feat: Add Ruby code - chapter "Backtracking" (#1373)

* [feat] add ruby code - chapter backtracking

* feat: add ruby code block - chapter backtracking

codes/ruby/chapter_backtracking/n_queens.rb

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+=begin
+File: n_queens.rb
+Created Time: 2024-05-21
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 回溯算法：n 皇后 ###
+def backtrack(row, n, state, res, cols, diags1, diags2)
+  # 当放置完所有行时，记录解
+  if row == n
+    res << state.map { |row| row.dup }
+    return
+  end
+
+  # 遍历所有列
+  for col in 0...n
+    # 计算该格子对应的主对角线和次对角线
+    diag1 = row - col + n - 1
+    diag2 = row + col
+    # 剪枝：不允许该格子所在列、主对角线、次对角线上存在皇后
+    if !cols[col] && !diags1[diag1] && !diags2[diag2]
+      # 尝试：将皇后放置在该格子
+      state[row][col] = "Q"
+      cols[col] = diags1[diag1] = diags2[diag2] = true
+      # 放置下一行
+      backtrack(row + 1, n, state, res, cols, diags1, diags2)
+      # 回退：将该格子恢复为空位
+      state[row][col] = "#"
+      cols[col] = diags1[diag1] = diags2[diag2] = false
+    end
+  end
+end
+
+### 求解 n 皇后 ###
+def n_queens(n)
+  # 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位
+  state = Array.new(n) { Array.new(n, "#") }
+  cols = Array.new(n, false) # 记录列是否有皇后
+  diags1 = Array.new(2 * n - 1, false) # 记录主对角线上是否有皇后
+  diags2 = Array.new(2 * n - 1, false) # 记录次对角线上是否有皇后
+  res = []
+  backtrack(0, n, state, res, cols, diags1, diags2)
+
+  res
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 4
+  res = n_queens(n)
+
+  puts "输入棋盘长宽为 #{n}"
+  puts "皇后放置方案共有 #{res.length} 种"
+
+  for state in res
+    puts "--------------------"
+    for row in state
+      p row
+    end
+  end
+end

codes/ruby/chapter_backtracking/permutations_i.rb

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+=begin
+File: permutations_i.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 回溯算法：全排列 I ###
+def backtrack(state, choices, selected, res)
+  # 当状态长度等于元素数量时，记录解
+  if state.length == choices.length
+    res << state.dup
+    return
+  end
+
+  # 遍历所有选择
+  choices.each_with_index do |choice, i|
+    # 剪枝：不允许重复选择元素
+    unless selected[i]
+      # 尝试：做出选择，更新状态
+      selected[i] = true
+      state << choice
+      # 进行下一轮选择
+      backtrack(state, choices, selected, res)
+      # 回退：撤销选择，恢复到之前的状态
+      selected[i] = false
+      state.pop
+    end
+  end
+end
+
+### 全排列 I ###
+def permutations_i(nums)
+  res = []
+  backtrack([], nums, Array.new(nums.length, false), res)
+  res
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [1, 2, 3]
+
+  res = permutations_i(nums)
+
+  puts "输入数组 nums = #{nums}"
+  puts "所有排列 res = #{res}"
+end

codes/ruby/chapter_backtracking/permutations_ii.rb

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+=begin
+File: permutations_ii.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 回溯算法：全排列 II ###
+def backtrack(state, choices, selected, res)
+  # 当状态长度等于元素数量时，记录解
+  if state.length == choices.length
+    res << state.dup
+    return
+  end
+
+  # 遍历所有选择
+  duplicated = Set.new
+  choices.each_with_index do |choice, i|
+    # 剪枝：不允许重复选择元素 且 不允许重复选择相等元素
+    if !selected[i] && !duplicated.include?(choice)
+      # 尝试：做出选择，更新状态
+      duplicated.add(choice)
+      selected[i] = true
+      state << choice
+      # 进行下一轮选择
+      backtrack(state, choices, selected, res)
+      # 回退：撤销选择，恢复到之前的状态
+      selected[i] = false
+      state.pop
+    end
+  end
+end
+
+### 全排列 II ###
+def permutations_ii(nums)
+  res = []
+  backtrack([], nums, Array.new(nums.length, false), res)
+  res
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [1, 2, 2]
+
+  res = permutations_ii(nums)
+
+  puts "输入数组 nums = #{nums}"
+  puts "所有排列 res = #{res}"
+end

codes/ruby/chapter_backtracking/preorder_traversal_i_compact.rb

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+=begin
+File: preorder_traversal_i_compact.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+require_relative '../utils/tree_node'
+require_relative '../utils/print_util'
+
+### 前序遍历：例题一 ###
+def pre_order(root)
+  return unless root
+
+  # 记录解
+  $res << root if root.val == 7
+
+  pre_order(root.left)
+  pre_order(root.right)
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  root = arr_to_tree([1, 7, 3, 4, 5, 6, 7])
+  puts "\n初始化二叉树"
+  print_tree(root)
+
+  # 前序遍历
+  $res = []
+  pre_order(root)
+
+  puts "\n输出所有值为 7 的节点"
+  p $res.map { |node| node.val }
+end

