|
48 | 48 |
|
49 | 49 | <!-- 这里可写通用的实现逻辑 --> |
50 | 50 |
|
| 51 | +**方法一:状态压缩 + 动态规划** |
| 52 | + |
| 53 | +我们注意到,网格的行数不超过 5ドル,ドル那么一列中最多有 3ドル^5=243$ 种不同的颜色方案。 |
| 54 | + |
| 55 | +因此,我们定义 $f[i][j]$ 表示前 $i$ 列中,第 $i$ 列的涂色状态为 $j$ 的方案数。状态 $f[i][j]$ 由 $f[i - 1][k]$ 转移而来,其中 $k$ 是第 $i - 1$ 列的涂色状态,且 $k$ 和 $j$ 满足不同颜色相邻的要求。即: |
| 56 | + |
| 57 | +$$ |
| 58 | +f[i][j] = \sum_{k \in \text{valid}(j)} f[i - 1][k] |
| 59 | +$$ |
| 60 | + |
| 61 | +其中 $\text{valid}(j)$ 表示状态 $j$ 的所有合法前驱状态。 |
| 62 | + |
| 63 | +最终的答案即为 $f[n][j]$ 的总和,其中 $j$ 是任意合法的状态。 |
| 64 | + |
| 65 | +我们注意到,$f[i][j]$ 只和 $f[i - 1][k]$ 有关,因此我们可以使用滚动数组优化空间复杂度。 |
| 66 | + |
| 67 | +时间复杂度 $O((m + n) \times 3^{2m}),ドル空间复杂度 $O(3^m)$。其中 $m$ 和 $n$ 分别是网格的行数和列数。 |
| 68 | + |
51 | 69 | <!-- tabs:start --> |
52 | 70 |
|
53 | 71 | ### **Python3** |
54 | 72 |
|
55 | 73 | <!-- 这里可写当前语言的特殊实现逻辑 --> |
56 | 74 |
|
57 | 75 | ```python |
| 76 | +class Solution: |
| 77 | + def colorTheGrid(self, m: int, n: int) -> int: |
| 78 | + def f1(x: int) -> bool: |
| 79 | + last = -1 |
| 80 | + for _ in range(m): |
| 81 | + if x % 3 == last: |
| 82 | + return False |
| 83 | + last = x % 3 |
| 84 | + x //= 3 |
| 85 | + return True |
| 86 | + |
| 87 | + def f2(x: int, y: int) -> bool: |
| 88 | + for _ in range(m): |
| 89 | + if x % 3 == y % 3: |
| 90 | + return False |
| 91 | + x, y = x // 3, y // 3 |
| 92 | + return True |
58 | 93 |
|
| 94 | + mod = 10**9 + 7 |
| 95 | + mx = 3**m |
| 96 | + valid = {i for i in range(mx) if f1(i)} |
| 97 | + d = defaultdict(list) |
| 98 | + for x in valid: |
| 99 | + for y in valid: |
| 100 | + if f2(x, y): |
| 101 | + d[x].append(y) |
| 102 | + f = [int(i in valid) for i in range(mx)] |
| 103 | + for _ in range(n - 1): |
| 104 | + g = [0] * mx |
| 105 | + for i in valid: |
| 106 | + for j in d[i]: |
| 107 | + g[i] = (g[i] + f[j]) % mod |
| 108 | + f = g |
| 109 | + return sum(f) % mod |
59 | 110 | ``` |
60 | 111 |
|
61 | 112 | ### **Java** |
62 | 113 |
|
63 | 114 | <!-- 这里可写当前语言的特殊实现逻辑 --> |
64 | 115 |
|
65 | 116 | ```java |
| 117 | +class Solution { |
| 118 | + private int m; |
| 119 | + |
| 120 | + public int colorTheGrid(int m, int n) { |
| 121 | + this.m = m; |
| 122 | + final int mod = (int) 1e9 + 7; |
| 123 | + int mx = (int) Math.pow(3, m); |
| 124 | + Set<Integer> valid = new HashSet<>(); |
| 125 | + for (int i = 0; i < mx; ++i) { |
| 126 | + if (f1(i)) { |
| 127 | + valid.add(i); |
| 128 | + } |
| 129 | + } |
| 130 | + Map<Integer, List<Integer>> d = new HashMap<>(); |
| 131 | + for (int i : valid) { |
| 132 | + for (int j : valid) { |
| 133 | + if (f2(i, j)) { |
| 134 | + d.computeIfAbsent(i, k -> new ArrayList<>()).add(j); |
| 135 | + } |
| 136 | + } |
| 137 | + } |
| 138 | + int[] f = new int[mx]; |
| 139 | + for (int i = 0; i < mx; ++i) { |
| 140 | + f[i] = valid.contains(i) ? 1 : 0; |
| 141 | + } |
| 142 | + for (int k = 1; k < n; ++k) { |
| 143 | + int[] g = new int[mx]; |
| 144 | + for (int i : valid) { |
| 145 | + for (int j : d.getOrDefault(i, List.of())) { |
| 146 | + g[i] = (g[i] + f[j]) % mod; |
