1411. Number of Ways to Paint N × 3 Grid

Description

You have a grid of size n x 3 and you want to paint each cell of the grid with exactly one of the three colors: Red, Yellow, or Green while making sure that no two adjacent cells have the same color (i.e., no two cells that share vertical or horizontal sides have the same color).

Given n the number of rows of the grid, return the number of ways you can paint this grid. As the answer may grow large, the answer must be computed modulo 109 + 7.

 

Example 1:

Input: n = 1
Output: 12
Explanation: There are 12 possible way to paint the grid as shown.

Example 2:

Input: n = 5000
Output: 30228214

 

Constraints:

• n == grid.length
• 1 <= n <= 5000

Solutions

Solution 1: Recursion

We classify all possible states for each row. According to the principle of symmetry, when a row only has \(3\) elements, all legal states are classified as: \(010\) type, \(012\) type.

• When the state is \(010\) type: The possible states for the next row are: \(101\), \(102\), \(121\), \(201\), \(202\). These \(5\) states can be summarized as \(3\) \(010\) types and \(2\) \(012\) types.
• When the state is \(012\) type: The possible states for the next row are: \(101\), \(120\), \(121\), \(201\). These \(4\) states can be summarized as \(2\) \(010\) types and \(2\) \(012\) types.

In summary, we can get: \(newf0 = 3 \times f0 + 2 \times f1\), \(newf1 = 2 \times f0 + 2 \times f1\).

The time complexity is \(O(n)\), where \(n\) is the number of rows in the grid. The space complexity is \(O(1)\).

Python3JavaC++GoTypeScript

class Solution:
    def numOfWays(self, n: int) -> int:
        mod = 10**9 + 7
        f0 = f1 = 6
        for _ in range(n - 1):
            g0 = (3 * f0 + 2 * f1) % mod
            g1 = (2 * f0 + 2 * f1) % mod
            f0, f1 = g0, g1
        return (f0 + f1) % mod

class Solution {
    public int numOfWays(int n) {
        int mod = (int) 1e9 + 7;
        long f0 = 6, f1 = 6;
        for (int i = 0; i < n - 1; ++i) {
            long g0 = (3 * f0 + 2 * f1) % mod;
            long g1 = (2 * f0 + 2 * f1) % mod;
            f0 = g0;
            f1 = g1;
        }
        return (int) (f0 + f1) % mod;
    }
}

using ll = long long;

class Solution {
public:
    int numOfWays(int n) {
        int mod = 1e9 + 7;
        ll f0 = 6, f1 = 6;
        while (--n) {
            ll g0 = (f0 * 3 + f1 * 2) % mod;
            ll g1 = (f0 * 2 + f1 * 2) % mod;
            f0 = g0;
            f1 = g1;
        }
        return (int) (f0 + f1) % mod;
    }
};

func numOfWays(n int) int {
    mod := int(1e9) + 7
    f0, f1 := 6, 6
    for n > 1 {
        n--
        g0 := (f0*3 + f1*2) % mod
        g1 := (f0*2 + f1*2) % mod
        f0, f1 = g0, g1
    }
    return (f0 + f1) % mod
}

function numOfWays(n: number): number {
    const mod: number = 10 ** 9 + 7;
    let f0: number = 6;
    let f1: number = 6;

    for (let i = 1; i < n; i++) {
        const g0: number = (3 * f0 + 2 * f1) % mod;
        const g1: number = (2 * f0 + 2 * f1) % mod;
        f0 = g0;
        f1 = g1;
    }

    return (f0 + f1) % mod;
}

Solution 2: State Compression + Dynamic Programming

We notice that the grid only has \(3\) columns, so there are at most \(3^3=27\) different coloring schemes in a row.

Therefore, we define \(f[i][j]\) to represent the number of schemes in the first \(i\) rows, where the coloring state of the \(i\)th row is \(j\). The state \(f[i][j]\) is transferred from \(f[i - 1][k]\), where \(k\) is the coloring state of the \(i - 1\)th row, and \(k\) and \(j\) meet the requirement of different colors being adjacent. That is:

\[ f[i][j] = \sum_{k \in \textit{valid}(j)} f[i - 1][k] \]

where \(\textit{valid}(j)\) represents all legal predecessor states of state \(j\).

The final answer is the sum of \(f[n][j]\), where \(j\) is any legal state.

We notice that \(f[i][j]\) is only related to \(f[i - 1][k]\), so we can use a rolling array to optimize the space complexity.

The time complexity is \(O((m + n) \times 3^{2m})\), and the space complexity is \(O(3^m)\). Here, \(m\) and \(n\) are the number of rows and columns of the grid, respectively.

