Translate all code to English (#1836)

* Review the EN heading format.

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* bug fixes

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* Revisit the CN-to-EN translation for Python code using Claude-4.5

* Revisit the CN-to-EN translation for Java code using Claude-4.5

* Revisit the CN-to-EN translation for Cpp code using Claude-4.5.

* Fix the dictionary.

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* Translate Go code to English.

* Update workflows to test EN code.

* Add EN translation for C.

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* Trigger the CI check.

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* Add the EN version of Dart code.

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* Fix the workflows.

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* Update the CI check for the English version  code.

* Update Python CI check.

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* Fix Ruby comments

en/codes/ruby/chapter_dynamic_programming/climbing_stairs_backtrack.rb

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+=begin
+File: climbing_stairs_backtrack.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Backtracking ###
+def backtrack(choices, state, n, res)
+  # When climbing to the n-th stair, add 1 to the solution count
+  res[0] += 1 if state == n
+  # Traverse all choices
+  for choice in choices
+    # Pruning: not allowed to go beyond the n-th stair
+    next if state + choice > n
+
+    # Attempt: make choice, update state
+    backtrack(choices, state + choice, n, res)
+  end
+  # Backtrack
+end
+
+### Climbing stairs: backtracking ###
+def climbing_stairs_backtrack(n)
+  choices = [1, 2] # Can choose to climb up 1 or 2 stairs
+  state = 0 # Start climbing from the 0-th stair
+  res = [0] # Use res[0] to record the solution count
+  backtrack(choices, state, n, res)
+  res.first
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 9
+
+  res = climbing_stairs_backtrack(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+end

en/codes/ruby/chapter_dynamic_programming/climbing_stairs_constraint_dp.rb

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+=begin
+File: climbing_stairs_constraint_dp.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Climbing stairs with constraint: DP ###
+def climbing_stairs_constraint_dp(n)
+  return 1 if n == 1 || n == 2
+
+  # Initialize dp table, used to store solutions to subproblems
+  dp = Array.new(n + 1) { Array.new(3, 0) }
+  # Initial state: preset the solution to the smallest subproblem
+  dp[1][1], dp[1][2] = 1, 0
+  dp[2][1], dp[2][2] = 0, 1
+  # State transition: gradually solve larger subproblems from smaller ones
+  for i in 3...(n + 1)
+    dp[i][1] = dp[i - 1][2]
+    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]
+  end
+
+  dp[n][1] + dp[n][2]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 9
+
+  res = climbing_stairs_constraint_dp(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+end

en/codes/ruby/chapter_dynamic_programming/climbing_stairs_dfs.rb

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+=begin
+File: climbing_stairs_dfs.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Search ###
+def dfs(i)
+  # Known dp[1] and dp[2], return them
+  return i if i == 1 || i == 2
+  # dp[i] = dp[i-1] + dp[i-2]
+  dfs(i - 1) + dfs(i - 2)
+end
+
+### Climbing stairs: search ###
+def climbing_stairs_dfs(n)
+  dfs(n)
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 9
+
+  res = climbing_stairs_dfs(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+end

en/codes/ruby/chapter_dynamic_programming/climbing_stairs_dfs_mem.rb

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+=begin
+File: climbing_stairs_dfs_mem.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Memoization search ###
+def dfs(i, mem)
+  # Known dp[1] and dp[2], return them
+  return i if i == 1 || i == 2
+  # If record dp[i] exists, return it directly
+  return mem[i] if mem[i] != -1
+
+  # dp[i] = dp[i-1] + dp[i-2]
+  count = dfs(i - 1, mem) + dfs(i - 2, mem)
+  # Record dp[i]
+  mem[i] = count
+end
+
+### Climbing stairs: memoization search ###
+def climbing_stairs_dfs_mem(n)
+  # mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record
+  mem = Array.new(n + 1, -1)
+  dfs(n, mem)
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 9
+
+  res = climbing_stairs_dfs_mem(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+end

en/codes/ruby/chapter_dynamic_programming/climbing_stairs_dp.rb

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+=begin
+File: climbing_stairs_dp.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Climbing stairs: dynamic programming ###
+def climbing_stairs_dp(n)
+  return n  if n == 1 || n == 2
+
+  # Initialize dp table, used to store solutions to subproblems
+  dp = Array.new(n + 1, 0)
+  # Initial state: preset the solution to the smallest subproblem
+  dp[1], dp[2] = 1, 2
+  # State transition: gradually solve larger subproblems from smaller ones
+  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }
+
+  dp[n]
+end
+
+### Climbing stairs: space-optimized DP ###
+def climbing_stairs_dp_comp(n)
+  return n if n == 1 || n == 2
+
+  a, b = 1, 2
+  (3...(n + 1)).each { a, b = b, a + b }
+
+  b
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  n = 9
+
+  res = climbing_stairs_dp(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+
+  res = climbing_stairs_dp_comp(n)
+  puts "Climbing #{n} stairs has #{res} solutions"
+end

