0038-外观数列

Raphael Liu Lv10
  2023-09-13 00:47:26 2023-09-13 00:47:26 Created   2023-09-13 10:41:01 2023-09-13 10:41:01 Updated    

给定一个正整数 n ，输出外观数列的第 n 项。

「外观数列」是一个整数序列，从数字 1 开始，序列中的每一项都是对前一项的描述。

你可以将其视作是由递归公式定义的数字字符串序列：

  • countAndSay(1) = "1"
  • countAndSay(n) 是对 countAndSay(n-1) 的描述，然后转换成另一个数字字符串。

前五项如下：

1.     1
2.     11
3.     21
4.     1211
5.     111221
第一项是数字 1 
描述前一项，这个数是 1 即 “ 一 个 1 ”，记作 "11"
描述前一项，这个数是 11 即 “ 二 个 1 ” ，记作 "21"
描述前一项，这个数是 21 即 “ 一 个 2 + 一 个 1 ” ，记作 "1211"
描述前一项，这个数是 1211 即 “ 一 个 1 + 一 个 2 + 二 个 1 ” ，记作 "111221"

要 描述 一个数字字符串，首先要将字符串分割为 最小 数量的组，每个组都由连续的最多 相同字符
组成。然后对于每个组，先描述字符的数量，然后描述字符，形成一个描述组。要将描述转换为数字字符串，先将每组中的字符数量用数字替换，再将所有描述组连接起来。

例如，数字字符串 "3322251" 的描述如下图：

示例 1：

**输入：** n = 1
**输出：** "1"
**解释：** 这是一个基本样例。

示例 2：

**输入：** n = 4
**输出：** "1211"
**解释：**
countAndSay(1) = "1"
countAndSay(2) = 读 "1" = 一 个 1 = "11"
countAndSay(3) = 读 "11" = 二 个 1 = "21"
countAndSay(4) = 读 "21" = 一 个 2 + 一 个 1 = "12" + "11" = "1211"

提示：

  • 1 <= n <= 30

📺 视频题解

38.外观数列-仲耀晖.mp4

📖 文字题解

方法一：遍历生成

思路

通过阅读题目，我们可以了解到所谓的「外观数列」，其实本质上只是依次统计字符串中连续相同字符的个数。

  • 例如字符串 $\texttt{1112234445666}$ 我们依次统计连续相同字符的个数为: $3$ 个连续的字符 $\texttt{1}$, $2$ 个连续的 $\texttt{2}$，$1$ 个连续的字符 $\texttt{3}$，$3$ 个连续的字符 $\texttt{4}$，$1$ 个连续的字符 $\texttt{5}$，$3$ 个连续的字符 $\texttt{6}$。我们对其进行整理为 $\texttt{(3)1(2)2(1)3(3)4(1)5(3)6}$，我们将括号内的数字也转换为字符串为 $\texttt{312213341536}$。

算法

题目中给定的递归公式定义的数字字符串序列如下：

  • $\texttt{countAndSay(1) = “1”}$；

  • $\texttt{countAndSay(n)}$ 是对 $\texttt{countAndSay(n-1)}$ 的描述，然后转换成另一个数字字符串。

我们定义字符串 $S_{i}$ 表示 $\texttt{countAndSay(i)}$，我们如果要求得 $S_{n}$，则我们需先求出 $S_{n-1}$，然后按照上述描述的方法生成，即从左到右依次扫描字符串 $S_{n-1}$ 中连续相同的字符的最大数目，然后将字符的统计数目转化为数字字符串再连接上对应的字符。我们已知 $S_{1}$ 为 $\texttt{“1”}$，我们按照上述方法依次生成 $S_{2},S_{3},\ldots,S_{n-1},S_{n}$ 即可。

代码

[sol1-C++]
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class Solution {
public:
    string countAndSay(int n) {
        string prev = "1";
        for (int i = 2; i <= n; ++i) {
            string curr = "";
            int start = 0;
            int pos = 0;

            while (pos < prev.size()) {
                while (pos < prev.size() && prev[pos] == prev[start]) {
                    pos++;
                }
                curr += to_string(pos - start) + prev[start];
                start = pos;
            }
            prev = curr;
        }
        
        return prev;
    }
};
[sol1-Java]
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class Solution {
    public String countAndSay(int n) {
        String str = "1";
        for (int i = 2; i <= n; ++i) {
            StringBuilder sb = new StringBuilder();
            int start = 0;
            int pos = 0;

            while (pos < str.length()) {
                while (pos < str.length() && str.charAt(pos) == str.charAt(start)) {
                    pos++;
                }
                sb.append(Integer.toString(pos - start)).append(str.charAt(start));
                start = pos;
            }
            str = sb.toString();
        }
        
