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dynamic_programming_breaks.rb

class SparseMatrix
  def initialize
    @data = {}
  end

  def set(row_idx, col_idx, value)
    data[key(row_idx, col_idx)] = value
  end

  def get(row_idx, col_idx)
    data[key(row_idx, col_idx)]
  end

  private

  attr_reader :data

  def key(row_idx, col_idx)
    "#{row_idx}-#{col_idx}"
  end
end

class Minima
  attr_accessor :start_index, :value

  def initialize(value: 10**20)
    @start_index = 0
    @value = value
  end
end

class Minimas
  def initialize(count_nodes:)
    @data = [Minima.new(value: 0)] + Array.new(count_nodes){ Minima.new }
  end

  def update_minima(cost, start_node_idx, end_node_idx)
    return if @data[end_node_idx + 1].value <= cost

    @data[end_node_idx + 1].value = cost
    @data[end_node_idx].start_index = start_node_idx
  end

  def start_nodes
    @data.map(&:start_index)
  end

  def [](idx)
    @data[idx].value
  end

  private
  attr_reader :data
end

class Dynamic
  attr_reader :word_lengths, :max_line_length, :space_length, :word_lengths_count, :line_spaces

  def initialize(words:, max_line_length:)
    @word_lengths = words.map(&:length)
    @word_lengths_count = @word_lengths.size
    @max_line_length = max_line_length
    @space_length = 1
  end

  def breaks
    @line_spaces = precalculated_possible_line_spaces

    @minimas = Minimas.new(count_nodes: word_lengths_count)

    (0...word_lengths_count).each do |end_word_idx|
      process_possible_lines_ending_with(end_word_idx)
    end

    @minimas.start_nodes
  end

  private

  def process_possible_lines_ending_with(end_word_idx)
    start_word_idx = end_word_idx

    while start_word_idx >= 0
      process_possible_line(start_word_idx, end_word_idx)

      start_word_idx -= 1
    end
  end

  def process_possible_line(start_word_idx, end_word_idx)
    cost = penality_for_line(start_word_idx, end_word_idx)

    @minimas.update_minima(cost, start_word_idx, end_word_idx)
  end

  def penality_for_line(start_word_idx, end_word_idx)
    return 10 ** 10 if line_spaces.get(start_word_idx, end_word_idx) < 0

    @minimas[start_word_idx] + line_spaces.get(start_word_idx, end_word_idx) ** 2
  end

  def precalculated_possible_line_spaces
    word_lengths.each_with_index.each_with_object(SparseMatrix.new) do |(start_word_length, start_word_idx), line_space|
      remaining_space_in_line = max_line_length - start_word_length
      line_space.set start_word_idx, start_word_idx, remaining_space_in_line

      ((start_word_idx + 1)...word_lengths_count).each do |end_word_idx|
        remaining_space_in_line = line_space.get(start_word_idx, end_word_idx - 1) - word_lengths[end_word_idx] - space_length
        line_space.set start_word_idx, end_word_idx, remaining_space_in_line
      end
    end
  end
end

graph_clustered.svg

greety_breaks.rb

class GreetyBreaks
  attr_reader :word_lengths, :max_line_length, :space_length

  def initialize(words:, max_line_length:)
    @word_lengths = words.map(&:length)
    @max_line_length = max_line_length
    @space_length = 1
  end

  def breaks
    current_length = -space_length

    word_lengths.each_with_index.each_with_object([]) do |(word_length, idx), breaks|
      if word_length + current_length >= max_line_length
        breaks.push idx
        current_length = -space_length
      end

      current_length += space_length + word_length
    end
  end
end

greety_line_breaks.svg

knuth_plass_breaks.svg

monge-matrix-convex.svg

monge-matrix.svg

shortest_path_breaks.rb

class Minima
  attr_accessor :start_index, :value

  def initialize(value: 10 ** 20)
    @start_index = 0
    @value = value
  end
end

class Minimas
  def initialize(count_nodes:)
    @data = [Minima.new(value: 0)] + Array.new(count_nodes) { Minima.new }
  end

  def update_minima(cost, start_node_idx, end_node_idx)
    return if @data[end_node_idx].value <= cost

    @data[end_node_idx].value = cost
    @data[end_node_idx].start_index = start_node_idx
  end

  def start_nodes
    @data.map(&:start_index)
  end

  def [](idx)
    @data[idx].value
  end

  private

  attr_reader :data
end

class ShortestPath
  attr_reader :word_lengths, :max_line_length, :space_length, :word_lengths_count, :line_spaces

  def initialize(words:, max_line_length:)
    @word_lengths = words.map(&:length)
    @word_lengths_count = @word_lengths.size
    @max_line_length = max_line_length
    @space_length = 1
  end

  def breaks
    pre_calculate_word_offsets

    @minimas = Minimas.new(count_nodes: word_lengths_count)

    (0...word_lengths_count).each do |start_word_idx|
      process_possible_lines(start_word_idx, start_word_idx + 1)
    end

