column (left to right).
  • When solving for the current column, try to place a queen on each possible row, and solve the sub-problem (starting with the next column) recursively.

• Pseudocode:

      Algorithm: solve (n, c)
      Input: n = the number of queens to assign, c = the column to start with
   
              // ... Bottom-out conditions ... (we'll do this later) 
  
         1.   for each row r
         2.     if [r,c] is a non-attacked square
         3.        place queen n on cell [r,c]
         4.        found = solve (n-1, c+1)               // Assign remaining n-1 recursively
         5.        if (found)
         6.          return true
         7.        endif
         8.     endif
         9.   endfor
              // We reach here if none of the rows in column c worked out.
         10.  return false
      

• Thus, we try every row for a queen and try to solve the sub-problem with fewer queens.
       => If if that gets solved, we can place the queen there and recurse back up

• The bottom-out conditions:
  • Check if there are no more queens to assign
         => Problem gets solved.
  • Check if there are no more columns to try
         => No solution exists.

• We will build a class called ChessBoard with methods that hide most of the messy board-manipulation details:
  • addQueen (row,col): add a queen in square [row,col].
  • removeQueen (row,col): remove the queen in square [row,col].
  • boolean isForbidden (row,col): see if [row,col] is an attacked square.
  • display(): to bring up a GUI with the board displayed.

import java.awt.*;
import java.util.*;

public class NQueens {

    static ChessBoard board;


    public static void main (String[] argv)
    {
        solveQueens (8, 8);
    }


    static void solveQueens (int numQueens, int size)
    {
        board = new ChessBoard (size);

        boolean solutionExists = solve (numQueens, 0);

        if (! solutionExists) {
            System.out.println ("No solution for " + numQueens + " on a " + size + " x " + size + " board");
            return;
        }

        System.out.println ("Solution found for " + numQueens + " on a " + size + " x " + size + " board");

        System.out.println (board);
        board.display();
    }
    

    static boolean solve (int numQueens, int whichCol)
    {
        // Bottom-out condition 1:
        if (numQueens == 0) {
            // None to assign - done.
            return true;
        }
        

        // Bottom-out condition 2:
        if (whichCol >= board.size()) {
            // No columns left to try - done.
            return false;
        }
        
        // Try every un-forbidden spot in each row.
        for (int row=0; row < board.size(); row++) {
            if ( ! board.isForbidden(row,whichCol) ) {

                // Try this location.
                board.addQueen (row, whichCol);
                
                boolean solutionExists = solve (numQueens-1, whichCol+1);

                if (solutionExists) {
                    return true;
                }

                // Else, un-do
                board.removeQueen (row, whichCol);
            }

        }
        
        // Couldn't find a solution.
        return false;
    }

}