        return str;
    }
}
[sol1-C#]
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public class Solution {
    public string CountAndSay(int n) {
        string str = "1";
        for (int i = 2; i <= n; ++i) {
            StringBuilder sb = new StringBuilder();
            int start = 0;
            int pos = 0;

            while (pos < str.Length) {
                while (pos < str.Length && str[pos] == str[start]) {
                    pos++;
                }
                sb.Append(pos - start).Append(str[start]);
                start = pos;
            }
            str = sb.ToString();
        }
        
        return str;
    }
}
[sol1-Python3]
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class Solution:
    def countAndSay(self, n: int) -> str:
        prev = "1"
        for i in range(n-1):
            curr = ""
            pos = 0
            start = 0

            while pos < len(prev):
                while pos < len(prev) and prev[pos] == prev[start]:
                    pos += 1
                curr += str(pos - start) + prev[start]
                start = pos
            prev = curr
        
        return prev
[sol1-JavaScript]
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var countAndSay = function(n) {
    let str = "1";
    for (let i = 2; i <= n; ++i) {
        const sb = [];
        let start = 0;
        let pos = 0;

        while (pos < str.length) {
            while (pos < str.length && str[pos] === str[start]) {
                pos++;
            }
            sb.push('' + (pos - start) + str[start]);
            start = pos;
        }
        str = sb.join('');
    }
    
    return str;
};
[sol1-Golang]
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func countAndSay(n int) string {
    prev := "1"
    for i := 2; i <= n; i++ {
        cur := &strings.Builder{}
        for j, start := 0, 0; j < len(prev); start = j {
            for j < len(prev) && prev[j] == prev[start] {
                j++
            }
            cur.WriteString(strconv.Itoa(j - start))
            cur.WriteByte(prev[start])
        }
        prev = cur.String()
    }
    return prev
}

复杂度分析

  • 时间复杂度：$O(N \times M)$，其中 $N$ 为给定的正整数，$M$ 为生成的字符串中的最大长度。

  • 空间复杂度：$O(M)$。其中 $M$ 为生成的字符串中的最大长度。

方法二：枚举查表

思路与算法

题目中给定的数量限制为 $1 \le n \le 30$，由于数量很小，我们只需要依次生成前 $30$ 个 $\texttt{countAndSay}$ 字符串并将其存储为静态数组，每次直接查询数组中的第 $n$ 个元素即可。