    @minimas.start_nodes
  end

  private

  def pre_calculate_word_offsets
    @offsets = [0]

    word_lengths.each do |word_length|
      @offsets.append(@offsets[-1] + word_length)
    end
  end

  def process_possible_lines(start_word_idx, first_end_node_idx)
    end_node_idx = first_end_node_idx

    while end_node_idx <= word_lengths_count
      break unless process_possible_line(end_node_idx, start_word_idx)

      end_node_idx += 1
    end
  end

  def process_possible_line(end_node_idx, start_word_idx)
    current_line_length = calculate_current_line_length(end_node_idx, start_word_idx)
    no_space_left_in_line = current_line_length > max_line_length

    return false if no_space_left_in_line

    penality = (max_line_length - current_line_length) ** 2

    cost = @minimas[start_word_idx] + penality

    @minimas.update_minima(cost, start_word_idx, end_node_idx)

    true
  end

  def calculate_current_line_length(end_node_idx, start_word_idx)
    spaces = end_node_idx - start_word_idx - 1
    @offsets[end_node_idx] - @offsets[start_word_idx] + spaces
  end
end

smawk.rb

class Minima
  attr_accessor :start_index, :value

  def initialize
    @start_index = 0
    @value = 10 ** 20
  end
end

class SMAWK
  attr_reader :minima_provider, :minimas, :cnt_elements

  def initialize(minima_provider, cnt_elements)
    @minima_provider = minima_provider
    @minimas = Array.new(cnt_elements + 1) { Minima.new }
    @cnt_elements = cnt_elements

    @minimas[0].value = 0
  end

  def cost(start_idx, end_idx)
    minima_provider.call(minimas[start_idx].value, start_idx, end_idx)
  end

  def start_index(end_index)
    minimas[end_index].start_index
  end

  def calc
    backward_facing_cursor = cnt_elements + 1
    forward_facing_cursor = 0
    offset = 0

    loop do
      rth_column = [backward_facing_cursor, 2 ** (forward_facing_cursor + 1)].min
      edge_between_head_and_tail = 2 ** forward_facing_cursor + offset

      calc_sub_matrix(offset...edge_between_head_and_tail, edge_between_head_and_tail...(rth_column + offset))

      right_most_col_of_sub_matrix = minimas[rth_column - 1 + offset]
      minimum_right_most_col_of_sub_matrix = right_most_col_of_sub_matrix.value

      exited_by_break = ((2 ** forward_facing_cursor)...(rth_column - 1)).each do |start_word_idx|
        current_score = cost(start_word_idx + offset, rth_column - 1 + offset)

        if current_score <= minimum_right_most_col_of_sub_matrix
          backward_facing_cursor -= start_word_idx
          forward_facing_cursor = 0
          offset += start_word_idx
          break true
        end

        false
      end

      next if exited_by_break == true

      break if rth_column == backward_facing_cursor

      forward_facing_cursor += 1
    end
  end

  def calc_sub_matrix(rows, columns)
    rows = rows.to_a
    columns = columns.to_a

    rows = evaluate_sub_matrix_rows(columns, rows)

    handle_leftover_columns(columns, rows)

    calculate_sub_matrix_minima(columns, rows)
  end

  private

  def calculate_sub_matrix_minima(columns, rows)
    current_row = current_column = 0
    while current_column < columns.size
      start_row_index = if current_column + 1 < columns.size
                          minimas[columns[current_column + 1]].start_index
                        else
                          rows[-1]
                        end

      current_column_cost = cost(rows[current_row], columns[current_column])

      if current_column_cost < minimas[columns[current_column]].value
        minimas[columns[current_column]].value = current_column_cost
        minimas[columns[current_column]].start_index = rows[current_row]
      end

      if rows[current_row] < start_row_index
        current_row += 1
      else
        current_column += 2
      end
    end
  end

  def handle_leftover_columns(columns, rows)
    return if columns.empty?

    every_second_column = columns.each_slice(2).map { |v| v[1] }.to_a.compact
    new_sub_matrix = every_second_column
    calc_sub_matrix(rows, new_sub_matrix)
  end

  def evaluate_sub_matrix_rows(columns, rows)
    [].tap do |stack|
      current_row = 0
      while current_row < rows.size
        if stack.size.positive?
          current_column_cost = columns[stack.size - 1]

          if cost(stack[-1], current_column_cost) < cost(rows[current_row], current_column_cost)
            stack.append(rows[current_row]) if stack.size < columns.size
            current_row += 1
          else
            stack.pop
          end
        else
          stack.append(rows[current_row])
          current_row += 1
        end
      end
    end
  end
end

class LineBreaks
  NEAR_INFINITY = 10 ** 10

  attr_reader :word_offsets, :max_line_length, :space_length, :word_offsets_count, :line_spaces

  def initialize(words:, max_line_length:)
    @word_offsets = (0...(words.size-1)).each.reduce([0]){|res, index| res.push(words[index].length + res[-1])}

    @word_offsets_count = @word_offsets.size

    @max_line_length = max_line_length
    @space_length = 1

    @smawk = SMAWK.new(lambda { |current_minima, start_idx, end_idx|
      cost(current_minima, start_idx, end_idx)
    }, words.size - 1 )
  end

  def concave_modified_infinity(current_line_width, max_line_length)
    concave_modifier = current_line_width - max_line_length
    NEAR_INFINITY * concave_modifier
  end

  def cost(current_minima, start_idx, end_idx)
    white_spaces = end_idx - start_idx - 1

    start_word = word_offsets[start_idx]
    end_word = word_offsets[end_idx]

    current_line_width = end_word - start_word + white_spaces

    return concave_modified_infinity(current_line_width, max_line_length) if current_line_width > max_line_length

    current_minima + (max_line_length - current_line_width) ** 2
  end

  def breaks
    @smawk.calc

    (0...word_offsets_count).map { |idx| @smawk.start_index(idx) }
  end
end

words_as_directed_graph.svg

words_as_graph_with_break_edges.svg

Explanations: https://medium.com/@dieter.s/visualizations-with-ruby-cdc16c0cee2e