en/codes/ruby/chapter_dynamic_programming/coin_change.rb

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+=begin
+File: coin_change.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Coin change: dynamic programming ###
+def coin_change_dp(coins, amt)
+  n = coins.length
+  _MAX = amt + 1
+  # Initialize dp table
+  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }
+  # State transition: first row and first column
+  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }
+  # State transition: rest of the rows and columns
+  for i in 1...(n + 1)
+    for a in 1...(amt + 1)
+      if coins[i - 1] > a
+        # If exceeds target amount, don't select coin i
+        dp[i][a] = dp[i - 1][a]
+      else
+        # The smaller value between not selecting and selecting coin i
+        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min
+      end
+    end
+  end
+  dp[n][amt] != _MAX ? dp[n][amt] : -1
+end
+
+### Coin change: space-optimized DP ###
+def coin_change_dp_comp(coins, amt)
+  n = coins.length
+  _MAX = amt + 1
+  # Initialize dp table
+  dp = Array.new(amt + 1, _MAX)
+  dp[0] = 0
+  # State transition
+  for i in 1...(n + 1)
+    # Traverse in forward order
+    for a in 1...(amt + 1)
+      if coins[i - 1] > a
+        # If exceeds target amount, don't select coin i
+        dp[a] = dp[a]
+      else
+        # The smaller value between not selecting and selecting coin i
+        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min
+      end
+    end
+  end
+  dp[amt] != _MAX ? dp[amt] : -1
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  coins = [1, 2, 5]
+  amt = 4
+
+  # Dynamic programming
+  res = coin_change_dp(coins, amt)
+  puts "Minimum coins needed to make target amount is #{res}"
+
+  # Space-optimized dynamic programming
+  res = coin_change_dp_comp(coins, amt)
+  puts "Minimum coins needed to make target amount is #{res}"
+end

en/codes/ruby/chapter_dynamic_programming/coin_change_ii.rb

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+=begin
+File: coin_change_ii.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Coin change II: dynamic programming ###
+def coin_change_ii_dp(coins, amt)
+  n = coins.length
+  # Initialize dp table
+  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }
+  # Initialize first column
+  (0...(n + 1)).each { |i| dp[i][0] = 1 }
+  # State transition
+  for i in 1...(n + 1)
+    for a in 1...(amt + 1)
+      if coins[i - 1] > a
+        # If exceeds target amount, don't select coin i
+        dp[i][a] = dp[i - 1][a]
+      else
+        # Sum of the two options: not selecting and selecting coin i
+        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
+      end
+    end
+  end
+  dp[n][amt]
+end
+
+### Coin change II: space-optimized DP ###
+def coin_change_ii_dp_comp(coins, amt)
+  n = coins.length
+  # Initialize dp table
+  dp = Array.new(amt + 1, 0)
+  dp[0] = 1
+  # State transition
+  for i in 1...(n + 1)
+    # Traverse in forward order
+    for a in 1...(amt + 1)
+      if coins[i - 1] > a
+        # If exceeds target amount, don't select coin i
+        dp[a] = dp[a]
+      else
+        # Sum of the two options: not selecting and selecting coin i
+        dp[a] = dp[a] + dp[a - coins[i - 1]]
+      end
+    end
+  end
+  dp[amt]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  coins = [1, 2, 5]
+  amt = 5
+
+  # Dynamic programming
+  res = coin_change_ii_dp(coins, amt)
+  puts "Number of coin combinations to make target amount is #{res}"
+
+  # Space-optimized dynamic programming
+  res = coin_change_ii_dp_comp(coins, amt)
+  puts "Number of coin combinations to make target amount is #{res}"
+end