代码

[sol2-C++]
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class Solution {
public:
    string countAndSay(int n) {
        vector<string> arr = 
        {
            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        };
        return arr[n];        
    }
};
[sol2-Java]
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class Solution {
    public String countAndSay(int n) {
        String[] arr = {
            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        };
        return arr[n];
    }
}
[sol2-C#]
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public class Solution {
    public string CountAndSay(int n) {
        string[] arr = {
            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        };
        return arr[n];
    }
}
[sol2-Python3]
1
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class Solution:
    def countAndSay(self, n: int) -> str:
        arr = ["","1","11","21","1211","111221","312211","13112221","1113213211","31131211131221","13211311123113112211","11131221133112132113212221","3113112221232112111312211312113211","1321132132111213122112311311222113111221131221","11131221131211131231121113112221121321132132211331222113112211","311311222113111231131112132112311321322112111312211312111322212311322113212221","132113213221133112132113311211131221121321131211132221123113112221131112311332111213211322211312113211","11131221131211132221232112111312212321123113112221121113122113111231133221121321132132211331121321231231121113122113322113111221131221","31131122211311123113321112131221123113112211121312211213211321322112311311222113311213212322211211131221131211132221232112111312111213111213211231131122212322211331222113112211","1321132132211331121321231231121113112221121321132122311211131122211211131221131211132221121321132132212321121113121112133221123113112221131112311332111213122112311311123112111331121113122112132113213211121332212311322113212221","11131221131211132221232112111312111213111213211231132132211211131221131211221321123113213221123113112221131112311332211211131221131211132211121312211231131112311211232221121321132132211331121321231231121113112221121321133112132112312321123113112221121113122113121113123112112322111213211322211312113211","311311222113111231133211121312211231131112311211133112111312211213211312111322211231131122211311122122111312211213211312111322211213211321322113311213212322211231131122211311123113223112111311222112132113311213211221121332211211131221131211132221232112111312111213111213211231132132211211131221232112111312211213111213122112132113213221123113112221131112311311121321122112132231121113122113322113111221131221","132113213221133112132123123112111311222112132113311213211231232112311311222112111312211311123113322112132113213221133122112231131122211211131221131112311332211211131221131211132221232112111312111213322112132113213221133112132113221321123113213221121113122123211211131221222112112322211231131122211311123113321112131221123113111231121113311211131221121321131211132221123113112211121312211231131122211211133112111311222112111312211312111322211213211321322113311213211331121113122122211211132213211231131122212322211331222113112211","111312211312111322212321121113121112131112132112311321322112111312212321121113122112131112131221121321132132211231131122211331121321232221121113122113121113222123112221221321132132211231131122211331121321232221123113112221131112311332111213122112311311123112112322211211131221131211132221232112111312211322111312211213211312111322211231131122111213122112311311221132211221121332211213211321322113311213212312311211131122211213211331121321123123211231131122211211131221131112311332211213211321223112111311222112132113213221123123211231132132211231131122211311123113322112111312211312111322212321121113122123211231131122113221123113221113122112132113213211121332212311322113212221","3113112221131112311332111213122112311311123112111331121113122112132113121113222112311311221112131221123113112221121113311211131122211211131221131211132221121321132132212321121113121112133221123113112221131112311332111213213211221113122113121113222112132113213221232112111312111213322112132113213221133112132123123112111311222112132113311213211221121332211231131122211311123113321112131221123113112221132231131122211211131221131112311332211213211321223112111311222112132113212221132221222112112322211211131221131211132221232112111312111213111213211231132132211211131221232112111312211213111213122112132113213221123113112221133112132123222112111312211312112213211231132132211211131221131211132221121311121312211213211312111322211213211321322113311213212322211231131122211311123113321112131221123113112211121312211213211321222113222112132113223113112221121113122113121113123112112322111213211322211312113211","132113213221133112132123123112111311222112132113311213211231232112311311222112111312211311123113322112132113212231121113112221121321132132211231232112311321322112311311222113111231133221121113122113121113221112131221123113111231121123222112132113213221133112132123123112111312111312212231131122211311123113322112111312211312111322111213122112311311123112112322211211131221131211132221232112111312111213111213211231132132211211131221232112111312212221121123222112132113213221133112132123123112111311222112132113213221132213211321322112311311222113311213212322211211131221131211221321123113213221121113122113121132211332113221122112133221123113112221131112311332111213122112311311123112111331121113122112132113121113222112311311221112131221123113112221121113311211131122211211131221131211132221121321132132212321121113121112133221123113112221131112212211131221121321131211132221123113112221131112311332211211133112111311222112111312211311123113322112111312211312111322212321121113121112133221121321132132211331121321231231121113112221121321132122311211131122211211131221131211322113322112111312211322132113213221123113112221131112311311121321122112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        return arr[n]
[sol2-JavaScript]
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var countAndSay = function(n) {
    const arr = [
        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    ];
    return arr[n];
};
[sol2-Golang]
1
2
3
4
5
var arr = []string{"", "1", "11", "21", "1211", "111221", "312211", "13112221", "1113213211", "31131211131221", "13211311123113112211", "11131221133112132113212221", "3113112221232112111312211312113211", "1321132132111213122112311311222113111221131221", "11131221131211131231121113112221121321132132211331222113112211", "311311222113111231131112132112311321322112111312211312111322212311322113212221", "132113213221133112132113311211131221121321131211132221123113112221131112311332111213211322211312113211", "11131221131211132221232112111312212321123113112221121113122113111231133221121321132132211331121321231231121113122113322113111221131221", "31131122211311123113321112131221123113112211121312211213211321322112311311222113311213212322211211131221131211132221232112111312111213111213211231131122212322211331222113112211", "1321132132211331121321231231121113112221121321132122311211131122211211131221131211132221121321132132212321121113121112133221123113112221131112311332111213122112311311123112111331121113122112132113213211121332212311322113212221", "11131221131211132221232112111312111213111213211231132132211211131221131211221321123113213221123113112221131112311332211211131221131211132211121312211231131112311211232221121321132132211331121321231231121113112221121321133112132112312321123113112221121113122113121113123112112322111213211322211312113211", "311311222113111231133211121312211231131112311211133112111312211213211312111322211231131122211311122122111312211213211312111322211213211321322113311213212322211231131122211311123113223112111311222112132113311213211221121332211211131221131211132221232112111312111213111213211231132132211211131221232112111312211213111213122112132113213221123113112221131112311311121321122112132231121113122113322113111221131221", 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func countAndSay(n int) string {
    return arr[n]
}

复杂度分析

  • 时间复杂度：$O(1)$。

  • 空间复杂度：$O(C \times M)$。其中 $C$ 是 $N$ 是上界，在本题中 $C = 30$，$M$ 为生成的字符串中的最大长度。

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0038-外观数列
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