From d61838a4ebfae1a214b03de21d09049a9fd5c62c Mon Sep 17 00:00:00 2001
From: krahets <krahets@163.com>
Date: Sat, 4 May 2024 19:28:48 +0800
Subject: [PATCH] Sync zh and zh-hant versions.

---
 .../codes/go/chapter_backtracking/n_queens.go |  1 +
 .../chapter_backtracking/permutations_ii.go   |  2 +-
 .../codes/ruby/chapter_sorting/bubble_sort.rb | 51 ++++++++++++++
 .../codes/ruby/chapter_sorting/bucket_sort.rb | 43 ++++++++++++
 .../ruby/chapter_sorting/counting_sort.rb     | 62 ++++++++++++++++
 .../codes/ruby/chapter_sorting/heap_sort.rb   | 45 ++++++++++++
 .../codes/ruby/chapter_sorting/merge_sort.rb  | 60 ++++++++++++++++
 .../codes/ruby/chapter_sorting/radix_sort.rb  | 70 +++++++++++++++++++
 .../ruby/chapter_sorting/selection_sort.rb    | 29 ++++++++
 .../chapter_backtracking/n_queens_problem.md  |  2 +-
 10 files changed, 363 insertions(+), 2 deletions(-)
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/bubble_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/bucket_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/counting_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/heap_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/merge_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/radix_sort.rb
 create mode 100644 zh-hant/codes/ruby/chapter_sorting/selection_sort.rb

diff --git a/zh-hant/codes/go/chapter_backtracking/n_queens.go b/zh-hant/codes/go/chapter_backtracking/n_queens.go
index 7d8fd5e82..516af5eec 100644
--- a/zh-hant/codes/go/chapter_backtracking/n_queens.go
+++ b/zh-hant/codes/go/chapter_backtracking/n_queens.go
@@ -15,6 +15,7 @@ func backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, d
 
 		}
 		*res = append(*res, newState)
+		return
 	}
 	// 走訪所有列
 	for col := 0; col < n; col++ {
diff --git a/zh-hant/codes/go/chapter_backtracking/permutations_ii.go b/zh-hant/codes/go/chapter_backtracking/permutations_ii.go
index 421824f44..15065625e 100644
--- a/zh-hant/codes/go/chapter_backtracking/permutations_ii.go
+++ b/zh-hant/codes/go/chapter_backtracking/permutations_ii.go
@@ -23,7 +23,7 @@ func backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {
 			(*selected)[i] = true
 			*state = append(*state, choice)
 			// 進行下一輪選擇
-			backtrackI(state, choices, selected, res)
+			backtrackII(state, choices, selected, res)
 			// 回退：撤銷選擇，恢復到之前的狀態
 			(*selected)[i] = false
 			*state = (*state)[:len(*state)-1]
diff --git a/zh-hant/codes/ruby/chapter_sorting/bubble_sort.rb b/zh-hant/codes/ruby/chapter_sorting/bubble_sort.rb
new file mode 100644
index 000000000..bdd52c751
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/bubble_sort.rb
@@ -0,0 +1,51 @@
+=begin
+File: bubble_sort.rb
+Created Time: 2024-05-02
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 泡沫排序 ###
+def bubble_sort(nums)
+  n = nums.length
+  # 外迴圈：未排序區間為 [0, i]
+  for i in (n - 1).downto(1)
+    # 內迴圈：將未排序區間 [0, i] 中的最大元素交換至該區間的最右端
+    for j in 0...i
+      if nums[j] > nums[j + 1]
+        # 交換 nums[j] 與 nums[j + 1]
+        nums[j], nums[j + 1] = nums[j + 1], nums[j]
+      end
+    end
+  end
+end
+
+### 泡沫排序（標誌最佳化）###
