思路
- 验证已填格子是否合法，非法则回 false
- 选一个空格子，如果没有空格子则返回 true
- 对空格子进行填数，依次尝试 1-9
- 猜解剩余格子，重复步骤 1
- 步骤 4 猜解成功则返回 true，否则回到步骤 3 尝试下一个填数
- 1-9 均返回 false，则返回 false
实现
- 约定
- 使用 9 x 9 的二维数组存储所有格子
- 每个格子维护一个可填数的 mask，采用 bitmap 方式存储，所以 0b111_111_111 可填 1-9，0b000_000_000 无可填数
- 为其中一个格子填数时，更新其所对应的行、列、区块上的每个格子的 mask
- 当有格子的 mask 为 0b000_000_000 时，数独不可解
- 当行、列、区块上 mask 有冲突时，数独不可解
- 因涉及递归，所以每个格子实际存储为一个 mask 栈
代码
// 测试 public class SudoKu { public static String sudokuString(BiFunction<Integer, Integer, Integer> supplier) { StringBuilder str = new StringBuilder(" _0_1_2_3_4_5_6_7_8\n"); for (int r = 0; r < 9; r++) { str.append(r).append(" | "); for (int c = 0; c < 9; c++) { int v = supplier.apply(r, c); str.append(v).append(" "); } str.append('\n'); } return str.toString(); } public static void main(String[] args) { List<Integer> inputs = Arrays.asList(821007900, 7000000, 400003000, 908040000, 000000001, 374201000, 160040, 60000000, 709008600); int[][] cells = new int[9][9]; for (int rowIndex = 0; rowIndex < inputs.size(); rowIndex++) { int input = inputs.get(rowIndex); for (int ci = 8; input > 0; input /= 10, ci--) { cells[rowIndex][ci] = input % 10; } } SudoKuResolver rc = new SudoKuResolver(cells); System.out.printf("Input >>\n%s", sudokuString(rc::getCellValue)); System.out.printf("resolving...\n"); long s = System.currentTimeMillis(); boolean resolved = rc.resolve(); long e = System.currentTimeMillis(); long cost = e - s; System.out.printf("finished, cost: %s\n" , cost); System.out.printf("Output >>\n%s", resolved ? sudokuString(rc::getCellValue) : "不可解"); } } // 解析 class SudoKuResolver { public final static int[] MIN_INDEX = new int[] { 0, 0, 0, 3, 3, 3, 6, 6, 6 }; public final static int[] MAX_INDEX = new int[] { 2, 2, 2, 5, 5, 5, 8, 8, 8 }; private static final Logger LOGGER = LoggerFactory.getLogger(SudoKuResolver.class); private final CellMaskBucket[][] cellBuckets; private int depth; public SudoKuResolver(int[][] cells) { this.cellBuckets = new CellMaskBucket[9][9]; for (int i = 0; i < 9; i++) { for (int j = 0; j < 9; j++) { cellBuckets[i][j] = new CellMaskBucket(i, j); cellBuckets[i][j].setMask(0b111_111_111); } } for (int i = 0; i < 9; i++) { for (int j = 0; j < 9; j++) { int v = cells[i][j]; if (v != 0) { CellMaskBucket bucket = cellBuckets[i][j]; bucket.setMask(toMask(cells[i][j])); } } } } public boolean resolve() { // 是否有效 if (!isValid()) { return false; } // 选出格子 CellMaskBucket cell = pickCell(); if (cell == null) { return true; } // 使用调用栈保存拆解步骤 int m = cell.getMask(); if (LOGGER.isDebugEnabled()) { int trys[] = computeTrys(m); LOGGER.debug("{}: try [ {} ]", cell, Strings.join(",", trys)); } while (Integer.bitCount(m) > 0) { int zc = Integer.numberOfTrailingZeros(m); int tv = 1 << zc; push(); setCellMask(cell, tv); if (LOGGER.isDebugEnabled()) { LOGGER.debug("{}: try {}\n{}", cell, zc + 1, SudoKu.sudokuString(this::getCellValue)); } if (resolve()) { return true; } else { m &= ~tv; pop(); if (LOGGER.isDebugEnabled()) { LOGGER.debug("{}: not {}", cell, zc + 1); } } } // 都不可解，返回 false return false; } void push() { if (++depth > 80) { throw new ArrayIndexOutOfBoundsException(depth--); } for (int i = 0; i < 9; i++) { for (int j = 0; j < 9; j++) { cellBuckets[i][j].bucket[depth] = cellBuckets[i][j].bucket[depth - 1]; } } } void pop() { if (--depth < 0) { throw new ArrayIndexOutOfBoundsException(depth++); } } void setCellMask(CellMaskBucket bucket, int mask) { bucket.setMask(mask); rcbTravel(bucket.row, bucket.col, (i, j) -> { int om = cellBuckets[i][j].getMask(); if (Integer.bitCount(om) > 1) { this.cellBuckets[i][j].setMask(om & (~mask)); } }); } void rcbTravel(int r, int c, BiConsumer<Integer, Integer> act) { // 行 for (int i = 0; i < 9; i++) { if (i == c) { continue; } act.accept(r, i); } // 列 for (int i = 0; i < 9; i++) { if (i == r) { continue; } act.accept(i, c); } // 区块 for (int i = MIN_INDEX[r]; i <= MAX_INDEX[r]; i++) { for (int j = MIN_INDEX[c]; j <= MAX_INDEX[c]; j++) { if (i == r || j == c) { continue; } act.accept(i, j); } } } CellMaskBucket pickCell() { for (int i = 0; i < 9; i++) { for (int j = 0; j < 9; j++) { CellMaskBucket cs = cellBuckets[i][j]; if (Integer.bitCount(cs.getMask()) > 1) { return cs; } } } return null; } boolean isValid() { int rms, cms, re, ce, bms, be; for (int i = 0; i < 9; i++) { rms = cms = re = ce = bms = be = 0; for (int j = 0; j < 9; j++) { // 行 int rm = cellBuckets[i][j].getMask(); if (Integer.bitCount(rm) == 1) { re += 1; rms += rm; if (Integer.bitCount(rms) != re) { return false; } } // 列 int cm = cellBuckets[j][i].getMask(); if (Integer.bitCount(cm) == 1) { ce += 1; cms += cm; if (Integer.bitCount(cms) != ce) { return false; } } // block int r = (i / 3) * 3 + (j / 3); int c = (i % 3) * 3 + (j % 3); int bm = cellBuckets[r][c].getMask(); if (Integer.bitCount(bm) == 1) { be += 1; bms += bm; if (Integer.bitCount(bms) != be) { return false; } } } } return true; } public int getCellValue(int r, int c) { int m = this.cellBuckets[r][c].getMask(); return toDecimal(m); } class CellMaskBucket { private final int row, col; private final int[] bucket = new int[81]; CellMaskBucket(int row, int col) { this.row = row; this.col = col; } void setMask(int v) { bucket[depth] = v; } int getMask() { return bucket[depth]; } @Override public String toString() { return "(" + row + ", " + col + ")"; } } static int toDecimal(int mask) { if (Integer.bitCount(mask) == 1) { return Integer.numberOfTrailingZeros(mask << 1); } return 0; } static int toMask(int dec) { return dec == 0 ? 0b111_111_111 : 1 << (dec - 1); } static int[] computeTrys(int m) { int[] r = new int[Integer.bitCount(m)]; for (int i = 0; i < r.length; i++) { int z = Integer.numberOfTrailingZeros(m); r[i] = z + 1; m >>= r[i]; m <<= r[i]; } return r; } }