en/codes/ruby/chapter_dynamic_programming/edit_distance.rb

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+=begin
+File: edit_distance.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Edit distance: brute force search ###
+def edit_distance_dfs(s, t, i, j)
+  # If both s and t are empty, return 0
+  return 0 if i == 0 && j == 0
+  # If s is empty, return length of t
+  return j if i == 0
+  # If t is empty, return length of s
+  return i if j == 0
+  # If two characters are equal, skip both characters
+  return edit_distance_dfs(s, t, i - 1, j - 1) if s[i - 1] == t[j - 1]
+  # Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+  insert = edit_distance_dfs(s, t, i, j - 1)
+  delete = edit_distance_dfs(s, t, i - 1, j)
+  replace = edit_distance_dfs(s, t, i - 1, j - 1)
+  # Return minimum edit steps
+  [insert, delete, replace].min + 1
+end
+
+def edit_distance_dfs_mem(s, t, mem, i, j)
+  # If both s and t are empty, return 0
+  return 0 if i == 0 && j == 0
+  # If s is empty, return length of t
+  return j if i == 0
+  # If t is empty, return length of s
+  return i if j == 0
+  # If there's a record, return it directly
+  return mem[i][j] if mem[i][j] != -1
+  # If two characters are equal, skip both characters
+  return edit_distance_dfs_mem(s, t, mem, i - 1, j - 1) if s[i - 1] == t[j - 1]
+  # Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+  insert = edit_distance_dfs_mem(s, t, mem, i, j - 1)
+  delete = edit_distance_dfs_mem(s, t, mem, i - 1, j)
+  replace = edit_distance_dfs_mem(s, t, mem, i - 1, j - 1)
+  # Record and return minimum edit steps
+  mem[i][j] = [insert, delete, replace].min + 1
+end
+
+### Edit distance: dynamic programming ###
+def edit_distance_dp(s, t)
+  n, m = s.length, t.length
+  dp = Array.new(n + 1) { Array.new(m + 1, 0) }
+  # State transition: first row and first column
+  (1...(n + 1)).each { |i| dp[i][0] = i }
+  (1...(m + 1)).each { |j| dp[0][j] = j }
+  # State transition: rest of the rows and columns
+  for i in 1...(n + 1)
+    for j in 1...(m +1)
+      if s[i - 1] == t[j - 1]
+        # If two characters are equal, skip both characters
+        dp[i][j] = dp[i - 1][j - 1]
+      else
+        # Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1
+      end
+    end
+  end
+  dp[n][m]
+end
+
+### Edit distance: space-optimized DP ###
+def edit_distance_dp_comp(s, t)
+  n, m = s.length, t.length
+  dp = Array.new(m + 1, 0)
+  # State transition: first row
+  (1...(m + 1)).each { |j| dp[j] = j }
+  # State transition: rest of the rows
+  for i in 1...(n + 1)
+    # State transition: first column
+    leftup = dp.first # Temporarily store dp[i-1, j-1]
+    dp[0] += 1
+    # State transition: rest of the columns
+    for j in 1...(m + 1)
+      temp = dp[j]
+      if s[i - 1] == t[j - 1]
+        # If two characters are equal, skip both characters
+        dp[j] = leftup
+      else
+        # Minimum edit steps = minimum edit steps of insert, delete, replace + 1
+        dp[j] = [dp[j - 1], dp[j], leftup].min + 1
+      end
+      leftup = temp # Update for next round's dp[i-1, j-1]
+    end
+  end
+  dp[m]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  s = 'bag'
+  t = 'pack'
+  n, m = s.length, t.length
+
+  # Brute-force search
+  res = edit_distance_dfs(s, t, n, m)
+  puts "Changing #{s} to #{t} requires minimum #{res} edits"
+
+  # Memoization search
+  mem = Array.new(n + 1) { Array.new(m + 1, -1) }
+  res = edit_distance_dfs_mem(s, t, mem, n, m)
+  puts "Changing #{s} to #{t} requires minimum #{res} edits"
+
+  # Dynamic programming
+  res = edit_distance_dp(s, t)
+  puts "Changing #{s} to #{t} requires minimum #{res} edits"
+
+  # Space-optimized dynamic programming
+  res = edit_distance_dp_comp(s, t)
+  puts "Changing #{s} to #{t} requires minimum #{res} edits"
+end