Python3JavaC++GoTypeScript

class Solution:
    def numOfWays(self, n: int) -> int:
        def f1(x: int) -> bool:
            last = -1
            for _ in range(3):
                if x % 3 == last:
                    return False
                last = x % 3
                x //= 3
            return True

        def f2(x: int, y: int) -> bool:
            for _ in range(3):
                if x % 3 == y % 3:
                    return False
                x //= 3
                y //= 3
            return True

        mod = 10**9 + 7
        m = 27
        valid = {i for i in range(m) if f1(i)}
        d = defaultdict(list)
        for i in valid:
            for j in valid:
                if f2(i, j):
                    d[i].append(j)
        f = [int(i in valid) for i in range(m)]
        for _ in range(n - 1):
            g = [0] * m
            for i in valid:
                for j in d[i]:
                    g[j] = (g[j] + f[i]) % mod
            f = g
        return sum(f) % mod

class Solution {
    public int numOfWays(int n) {
        final int mod = (int) 1e9 + 7;
        int m = 27;
        Set<Integer> valid = new HashSet<>();
        int[] f = new int[m];
        for (int i = 0; i < m; ++i) {
            if (f1(i)) {
                valid.add(i);
                f[i] = 1;
            }
        }
        Map<Integer, List<Integer>> d = new HashMap<>();
        for (int i : valid) {
            for (int j : valid) {
                if (f2(i, j)) {
                    d.computeIfAbsent(i, k -> new ArrayList<>()).add(j);
                }
            }
        }
        for (int k = 1; k < n; ++k) {
            int[] g = new int[m];
            for (int i : valid) {
                for (int j : d.getOrDefault(i, List.of())) {
                    g[j] = (g[j] + f[i]) % mod;
                }
            }
            f = g;
        }
        int ans = 0;
        for (int x : f) {
            ans = (ans + x) % mod;
        }
        return ans;
    }

    private boolean f1(int x) {
        int last = -1;
        for (int i = 0; i < 3; ++i) {
            if (x % 3 == last) {
                return false;
            }
            last = x % 3;
            x /= 3;
        }
        return true;
    }

    private boolean f2(int x, int y) {
        for (int i = 0; i < 3; ++i) {
            if (x % 3 == y % 3) {
                return false;
            }
            x /= 3;
            y /= 3;
        }
        return true;
    }
}

class Solution {
public:
    int numOfWays(int n) {
        int m = 27;

        auto f1 = [&](int x) {
            int last = -1;
            for (int i = 0; i < 3; ++i) {
                if (x % 3 == last) {
                    return false;
                }
                last = x % 3;
                x /= 3;
            }
            return true;
        };
        auto f2 = [&](int x, int y) {
            for (int i = 0; i < 3; ++i) {
                if (x % 3 == y % 3) {
                    return false;
                }
                x /= 3;
                y /= 3;
            }
            return true;
        };

        const int mod = 1e9 + 7;
        unordered_set<int> valid;
        vector<int> f(m);
        for (int i = 0; i < m; ++i) {
            if (f1(i)) {
                valid.insert(i);
                f[i] = 1;
            }
        }
        unordered_map<int, vector<int>> d;
        for (int i : valid) {
            for (int j : valid) {
                if (f2(i, j)) {
                    d[i].push_back(j);
                }
            }
        }
        for (int k = 1; k < n; ++k) {
            vector<int> g(m);
            for (int i : valid) {
                for (int j : d[i]) {
                    g[j] = (g[j] + f[i]) % mod;
                }
            }
            f = move(g);
        }
        int ans = 0;
        for (int x : f) {
            ans = (ans + x) % mod;
        }
        return ans;
    }
};

func numOfWays(n int) (ans int) {
    f1 := func(x int) bool {
        last := -1
        for i := 0; i < 3; i++ {
            if x%3 == last {
                return false
            }
            last = x % 3
            x /= 3
        }
        return true
    }
    f2 := func(x, y int) bool {
        for i := 0; i < 3; i++ {
            if x%3 == y%3 {
                return false
            }
            x /= 3
            y /= 3
        }
        return true
    }
    m := 27
    valid := map[int]bool{}
    f := make([]int, m)
    for i := 0; i < m; i++ {
        if f1(i) {
            valid[i] = true
            f[i] = 1
        }
    }
    d := map[int][]int{}
    for i := range valid {
        for j := range valid {
            if f2(i, j) {
                d[i] = append(d[i], j)
            }
        }
    }
    const mod int = 1e9 + 7
    for k := 1; k < n; k++ {
        g := make([]int, m)
        for i := range valid {
            for _, j := range d[i] {
                g[i] = (g[i] + f[j]) % mod
            }
        }
        f = g
    }
    for _, x := range f {
        ans = (ans + x) % mod
    }
    return
}

function numOfWays(n: number): number {
    const f1 = (x: number): boolean => {
        let last = -1;
        for (let i = 0; i < 3; ++i) {
            if (x % 3 === last) {
                return false;
            }
            last = x % 3;
            x = Math.floor(x / 3);
        }
        return true;
    };
    const f2 = (x: number, y: number): boolean => {
        for (let i = 0; i < 3; ++i) {
            if (x % 3 === y % 3) {
                return false;
            }
            x = Math.floor(x / 3);
            y = Math.floor(y / 3);
        }
        return true;
    };
    const m = 27;
    const valid = new Set<number>();
    const f: number[] = Array(m).fill(0);
    for (let i = 0; i < m; ++i) {
        if (f1(i)) {
            valid.add(i);
            f[i] = 1;
        }
    }
    const d: Map<number, number[]> = new Map();
    for (const i of valid) {
        for (const j of valid) {
            if (f2(i, j)) {
                d.set(i, (d.get(i) || []).concat(j));
            }
        }
    }
    const mod = 10 ** 9 + 7;
    for (let k = 1; k < n; ++k) {
        const g: number[] = Array(m).fill(0);
        for (const i of valid) {
            for (const j of d.get(i) || []) {
                g[i] = (g[i] + f[j]) % mod;
            }
        }
        f.splice(0, f.length, ...g);
    }
    let ans = 0;
    for (const x of f) {
        ans = (ans + x) % mod;
    }
    return ans;
}

Comments