codes/ruby/chapter_backtracking/preorder_traversal_ii_compact.rb

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+=begin
+File: preorder_traversal_ii_compact.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+require_relative '../utils/tree_node'
+require_relative '../utils/print_util'
+
+### 前序遍历：例题二 ###
+def pre_order(root)
+  return unless root
+
+  # 尝试
+  $path << root
+
+  # 记录解
+  $res << $path.dup if root.val == 7
+
+  pre_order(root.left)
+  pre_order(root.right)
+
+  # 回退
+  $path.pop
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  root = arr_to_tree([1, 7, 3, 4, 5, 6, 7])
+  puts "\n初始化二叉树"
+  print_tree(root)
+
+  # 前序遍历
+  $path, $res = [], []
+  pre_order(root)
+
+  puts "\n输出所有根节点到节点 7 的路径"
+  for path in $res
+    p path.map { |node| node.val }
+  end
+end

codes/ruby/chapter_backtracking/preorder_traversal_iii_compact.rb

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+=begin
+File: preorder_traversal_iii_compact.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+require_relative '../utils/tree_node'
+require_relative '../utils/print_util'
+
+### 前序遍历：例题三 ###
+def pre_order(root)
+  # 剪枝
+  return if !root || root.val == 3
+
+  # 尝试
+  $path.append(root)
+
+  # 记录解
+  $res << $path.dup if root.val == 7
+
+  pre_order(root.left)
+  pre_order(root.right)
+
+  # 回退
+  $path.pop
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  root = arr_to_tree([1, 7, 3, 4, 5, 6, 7])
+  puts "\n初始化二叉树"
+  print_tree(root)
+
+  # 前序遍历
+  $path, $res = [], []
+  pre_order(root)
+
+  puts "\n输出所有根节点到节点 7 的路径，路径中不包含值为 3 的节点"
+  for path in $res
+    p path.map { |node| node.val }
+  end
+end

codes/ruby/chapter_backtracking/preorder_traversal_iii_template.rb

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+=begin
+File: preorder_traversal_iii_template.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+require_relative '../utils/tree_node'
+require_relative '../utils/print_util'
+
+### 判断当前状态是否为解 ###
+def is_solution?(state)
+  !state.empty? && state.last.val == 7
+end
+
+### 记录解 ###
+def record_solution(state, res)
+  res << state.dup
+end
+
+### 判断在当前状态下，该选择是否合法 ###
+def is_valid?(state, choice)
+  choice && choice.val != 3
+end
+
+### 更新状态 ###
+def make_choice(state, choice)
+  state << choice
+end
+
+### 恢复状态 ###
+def undo_choice(state, choice)
+  state.pop
+end
+
+### 回溯算法：例题三 ###
+def backtrack(state, choices, res)
+  # 检查是否为解
+  record_solution(state, res) if is_solution?(state)
+
+  # 遍历所有选择
+  for choice in choices
+    # 剪枝：检查选择是否合法
+    if is_valid?(state, choice)
+      # 尝试：做出选择，更新状态
+      make_choice(state, choice)
+      # 进行下一轮选择
+      backtrack(state, [choice.left, choice.right], res)
+      # 回退：撤销选择，恢复到之前的状态
+      undo_choice(state, choice)
+    end
+  end
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  root = arr_to_tree([1, 7, 3, 4, 5, 6, 7])
+  puts "\n初始化二叉树"
+  print_tree(root)
+
+  # 回溯算法
+  res = []
+  backtrack([], [root], res)
+
+  puts "\n输出所有根节点到节点 7 的路径，要求路径中不包含值为 3 的节点"
+  for path in res
+    p path.map { |node| node.val }
+  end
+end

codes/ruby/chapter_backtracking/subset_sum_i.rb

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+=begin
+File: subset_sum_i.rb
+Created Time: 2024-05-22
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 回溯算法：子集和 I ###
+def backtrack(state, target, choices, start, res)
+  # 子集和等于 target 时，记录解
+  if target.zero?
+    res << state.dup
+    return
+  end
+  # 遍历所有选择
+  # 剪枝二：从 start 开始遍历，避免生成重复子集
+  for i in start...choices.length
+    # 剪枝一：若子集和超过 target ，则直接结束循环
+    # 这是因为数组已排序，后边元素更大，子集和一定超过 target
+    break if target - choices[i] < 0
+    # 尝试：做出选择，更新 target, start
+    state << choices[i]
+    # 进行下一轮选择
+    backtrack(state, target - choices[i], choices, i, res)
+    # 回退：撤销选择，恢复到之前的状态
+    state.pop
+  end
+end
+
+### 求解子集和 I ###
+def subset_sum_i(nums, target)
+  state = [] # 状态（子集）
+  nums.sort! # 对 nums 进行排序
+  start = 0 # 遍历起始点
+  res = [] # 结果列表（子集列表）
+  backtrack(state, target, nums, start, res)
+  res
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  nums = [3, 4, 5]
+  target = 9
+  res = subset_sum_i(nums, target)
+
+  puts "输入数组 = #{nums}, target = #{target}"
+  puts "所有和等于 #{target} 的子集 res = #{res}"
+end