en/codes/ruby/chapter_dynamic_programming/knapsack.rb

@@ -0,0 +1,99 @@
+=begin
+File: knapsack.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### 0-1 knapsack: brute force search ###
+def knapsack_dfs(wgt, val, i, c)
+  # If all items have been selected or knapsack has no remaining capacity, return value 0
+  return 0 if i == 0 || c == 0
+  # If exceeds knapsack capacity, can only choose not to put it in
+  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] > c
+  # Calculate the maximum value of not putting in and putting in item i
+  no = knapsack_dfs(wgt, val, i - 1, c)
+  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]
+  # Return the larger value of the two options
+  [no, yes].max
+end
+
+### 0-1 knapsack: memoization search ###
+def knapsack_dfs_mem(wgt, val, mem, i, c)
+  # If all items have been selected or knapsack has no remaining capacity, return value 0
+  return 0 if i == 0 || c == 0
+  # If there's a record, return it directly
+  return mem[i][c] if mem[i][c] != -1
+  # If exceeds knapsack capacity, can only choose not to put it in
+  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] > c
+  # Calculate the maximum value of not putting in and putting in item i
+  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)
+  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]
+  # Record and return the larger value of the two options
+  mem[i][c] = [no, yes].max
+end
+
+### 0-1 knapsack: dynamic programming ###
+def knapsack_dp(wgt, val, cap)
+  n = wgt.length
+  # Initialize dp table
+  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }
+  # State transition
+  for i in 1...(n + 1)
+    for c in 1...(cap + 1)
+      if wgt[i - 1] > c
+        # If exceeds knapsack capacity, don't select item i
+        dp[i][c] = dp[i - 1][c]
+      else
+        # The larger value between not selecting and selecting item i
+        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max
+      end
+    end
+  end
+  dp[n][cap]
+end
+
+### 0-1 knapsack: space-optimized DP ###
+def knapsack_dp_comp(wgt, val, cap)
+  n = wgt.length
+  # Initialize dp table
+  dp = Array.new(cap + 1, 0)
+  # State transition
+  for i in 1...(n + 1)
+    # Traverse in reverse order
+    for c in cap.downto(1)
+      if wgt[i - 1] > c
+        # If exceeds knapsack capacity, don't select item i
+        dp[c] = dp[c]
+      else
+        # The larger value between not selecting and selecting item i
+        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max
+      end
+    end
+  end
+  dp[cap]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  wgt = [10, 20, 30, 40, 50]
+  val = [50, 120, 150, 210, 240]
+  cap = 50
+  n = wgt.length
+
+  # Brute-force search
+  res = knapsack_dfs(wgt, val, n, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+
+  # Memoization search
+  mem = Array.new(n + 1) { Array.new(cap + 1, -1) }
+  res = knapsack_dfs_mem(wgt, val, mem, n, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+
+  # Dynamic programming
+  res = knapsack_dp(wgt, val, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+
+  # Space-optimized dynamic programming
+  res = knapsack_dp_comp(wgt, val, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+end

en/codes/ruby/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rb

@@ -0,0 +1,39 @@
+=begin
+File: min_cost_climbing_stairs_dp.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Minimum cost climbing stairs: DP ###
+def min_cost_climbing_stairs_dp(cost)
+  n = cost.length - 1
+  return cost[n] if n == 1 || n == 2
+  # Initialize dp table, used to store solutions to subproblems
+  dp = Array.new(n + 1, 0)
+  # Initial state: preset the solution to the smallest subproblem
+  dp[1], dp[2] = cost[1], cost[2]
+  # State transition: gradually solve larger subproblems from smaller ones
+  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }
+  dp[n]
+end
+
+# Minimum cost climbing stairs: Space-optimized dynamic programming
+def min_cost_climbing_stairs_dp_comp(cost)
+  n = cost.length - 1
+  return cost[n] if n == 1 || n == 2
+  a, b = cost[1], cost[2]
+  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }
+  b
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1]
+  puts "Input stair cost list is #{cost}"
+
+  res = min_cost_climbing_stairs_dp(cost)
+  puts "Minimum cost to climb stairs is #{res}"
+
+  res = min_cost_climbing_stairs_dp_comp(cost)
+  puts "Minimum cost to climb stairs is #{res}"
+end