| 147 | + } |
| 148 | + } |
| 149 | + f = g; |
| 150 | + } |
| 151 | + int ans = 0; |
| 152 | + for (int x : f) { |
| 153 | + ans = (ans + x) % mod; |
| 154 | + } |
| 155 | + return ans; |
| 156 | + } |
| 157 | + |
| 158 | + private boolean f1(int x) { |
| 159 | + int last = -1; |
| 160 | + for (int i = 0; i < m; ++i) { |
| 161 | + if (x % 3 == last) { |
| 162 | + return false; |
| 163 | + } |
| 164 | + last = x % 3; |
| 165 | + x /= 3; |
| 166 | + } |
| 167 | + return true; |
| 168 | + } |
| 169 | + |
| 170 | + private boolean f2(int x, int y) { |
| 171 | + for (int i = 0; i < m; ++i) { |
| 172 | + if (x % 3 == y % 3) { |
| 173 | + return false; |
| 174 | + } |
| 175 | + x /= 3; |
| 176 | + y /= 3; |
| 177 | + } |
| 178 | + return true; |
| 179 | + } |
| 180 | +} |
| 181 | +``` |
| 182 | + |
| 183 | +### **C++** |
| 184 | + |
| 185 | +```cpp |
| 186 | +class Solution { |
| 187 | +public: |
| 188 | + int colorTheGrid(int m, int n) { |
| 189 | + auto f1 = [&](int x) { |
| 190 | + int last = -1; |
| 191 | + for (int i = 0; i < m; ++i) { |
| 192 | + if (x % 3 == last) { |
| 193 | + return false; |
| 194 | + } |
| 195 | + last = x % 3; |
| 196 | + x /= 3; |
| 197 | + } |
| 198 | + return true; |
| 199 | + }; |
| 200 | + auto f2 = [&](int x, int y) { |
| 201 | + for (int i = 0; i < m; ++i) { |
| 202 | + if (x % 3 == y % 3) { |
| 203 | + return false; |
| 204 | + } |
| 205 | + x /= 3; |
| 206 | + y /= 3; |
| 207 | + } |
| 208 | + return true; |
| 209 | + }; |
| 210 | + |
| 211 | + const int mod = 1e9 + 7; |
| 212 | + int mx = pow(3, m); |
| 213 | + unordered_set<int> valid; |
| 214 | + for (int i = 0; i < mx; ++i) { |
| 215 | + if (f1(i)) { |
| 216 | + valid.insert(i); |
| 217 | + } |
| 218 | + } |
| 219 | + unordered_map<int, vector<int>> d; |
| 220 | + for (int i : valid) { |
| 221 | + for (int j : valid) { |
| 222 | + if (f2(i, j)) { |
| 223 | + d[i].push_back(j); |
| 224 | + } |
| 225 | + } |
| 226 | + } |
| 227 | + vector<int> f(mx); |
| 228 | + for (int i = 0; i < mx; ++i) { |
| 229 | + f[i] = valid.count(i); |
| 230 | + } |
| 231 | + for (int k = 1; k < n; ++k) { |
| 232 | + vector<int> g(mx); |
| 233 | + for (int i : valid) { |
| 234 | + for (int j : d[i]) { |
| 235 | + g[i] = (g[i] + f[j]) % mod; |
| 236 | + } |
| 237 | + } |
| 238 | + f = move(g); |
| 239 | + } |
| 240 | + int ans = 0; |
| 241 | + for (int x : f) { |
| 242 | + ans = (ans + x) % mod; |
| 243 | + } |
| 244 | + return ans; |
| 245 | + } |
| 246 | +}; |
| 247 | +``` |
| 248 | + |
| 249 | +### **Go** |
| 250 | + |
| 251 | +```go |
| 252 | +func colorTheGrid(m int, n int) (ans int) { |
| 253 | + f1 := func(x int) bool { |
| 254 | + last := -1 |
| 255 | + for i := 0; i < m; i++ { |
| 256 | + if x%3 == last { |
| 257 | + return false |
| 258 | + } |
| 259 | + last = x % 3 |
| 260 | + x /= 3 |
| 261 | + } |
| 262 | + return true |
| 263 | + } |
| 264 | + f2 := func(x, y int) bool { |
| 265 | + for i := 0; i < m; i++ { |
| 266 | + if x%3 == y%3 { |
| 267 | + return false |
| 268 | + } |