+def bubble_sort_with_flag(nums)
+  n = nums.length
+  # 外迴圈：未排序區間為 [0, i]
+  for i in (n - 1).downto(1)
+    flag = false # 初始化標誌位
+
+    # 內迴圈：將未排序區間 [0, i] 中的最大元素交換至該區間的最右端
+    for j in 0...i
+      if nums[j] > nums[j + 1]
+        # 交換 nums[j] 與 nums[j + 1]
+        nums[j], nums[j + 1] = nums[j + 1], nums[j]
+        flag = true # 記錄交換元素
+      end
+    end
+
+    break unless flag # 此輪“冒泡”未交換任何元素，直接跳出
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [4, 1, 3, 1, 5, 2]
+  bubble_sort(nums)
+  puts "泡沫排序完成後 nums = #{nums}"
+
+  nums1 = [4, 1, 3, 1, 5, 2]
+  bubble_sort_with_flag(nums1)
+  puts "泡沫排序完成後 nums = #{nums1}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/bucket_sort.rb b/zh-hant/codes/ruby/chapter_sorting/bucket_sort.rb
new file mode 100644
index 000000000..5ce98050b
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/bucket_sort.rb
@@ -0,0 +1,43 @@
+=begin
+File: bucket_sort.rb
+Created Time: 2024-04-17
+Author: Martin Xu (martin.xus@gmail.com)
+=end
+
+### 桶排序 ###
+def bucket_sort(nums)
+  # 初始化 k = n/2 個桶，預期向每個桶分配 2 個元素
+  k = nums.length / 2
+  buckets = Array.new(k) { [] }
+  
+  # 1. 將陣列元素分配到各個桶中
+  nums.each do |num|
+    # 輸入資料範圍為 [0, 1)，使用 num * k 對映到索引範圍 [0, k-1]
+    i = (num * k).to_i
+    # 將 num 新增進桶 i
+    buckets[i] << num
+  end
+
+  # 2. 對各個桶執行排序
+  buckets.each do |bucket|
+    # 使用內建排序函式，也可以替換成其他排序演算法
+    bucket.sort!
+  end
+
+  # 3. 走訪桶合併結果
+  i = 0
+  buckets.each do |bucket|
+    bucket.each do |num|
+      nums[i] = num
+      i += 1
+    end
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  # 設輸入資料為浮點數，範圍為 [0, 1)
+  nums = [0.49, 0.96, 0.82, 0.09, 0.57, 0.43, 0.91, 0.75, 0.15, 0.37]
+  bucket_sort(nums)
+  puts "桶排序完成後 nums = #{nums}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/counting_sort.rb b/zh-hant/codes/ruby/chapter_sorting/counting_sort.rb
new file mode 100644
index 000000000..b0cae6bd9
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/counting_sort.rb
@@ -0,0 +1,62 @@
+=begin
+File: counting_sort.rb
+Created Time: 2024-05-02
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 計數排序 ###
+def counting_sort_naive(nums)
+  # 簡單實現，無法用於排序物件
+  # 1. 統計陣列最大元素 m
+  m = 0
+  nums.each { |num| m = [m, num].max }
+  # 2. 統計各數字的出現次數
+  # counter[num] 代表 num 的出現次數
+  counter = Array.new(m + 1, 0)
+  nums.each { |num| counter[num] += 1 }
+  # 3. 走訪 counter ，將各元素填入原陣列 nums
+  i = 0
+  for num in 0...(m + 1)
+    (0...counter[num]).each do
+      nums[i] = num
+      i += 1
+    end
+  end
+end
+
+### 計數排序 ###
+def counting_sort(nums)
+  # 完整實現，可排序物件，並且是穩定排序
+  # 1. 統計陣列最大元素 m
+  m = nums.max
+  # 2. 統計各數字的出現次數
+  # counter[num] 代表 num 的出現次數
+  counter = Array.new(m + 1, 0)
+  nums.each { |num| counter[num] += 1 }
+  # 3. 求 counter 的前綴和，將“出現次數”轉換為“尾索引”
+  # 即 counter[num]-1 是 num 在 res 中最後一次出現的索引
+  (0...m).each { |i| counter[i + 1] += counter[i] }
+  # 4. 倒序走訪 nums, 將各元素填入結果陣列 res
+  # 初始化陣列 res 用於記錄結果
+  n = nums.length
+  res = Array.new(n, 0)
+  (n - 1).downto(0).each do |i|
+    num = nums[i]
+    res[counter[num] - 1] = num # 將 num 放置到對應索引處
