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664. Strange Printer

Description

There is a strange printer with the following two special properties:

  • The printer can only print a sequence of the same character each time.
  • At each turn, the printer can print new characters starting from and ending at any place and will cover the original existing characters.

Given a string s, return the minimum number of turns the printer needed to print it.

 

Example 1:

Input: s = "aaabbb"
Output: 2
Explanation: Print "aaa" first and then print "bbb".

Example 2:

Input: s = "aba"
Output: 2
Explanation: Print "aaa" first and then print "b" from the second place of the string, which will cover the existing character 'a'.

 

Constraints:

  • 1 <= s.length <= 100
  • s consists of lowercase English letters.

Solutions

Solution 1: Dynamic Programming

Thinking

One print paints a contiguous segment with the same letter. \(n\le 100\) suits interval DP.

\(f[i][j]\) is the fewest prints for \(s[i..j]\). Equal endpoints reuse the last stroke; otherwise try every split. Fill shorter intervals first.

We define \(f[i][j]\) as the minimum operations to print \(s[i..j]\), with the initial value \(f[i][j]=\infty\), and the answer is \(f[0][n-1]\), where \(n\) is the length of string \(s\).

Consider \(f[i][j]\), if \(s[i] = s[j]\), we can print \(s[j]\) when print \(s[i]\), so we can ignore \(s[j]\) and continue to print \(s[i+1..j-1]\). If \(s[i] \neq s[j]\), we need to print the substring separately, i.e. \(s[i..k]\) and \(s[k+1..j]\), where \(k \in [i,j)\). So we can have the following transition equation:

\[ f[i][j]= \begin{cases} 1, & \textit{if } i=j \\ f[i][j-1], & \textit{if } s[i]=s[j] \\ \min_{i \leq k < j} \{f[i][k]+f[k+1][j]\}, & \textit{otherwise} \end{cases} \]

We can enumerate \(i\) from large to small and \(j\) from small to large, so that we can ensure that \(f[i][j-1]\), \(f[i][k]\) and \(f[k+1][j]\) have been calculated when we calculate \(f[i][j]\).

The time complexity is \(O(n^3)\) and the space complexity is \(O(n^2)\). Where \(n\) is the length of string \(s\).

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class Solution:
    def strangePrinter(self, s: str) -> int:
        n = len(s)
        f = [[inf] * n for _ in range(n)]
        for i in range(n - 1, -1, -1):
            f[i][i] = 1
            for j in range(i + 1, n):
                if s[i] == s[j]:
                    f[i][j] = f[i][j - 1]
                else:
                    for k in range(i, j):
                        f[i][j] = min(f[i][j], f[i][k] + f[k + 1][j])
        return f[0][-1]

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class Solution {
    public int strangePrinter(String s) {
        final int inf = 1 << 30;
        int n = s.length();
        int[][] f = new int[n][n];
        for (var g : f) {
            Arrays.fill(g, inf);
        }
        for (int i = n - 1; i >= 0; --i) {
            f[i][i] = 1;
            for (int j = i + 1; j < n; ++j) {
                if (s.charAt(i) == s.charAt(j)) {
                    f[i][j] = f[i][j - 1];
                } else {
                    for (int k = i; k < j; ++k) {
                        f[i][j] = Math.min(f[i][j], f[i][k] + f[k + 1][j]);
                    }
                }
            }
        }
        return f[0][n - 1];
    }
}

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class Solution {
public:
    int strangePrinter(string s) {
        int n = s.size();
        int f[n][n];
        memset(f, 0x3f, sizeof(f));
        for (int i = n - 1; ~i; --i) {
            f[i][i] = 1;
            for (int j = i + 1; j < n; ++j) {
                if (s[i] == s[j]) {
                    f[i][j] = f[i][j - 1];
                } else {
                    for (int k = i; k < j; ++k) {
                        f[i][j] = min(f[i][j], f[i][k] + f[k + 1][j]);
                    }
                }
            }
        }
        return f[0][n - 1];
    }
};

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func strangePrinter(s string) int {
    n := len(s)
    f := make([][]int, n)
    for i := range f {
        f[i] = make([]int, n)
        for j := range f[i] {
            f[i][j] = 1 << 30
        }
    }
    for i := n - 1; i >= 0; i-- {
        f[i][i] = 1
        for j := i + 1; j < n; j++ {
            if s[i] == s[j] {
                f[i][j] = f[i][j-1]
            } else {
                for k := i; k < j; k++ {
                    f[i][j] = min(f[i][j], f[i][k]+f[k+1][j])
                }
            }
        }
    }
    return f[0][n-1]
}

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function strangePrinter(s: string): number {
    const n = s.length;
    const f: number[][] = new Array(n).fill(0).map(() => new Array(n).fill(1 << 30));
    for (let i = n - 1; i >= 0; --i) {
        f[i][i] = 1;
        for (let j = i + 1; j < n; ++j) {
            if (s[i] === s[j]) {
                f[i][j] = f[i][j - 1];
            } else {
                for (let k = i; k < j; ++k) {
                    f[i][j] = Math.min(f[i][j], f[i][k] + f[k + 1][j]);
                }
            }
        }
    }
    return f[0][n - 1];
}

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