| 269 | + x /= 3 |
| 270 | + y /= 3 |
| 271 | + } |
| 272 | + return true |
| 273 | + } |
| 274 | + mx := int(math.Pow(3, float64(m))) |
| 275 | + valid := map[int]bool{} |
| 276 | + for i := 0; i < mx; i++ { |
| 277 | + if f1(i) { |
| 278 | + valid[i] = true |
| 279 | + } |
| 280 | + } |
| 281 | + d := map[int][]int{} |
| 282 | + for i := range valid { |
| 283 | + for j := range valid { |
| 284 | + if f2(i, j) { |
| 285 | + d[i] = append(d[i], j) |
| 286 | + } |
| 287 | + } |
| 288 | + } |
| 289 | + f := make([]int, mx) |
| 290 | + for i := 0; i < mx; i++ { |
| 291 | + if valid[i] { |
| 292 | + f[i] = 1 |
| 293 | + } |
| 294 | + } |
| 295 | + const mod int = 1e9 + 7 |
| 296 | + for k := 1; k < n; k++ { |
| 297 | + g := make([]int, mx) |
| 298 | + for i := range valid { |
| 299 | + for _, j := range d[i] { |
| 300 | + g[i] = (g[i] + f[j]) % mod |
| 301 | + } |
| 302 | + } |
| 303 | + f = g |
| 304 | + } |
| 305 | + for _, x := range f { |
| 306 | + ans = (ans + x) % mod |
| 307 | + } |
| 308 | + return |
| 309 | +} |
| 310 | +``` |
| 311 | + |
| 312 | +### **TypeScript** |
66 | 313 |
|
| 314 | +```ts |
| 315 | +function colorTheGrid(m: number, n: number): number { |
| 316 | + const f1 = (x: number): boolean => { |
| 317 | + let last = -1; |
| 318 | + for (let i = 0; i < m; ++i) { |
| 319 | + if (x % 3 === last) { |
| 320 | + return false; |
| 321 | + } |
| 322 | + last = x % 3; |
| 323 | + x = Math.floor(x / 3); |
| 324 | + } |
| 325 | + return true; |
| 326 | + }; |
| 327 | + const f2 = (x: number, y: number): boolean => { |
| 328 | + for (let i = 0; i < m; ++i) { |
| 329 | + if (x % 3 === y % 3) { |
| 330 | + return false; |
| 331 | + } |
| 332 | + x = Math.floor(x / 3); |
| 333 | + y = Math.floor(y / 3); |
| 334 | + } |
| 335 | + return true; |
| 336 | + }; |
| 337 | + const mx = 3 ** m; |
| 338 | + const valid = new Set<number>(); |
| 339 | + for (let i = 0; i < mx; ++i) { |
| 340 | + if (f1(i)) { |
| 341 | + valid.add(i); |
| 342 | + } |
| 343 | + } |
| 344 | + const d: Map<number, number[]> = new Map(); |
| 345 | + for (const i of valid) { |
| 346 | + for (const j of valid) { |
| 347 | + if (f2(i, j)) { |
| 348 | + d.set(i, (d.get(i) || []).concat(j)); |
| 349 | + } |
| 350 | + } |
| 351 | + } |
| 352 | + const f: number[] = Array(mx).fill(0); |
| 353 | + for (let i = 0; i < mx; ++i) { |
| 354 | + if (valid.has(i)) { |
| 355 | + f[i] = 1; |
| 356 | + } |
| 357 | + } |
| 358 | + const mod = 10 ** 9 + 7; |
| 359 | + for (let k = 1; k < n; ++k) { |
| 360 | + const g: number[] = Array(mx).fill(0); |
| 361 | + for (const i of valid) { |
| 362 | + for (const j of d.get(i) || []) { |
| 363 | + g[i] = (g[i] + f[j]) % mod; |
| 364 | + } |
| 365 | + } |
| 366 | + f.splice(0, f.length, ...g); |
| 367 | + } |
| 368 | + let ans = 0; |
| 369 | + for (const x of f) { |
| 370 | + ans = (ans + x) % mod; |
| 371 | + } |
| 372 | + return ans; |
| 373 | +} |
67 | 374 | ``` |
68 | 375 |
|
69 | 376 | ### **...** |
|
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