+ * The Tiling Problem involves filling a 2^n x 2^n board with a single missing + * square using L-shaped tiles (each tile covers exactly three squares). + * The algorithm recursively divides the board into four quadrants, places an + * L-shaped tile in the appropriate quadrant, and fills the remaining areas. + * + *
Applications: + * - Used in graphics and image processing. + * - Helpful in solving puzzles and tiling problems in competitive programming. + * + * @author Hardvan + */ +public final class TilingProblem { + private TilingProblem() { + } + + /** + * A counter used to label the L-shaped tiles placed on the board. + */ + private static int tile = 1; + + /** + * A 2D array representing the board to be tiled. + */ + private static int[][] board; + + /** + * Solves the tiling problem for a 2^n x 2^n board with one missing square. + * + * @param size The size of the board (must be a power of 2). + * @param missingRow The row index of the missing square. + * @param missingCol The column index of the missing square. + * @return A 2D array representing the tiled board with L-shaped tiles. + */ + public static int[][] solveTiling(int size, int missingRow, int missingCol) { + board = new int[size][size]; + fillBoard(size, 0, 0, missingRow, missingCol); + return board; + } + + /** + * Recursively fills the board with L-shaped tiles. + * + *
The board is divided into four quadrants. Depending on the location of + * the missing square, an L-shaped tile is placed at the center of the board + * to cover three of the four quadrants. The process is then repeated for + * each quadrant until the entire board is filled. + * + * @param size The current size of the sub-board. + * @param row The starting row index of the current sub-board. + * @param col The starting column index of the current sub-board. + * @param missingRow The row index of the missing square within the board. + * @param missingCol The column index of the missing square within the board. + */ + private static void fillBoard(int size, int row, int col, int missingRow, int missingCol) { + if (size == 1) { + return; + } + + int half = size / 2; + int t = tile++; + + // Top-left quadrant + if (missingRow < row + half && missingCol < col + half) { + fillBoard(half, row, col, missingRow, missingCol); + } else { + board[row + half - 1][col + half - 1] = t; + fillBoard(half, row, col, row + half - 1, col + half - 1); + } + + // Top-right quadrant + if (missingRow < row + half && missingCol >= col + half) { + fillBoard(half, row, col + half, missingRow, missingCol); + } else { + board[row + half - 1][col + half] = t; + fillBoard(half, row, col + half, row + half - 1, col + half); + } + + // Bottom-left quadrant + if (missingRow >= row + half && missingCol < col + half) { + fillBoard(half, row + half, col, missingRow, missingCol); + } else { + board[row + half][col + half - 1] = t; + fillBoard(half, row + half, col, row + half, col + half - 1); + } + + // Bottom-right quadrant + if (missingRow >= row + half && missingCol >= col + half) { + fillBoard(half, row + half, col + half, missingRow, missingCol); + } else { + board[row + half][col + half] = t; + fillBoard(half, row + half, col + half, row + half, col + half); + } + } +} diff --git a/src/test/java/com/thealgorithms/divideandconquer/TilingProblemTest.java b/src/test/java/com/thealgorithms/divideandconquer/TilingProblemTest.java new file mode 100644 index 000000000000..720e425f5ea3 --- /dev/null +++ b/src/test/java/com/thealgorithms/divideandconquer/TilingProblemTest.java @@ -0,0 +1,15 @@ +package com.thealgorithms.divideandconquer; + +import static org.junit.jupiter.api.Assertions.assertArrayEquals; + +import org.junit.jupiter.api.Test; + +public class TilingProblemTest { + + @Test + public void testTilingSize2() { + int[][] expected = {{1, 1}, {1, 0}}; + int[][] result = TilingProblem.solveTiling(2, 1, 1); + assertArrayEquals(expected, result); + } +}