en/codes/ruby/chapter_dynamic_programming/min_path_sum.rb

@@ -0,0 +1,93 @@
+=begin
+File: min_path_sum.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Minimum path sum: brute force search ###
+def min_path_sum_dfs(grid, i, j)
+  # If it's the top-left cell, terminate the search
+  return grid[i][j] if i == 0 && j == 0
+  # If row or column index is out of bounds, return +∞ cost
+  return Float::INFINITY if i < 0 || j < 0
+  # Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)
+  up = min_path_sum_dfs(grid, i - 1, j)
+  left = min_path_sum_dfs(grid, i, j - 1)
+  # Return the minimum path cost from top-left to (i, j)
+  [left, up].min + grid[i][j]
+end
+
+### Minimum path sum: memoization search ###
+def min_path_sum_dfs_mem(grid, mem, i, j)
+  # If it's the top-left cell, terminate the search
+  return grid[0][0] if i == 0 && j == 0
+  # If row or column index is out of bounds, return +∞ cost
+  return Float::INFINITY if i < 0 || j < 0
+  # If there's a record, return it directly
+  return mem[i][j] if mem[i][j] != -1
+  # Minimum path cost for left and upper cells
+  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)
+  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)
+  # Record and return the minimum path cost from top-left to (i, j)
+  mem[i][j] = [left, up].min + grid[i][j]
+end
+
+### Minimum path sum: dynamic programming ###
+def min_path_sum_dp(grid)
+  n, m = grid.length, grid.first.length
+  # Initialize dp table
+  dp = Array.new(n) { Array.new(m, 0) }
+  dp[0][0] = grid[0][0]
+  # State transition: first row
+  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }
+  # State transition: first column
+  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }
+  # State transition: rest of the rows and columns
+  for i in 1...n
+    for j in 1...m
+      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]
+    end
+  end
+  dp[n -1][m -1]
+end
+
+### Minimum path sum: space-optimized DP ###
+def min_path_sum_dp_comp(grid)
+  n, m = grid.length, grid.first.length
+  # Initialize dp table
+  dp = Array.new(m, 0)
+  # State transition: first row
+  dp[0] = grid[0][0]
+  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }
+  # State transition: rest of the rows
+  for i in 1...n
+    # State transition: first column
+    dp[0] = dp[0] + grid[i][0]
+    # State transition: rest of the columns
+    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }
+  end
+  dp[m - 1]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  grid = [[1, 3, 1, 5], [2, 2, 4, 2], [5, 3, 2, 1], [4, 3, 5, 2]]
+  n, m = grid.length, grid.first.length
+
+  # Brute-force search
+  res = min_path_sum_dfs(grid, n - 1, m - 1)
+  puts "Minimum path sum from top-left to bottom-right is #{res}"
+
+  # Memoization search
+  mem = Array.new(n) { Array.new(m, - 1) }
+  res = min_path_sum_dfs_mem(grid, mem, n - 1, m -1)
+  puts "Minimum path sum from top-left to bottom-right is #{res}"
+
+  # Dynamic programming
+  res = min_path_sum_dp(grid)
+  puts "Minimum path sum from top-left to bottom-right is #{res}"
+
+  # Space-optimized dynamic programming
+  res = min_path_sum_dp_comp(grid)
+  puts "Minimum path sum from top-left to bottom-right is #{res}"
+end

en/codes/ruby/chapter_dynamic_programming/unbounded_knapsack.rb

@@ -0,0 +1,61 @@
+=begin
+File: unbounded_knapsack.rb
+Created Time: 2024-05-29
+Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
+=end
+
+### Unbounded knapsack: dynamic programming ###
+def unbounded_knapsack_dp(wgt, val, cap)
+  n = wgt.length
+  # Initialize dp table
+  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }
+  # State transition
+  for i in 1...(n + 1)
+    for c in 1...(cap + 1)
+      if wgt[i - 1] > c
+        # If exceeds knapsack capacity, don't select item i
+        dp[i][c] = dp[i - 1][c]
+      else
+        # The larger value between not selecting and selecting item i
+        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max
+      end
+    end
+  end
+  dp[n][cap]
+end
+
+### Unbounded knapsack: space-optimized DP ###
+def unbounded_knapsack_dp_comp(wgt, val, cap)
+  n = wgt.length
+  # Initialize dp table
+  dp = Array.new(cap + 1, 0)
+  # State transition
+  for i in 1...(n + 1)
+    # Traverse in forward order
+    for c in 1...(cap + 1)
+      if wgt[i -1] > c
+        # If exceeds knapsack capacity, don't select item i
+        dp[c] = dp[c]
+      else
+        # The larger value between not selecting and selecting item i
+        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max
+      end
+    end
+  end
+  dp[cap]
+end
+
+### Driver Code ###
+if __FILE__ == $0
+  wgt = [1, 2, 3]
+  val = [5, 11, 15]
+  cap = 4
+
+  # Dynamic programming
+  res = unbounded_knapsack_dp(wgt, val, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+
+  # Space-optimized dynamic programming
+  res = unbounded_knapsack_dp_comp(wgt, val, cap)
+  puts "Maximum item value not exceeding knapsack capacity is #{res}"
+end