+    counter[num] -= 1 # 令前綴和自減 1 ，得到下次放置 num 的索引
+  end
+  # 使用結果陣列 res 覆蓋原陣列 nums
+  (0...n).each { |i| nums[i] = res[i] }
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [1, 0, 1, 2, 0, 4, 0, 2, 2, 4]
+
+  counting_sort_naive(nums)
+  puts "計數排序（無法排序物件）完成後 nums = #{nums}"
+
+  nums1 = [1, 0, 1, 2, 0, 4, 0, 2, 2, 4]
+  counting_sort(nums1)
+  puts "計數排序完成後 nums1 = #{nums1}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/heap_sort.rb b/zh-hant/codes/ruby/chapter_sorting/heap_sort.rb
new file mode 100644
index 000000000..d562ba9b4
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/heap_sort.rb
@@ -0,0 +1,45 @@
+=begin
+File: heap_sort.rb
+Created Time: 2024-04-10
+Author: junminhong (junminhong1110@gmail.com)
+=end
+
+### 堆積的長度為 n ，從節點 i 開始，從頂至底堆積化 ###
+def sift_down(nums, n, i)
+  while true
+    # 判斷節點 i, l, r 中值最大的節點，記為 ma
+    l = 2 * i + 1
+    r = 2 * i + 2
+    ma = i
+    ma = l if l < n && nums[l] > nums[ma]
+    ma = r if r < n && nums[r] > nums[ma]
+    # 若節點 i 最大或索引 l, r 越界，則無須繼續堆積化，跳出
+    break if ma == i
+    # 交換兩節點
+    nums[i], nums[ma] = nums[ma], nums[i]
+    # 迴圈向下堆積化
+    i = ma
+  end
+end
+
+### 堆積排序 ###
+def heap_sort(nums)
+  # 建堆積操作：堆積化除葉節點以外的其他所有節點
+  (nums.length / 2 - 1).downto(0) do |i|
+    sift_down(nums, nums.length, i)
+  end
+  # 從堆積中提取最大元素，迴圈 n-1 輪
+  (nums.length - 1).downto(1) do |i|
+    # 交換根節點與最右葉節點（交換首元素與尾元素）
+    nums[0], nums[i] = nums[i], nums[0]
+    # 以根節點為起點，從頂至底進行堆積化
+    sift_down(nums, i, 0)
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [4, 1, 3, 1, 5, 2]
+  heap_sort(nums)
+  puts "堆積排序完成後 nums = #{nums.inspect}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/merge_sort.rb b/zh-hant/codes/ruby/chapter_sorting/merge_sort.rb
new file mode 100644
index 000000000..317fbe8ab
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/merge_sort.rb
@@ -0,0 +1,60 @@
+=begin
+File: merge_sort.rb
+Created Time: 2024-04-10
+Author: junminhong (junminhong1110@gmail.com)
+=end
+
+### 合併左子陣列和右子陣列 ###
+def merge(nums, left, mid, right)
+  # 左子陣列區間為 [left, mid], 右子陣列區間為 [mid+1, right]
+  # 建立一個臨時陣列 tmp，用於存放合併後的結果
+  tmp = Array.new(right - left + 1, 0)
+  # 初始化左子陣列和右子陣列的起始索引
+  i, j, k = left, mid + 1, 0
+  # 當左右子陣列都還有元素時，進行比較並將較小的元素複製到臨時陣列中
+  while i <= mid && j <= right
+    if nums[i] <= nums[j]
+      tmp[k] = nums[i]
+      i += 1
+    else
+      tmp[k] = nums[j]
+      j += 1
+    end
+    k += 1
+  end
+  # 將左子陣列和右子陣列的剩餘元素複製到臨時陣列中
+  while i <= mid
+    tmp[k] = nums[i]
+    i += 1
+    k += 1
+  end
+  while j <= right
+    tmp[k] = nums[j]
+    j += 1
+    k += 1
+  end
+  # 將臨時陣列 tmp 中的元素複製回原陣列 nums 的對應區間
+  (0...tmp.length).each do |k|
+    nums[left + k] = tmp[k]
+  end
+end
+
+### 合併排序 ###
+def merge_sort(nums, left, right)
+  # 終止條件
+  # 當子陣列長度為 1 時終止遞迴
+  return if left >= right
+  # 劃分階段
+  mid = (left + right) / 2 # 計算中點
+  merge_sort(nums, left, mid) # 遞迴左子陣列
+  merge_sort(nums, mid + 1, right) # 遞迴右子陣列
+  # 合併階段
+  merge(nums, left, mid, right)
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [7, 3, 2, 6, 0, 1, 5, 4]
+  merge_sort(nums, 0, nums.length - 1)
