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2746. Decremental String Concatenation

Description

You are given a 0-indexed array words containing n strings.

Let's define a join operation join(x, y) between two strings x and y as concatenating them into xy. However, if the last character of x is equal to the first character of y, one of them is deleted.

For example join("ab", "ba") = "aba" and join("ab", "cde") = "abcde".

You are to perform n - 1 join operations. Let str0 = words[0]. Starting from i = 1 up to i = n - 1, for the ith operation, you can do one of the following:

  • Make stri = join(stri - 1, words[i])
  • Make stri = join(words[i], stri - 1)

Your task is to minimize the length of strn - 1.

Return an integer denoting the minimum possible length of strn - 1.

 

Example 1:

Input: words = ["aa","ab","bc"]
Output: 4
Explanation: In this example, we can perform join operations in the following order to minimize the length of str2: 
str0 = "aa"
str1 = join(str0, "ab") = "aab"
str2 = join(str1, "bc") = "aabc" 
It can be shown that the minimum possible length of str2 is 4.

Example 2:

Input: words = ["ab","b"]
Output: 2
Explanation: In this example, str0 = "ab", there are two ways to get str1: 
join(str0, "b") = "ab" or join("b", str0) = "bab". 
The first string, "ab", has the minimum length. Hence, the answer is 2.

Example 3:

Input: words = ["aaa","c","aba"]
Output: 6
Explanation: In this example, we can perform join operations in the following order to minimize the length of str2: 
str0 = "aaa"
str1 = join(str0, "c") = "aaac"
str2 = join("aba", str1) = "abaaac"
It can be shown that the minimum possible length of str2 is 6.
 

 

Constraints:

  • 1 <= words.length <= 1000
  • 1 <= words[i].length <= 50
  • Each character in words[i] is an English lowercase letter

Solutions

We notice that when concatenating strings, the first and last characters of the string will affect the length of the concatenated string. Therefore, we design a function \(dfs(i, a, b)\), which represents the minimum length of the concatenated string starting from the \(i\)-th string, and the first character of the previously concatenated string is \(a\), and the last character is \(b\).

The execution process of the function \(dfs(i, a, b)\) is as follows:

  • If \(i = n\), it means that all strings have been concatenated, return \(0\);
  • Otherwise, we consider concatenating the \(i\)-th string to the end or the beginning of the already concatenated string, and get the lengths \(x\) and \(y\) of the concatenated string, then \(dfs(i, a, b) = \min(x, y) + |words[i]|\).

To avoid repeated calculations, we use the method of memoization search. Specifically, we use a three-dimensional array \(f\) to store all the return values of \(dfs(i, a, b)\). When we need to calculate \(dfs(i, a, b)\), if \(f[i][a][b]\) has been calculated, we directly return \(f[i][a][b]\); otherwise, we calculate the value of \(dfs(i, a, b)\) according to the above recurrence relation, and store it in \(f[i][a][b]\).

In the main function, we directly return \(|words[0]| + dfs(1, words[0][0], words[0][|words[0]| - 1])\).

The time complexity is \(O(n \times C^2)\), and the space complexity is \(O(n \times C^2)\). Where \(C\) represents the maximum length of the string.

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class Solution:
    def minimizeConcatenatedLength(self, words: List[str]) -> int:
        @cache
        def dfs(i: int, a: str, b: str) -> int:
            if i >= len(words):
                return 0
            s = words[i]
            x = dfs(i + 1, a, s[-1]) - int(s[0] == b)
            y = dfs(i + 1, s[0], b) - int(s[-1] == a)
            return len(s) + min(x, y)

        return len(words[0]) + dfs(1, words[0][0], words[0][-1])

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class Solution {
    private Integer[][][] f;
    private String[] words;
    private int n;

    public int minimizeConcatenatedLength(String[] words) {
        n = words.length;
        this.words = words;
        f = new Integer[n][26][26];
        return words[0].length()
            + dfs(1, words[0].charAt(0) - 'a', words[0].charAt(words[0].length() - 1) - 'a');
    }

    private int dfs(int i, int a, int b) {
        if (i >= n) {
            return 0;
        }
        if (f[i][a][b] != null) {
            return f[i][a][b];
        }
        String s = words[i];
        int m = s.length();
        int x = dfs(i + 1, a, s.charAt(m - 1) - 'a') - (s.charAt(0) - 'a' == b ? 1 : 0);
        int y = dfs(i + 1, s.charAt(0) - 'a', b) - (s.charAt(m - 1) - 'a' == a ? 1 : 0);
        return f[i][a][b] = m + Math.min(x, y);
    }
}

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class Solution {
public:
    int minimizeConcatenatedLength(vector<string>& words) {
        int n = words.size();
        int f[n][26][26];
        memset(f, 0, sizeof(f));
        function<int(int, int, int)> dfs = [&](int i, int a, int b) {
            if (i >= n) {
                return 0;
            }
            if (f[i][a][b]) {
                return f[i][a][b];
            }
            auto s = words[i];
            int m = s.size();
            int x = dfs(i + 1, a, s[m - 1] - 'a') - (s[0] - 'a' == b);
            int y = dfs(i + 1, s[0] - 'a', b) - (s[m - 1] - 'a' == a);
            return f[i][a][b] = m + min(x, y);
        };
        return words[0].size() + dfs(1, words[0].front() - 'a', words[0].back() - 'a');
    }
};

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func minimizeConcatenatedLength(words []string) int {
    n := len(words)
    f := make([][26][26]int, n)
    var dfs func(i, a, b int) int
    dfs = func(i, a, b int) int {
        if i >= n {
            return 0
        }
        if f[i][a][b] > 0 {
            return f[i][a][b]
        }
        s := words[i]
        m := len(s)
        x := dfs(i+1, a, int(s[m-1]-'a'))
        y := dfs(i+1, int(s[0]-'a'), b)
        if int(s[0]-'a') == b {
            x--
        }
        if int(s[m-1]-'a') == a {
            y--
        }
        f[i][a][b] = m + min(x, y)
        return f[i][a][b]
    }
    return len(words[0]) + dfs(1, int(words[0][0]-'a'), int(words[0][len(words[0])-1]-'a'))
}

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function minimizeConcatenatedLength(words: string[]): number {
    const n = words.length;
    const f: number[][][] = Array(n)
        .fill(0)
        .map(() =>
            Array(26)
                .fill(0)
                .map(() => Array(26).fill(0)),
        );
    const dfs = (i: number, a: number, b: number): number => {
        if (i >= n) {
            return 0;
        }
        if (f[i][a][b] > 0) {
            return f[i][a][b];
        }
        const s = words[i];
        const m = s.length;
        const x =
            dfs(i + 1, a, s[m - 1].charCodeAt(0) - 97) - (s[0].charCodeAt(0) - 97 === b ? 1 : 0);
        const y =
            dfs(i + 1, s[0].charCodeAt(0) - 97, b) - (s[m - 1].charCodeAt(0) - 97 === a ? 1 : 0);
        return (f[i][a][b] = Math.min(x + m, y + m));
    };
    return (
        words[0].length +
        dfs(1, words[0][0].charCodeAt(0) - 97, words[0][words[0].length - 1].charCodeAt(0) - 97)
    );
}

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