add(coding/1415): the-k-th-lexicographical-string-of-all-happy-string…

…s-of-length-n

source/coding/1415-the-k-th-lexicographical-string-of-all-happy-strings-of-length-n.md

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+---
+title: 1415. The k-th Lexicographical String of All Happy Strings of Length n
+notebook: coding
+tags:
+- medium
+date: 2025-02-19 11:07:53
+updated: 2025-02-19 11:07:53
+---
+## Problem
+
+A **happy string** is a string that:
+
+- consists only of letters of the set `['a', 'b', 'c']`.
+- `s[i] != s[i + 1]` for all values of `i` from `1` to `s.length - 1` (string is 1-indexed).
+
+For example, strings **"abc", "ac", "b"** and **"abcbabcbcb"** are all happy strings and strings **"aa", "baa"** and **"ababbc"** are not happy strings.
+
+Given two integers `n` and `k`, consider a list of all happy strings of length `n` sorted in lexicographical order.
+
+Return _the kᵗʰ string_ of this list or return an **empty string** if there are less than `k` happy strings of length `n`.
+
+<https://leetcode.com/problems/the-k-th-lexicographical-string-of-all-happy-strings-of-length-n/>
+
+**Example 1:**
+
+> Input: `n = 1, k = 3`
+> Output: `"c"`
+> Explanation: The list `["a", "b", "c"]` contains all happy strings of length 1. The third string is `"c"`.
+
+**Example 2:**
+
+> Input: `n = 1, k = 4`
+> Output: `""`
+> Explanation: There are only 3 happy strings of length 1.
+
+**Example 3:**
+
+> Input: `n = 3, k = 9`
+> Output: `"cab"`
+> Explanation: There are 12 different happy string of length 3 `["aba", "abc", "aca", "acb", "bab", "bac", "bca", "bcb", "cab", "cac", "cba", "cbc"]`. You will find the 9ᵗʰ string = `"cab"`
+
+**Constraints:**
+
+- `1 <= n <= 10`
+- `1 <= k <= 100`
+
+## Test Cases
+
+``` python
+class Solution:
+    def getHappyString(self, n: int, k: int) -> str:
+```
+
+{% asset_code coding/1415-the-k-th-lexicographical-string-of-all-happy-strings-of-length-n/solution_test.py %}
+
+## Thoughts
+
+Happy string 除了第一位有 `'a'`、`'b'`、`'c'` 三个选择外，之后的每一位都只有两个选择——即与前一位不同的两个字母。如果用 `0` 表示这两个字母中较小的那个，用 `1` 表示较大的那个，显然 happy string 除去第一位之外，剩下的 `n - 1` 位对应了一个二进制数，一共有 `2ⁿ⁻¹` 个可能。而第一位共有三个可能，所以长度为 n 的 happy string 最多有 `3 * 2ⁿ⁻¹` 个。
+
+如果 `k > 3 * 2ⁿ⁻¹`，直接返回空字符串。
+
+否则用 `k - 1` 除以 `2ⁿ⁻¹`，商必然在 0 到 2 之间，分别对应 `'a'`、`'b'`、`'c'`；余数转成 `n - 1` 位的二进制表达，从最高位开始，根据前边对 `0` 和 `1` 的定义，就可以逐位构造出 happy string。显然得到的 happy string 就是所有长度为 n 的 happy strings 中按字典序第 k 大的。
+
+时间复杂度 `O(n)`，空间复杂度 `O(n)`。
+
+## Code
+
+{% asset_code coding/1415-the-k-th-lexicographical-string-of-all-happy-strings-of-length-n/solution.py %}

source/coding/1415-the-k-th-lexicographical-string-of-all-happy-strings-of-length-n/solution.py

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+class Solution:
+    def getHappyString(self, n: int, k: int) -> str:
+        base = 1 << (n - 1) # 2 ** (n - 1)
+        b, k = divmod(k - 1, base)
+        if b >= 3:
+            return ""
+
+        chars = 'abc'
+        candidates = ((1, 2), (0, 2), (0, 1))
+        d = b
+        parts = [chars[d]]
+        while base > 1:
+            base >>= 1
+            b, k = divmod(k, base)
+            d = candidates[d][b]
+            parts.append(chars[d])
+
+        return ''.join(parts)

source/coding/1415-the-k-th-lexicographical-string-of-all-happy-strings-of-length-n/solution_test.py

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+import pytest
+
+from solution import Solution
+
+
+@pytest.mark.parametrize('n, k, expected', [
+    (1, 3, "c"),
+    (1, 4, ""),
+    (3, 9, "cab"),
+])
+@pytest.mark.parametrize('sol', [Solution()])
+def test_solution(sol, n, k, expected):
+    assert sol.getHappyString(n, k) == expected