+  puts "合併排序完成後 nums = #{nums.inspect}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/radix_sort.rb b/zh-hant/codes/ruby/chapter_sorting/radix_sort.rb
new file mode 100644
index 000000000..b2b5350e5
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/radix_sort.rb
@@ -0,0 +1,70 @@
+=begin
+File: radix_sort.rb
+Created Time: 2024-05-03
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 獲取元素 num 的第 k 位，其中 exp = 10^(k-1) ###
+def digit(num, exp)
+  # 轉入 exp 而非 k 可以避免在此重複執行昂貴的次方計算
+  (num / exp) % 10
+end
+
+### 計數排序（根據 nums 第 k 位排序）###
+def counting_sort_digit(nums, exp)
+  # 十進位制的位範圍為 0~9 ，因此需要長度為 10 的桶陣列
+  counter = Array.new(10, 0)
+  n = nums.length
+  # 統計 0~9 各數字的出現次數
+  for i in 0...n
+    d = digit(nums[i], exp) # 獲取 nums[i] 第 k 位，記為 d
+    counter[d] += 1 # 統計數字 d 的出現次數
+  end
+  # 求前綴和，將“出現個數”轉換為“陣列索引”
+  (1...10).each { |i| counter[i] += counter[i - 1] }
+  # 倒序走訪，根據桶內統計結果，將各元素填入 res
+  res = Array.new(n, 0)
+  for i in (n - 1).downto(0)
+    d = digit(nums[i], exp)
+    j = counter[d] - 1 # 獲取 d 在陣列中的索引 j
+    res[j] = nums[i] # 將當前元素填入索引 j
+    counter[d] -= 1 # 將 d 的數量減 1
+  end
+  # 使用結果覆蓋原陣列 nums
+  (0...n).each { |i| nums[i] = res[i] }
+end
+
+### 基數排序 ###
+def radix_sort(nums)
+  # 獲取陣列的最大元素，用於判斷最大位數
+  m = nums.max
+  # 按照從低位到高位的順序走訪
+  exp = 1
+  while exp <= m
+    # 對陣列元素的第 k 位執行計數排序
+    # k = 1 -> exp = 1
+    # k = 2 -> exp = 10
+    # 即 exp = 10^(k-1)
+    counting_sort_digit(nums, exp)
+    exp *= 10
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  # 基數排序
+  nums = [
+    10546151,
+    35663510,
+    42865989,
+    34862445,
+    81883077,
+    88906420,
+    72429244,
+    30524779,
+    82060337,
+    63832996,
+  ]
+  radix_sort(nums)
+  puts "基數排序完成後 nums = #{nums}"
+end
diff --git a/zh-hant/codes/ruby/chapter_sorting/selection_sort.rb b/zh-hant/codes/ruby/chapter_sorting/selection_sort.rb
new file mode 100644
index 000000000..48a2a9fff
--- /dev/null
+++ b/zh-hant/codes/ruby/chapter_sorting/selection_sort.rb
@@ -0,0 +1,29 @@
+=begin
+File: selection_sort.rb
+Created Time: 2024-05-03
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 選擇排序 ###
+def selection_sort(nums)
+  n = nums.length
+  # 外迴圈：未排序區間為 [i, n-1]
+  for i in 0...(n - 1)
+    # 內迴圈：找到未排序區間內的最小元素
+    k = i
+    for j in (i + 1)...n
+      if nums[j] < nums[k]
+        k = j # 記錄最小元素的索引
+      end
+    end
+    # 將該最小元素與未排序區間的首個元素交換
+    nums[i], nums[k] = nums[k], nums[i]
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [4, 1, 3, 1, 5, 2]
+  selection_sort(nums)
+  puts "選擇排序完成後 nums = #{nums}"
+end
diff --git a/zh-hant/docs/chapter_backtracking/n_queens_problem.md b/zh-hant/docs/chapter_backtracking/n_queens_problem.md
index 87cd69b2c..43196a3e4 100644
--- a/zh-hant/docs/chapter_backtracking/n_queens_problem.md
+++ b/zh-hant/docs/chapter_backtracking/n_queens_problem.md
@@ -18,7 +18,7 @@
 
 也就是說，我們可以採取逐行放置策略：從第一行開始，在每行放置一個皇后，直至最後一行結束。
 
-下圖所示為 $4$ 皇后問題的逐行放置過程。受畫幅限制，下圖僅展開了第一行的其中一個搜尋分支，並且將不滿足列約束和對角線約束的方案都進行了剪枝。
+下圖所示為 4 皇后問題的逐行放置過程。受畫幅限制，下圖僅展開了第一行的其中一個搜尋分支，並且將不滿足列約束和對角線約束的方案都進行了剪枝。
 
 ![逐行放置策略](n_queens_problem.assets/n_queens_placing.png)