Problem:
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Farm Village

There is a village along a road. This village has $N$ houses numbered $1$ to $N$ in order along the road. Each house has a field that can make up to two units of the crop and needs just one unit of the crop. The total cost to distribute one unit of the crop to each house is the summation of carrying costs and growing costs.

The carrying cost: The cost to carry one unit of the crop between the $i$-th house and the ($i+1$)-th house is $d_i$. It takes the same cost in either direction to carry.

The growing cost: The cost to grow one unit of the crop in the $i$-th house's field is $g_i$.

Your task is to calculate the minimum total cost to supply one unit of the crop to each house.

Input

The input consists of a single test case formatted as follows.

$N$
$d_1$ ... $d_{N-1}$
$g_1$ ... $g_{N}$

The first line consists of an integer $N$ ($2 \leq 200,000$), which is the number of the houses. The second line consists of $N-1$ integers separated by spaces. The $i$-th integer $d_i$ ($1 \leq d_i \leq 10^9$, $1 \leq i \leq N-1$)represents the carrying cost between the $i$-th and the ($i+1$)-th houses. The third line consists of $N$ integers separated by spaces. The $i$-th integer $g_i$ ($1 \leq g_i \leq 10^9$, $1 \leq i \leq N$) represents the growing cost of the $i$-th house's field.

Output

Print the minimum cost to supply one unit of the crop to each house.

Sample Input 1

2
3
1 5

Output for Sample Input 1

5

Sample Input 2

3
100 100
1 2 3

Output for Sample Input 2

6

Sample Input 3

4
1 2 3
1 1 100 100

Output for Sample Input 3

12

Solution in Ruby (judge verdict: Time Limit Exceeded)

class FarmVillage
  def initialize(number_of_houses:, carrying_costs:, growing_costs:)
    @number_of_houses = number_of_houses
    @carrying_costs = carrying_costs.dup
    @growing_costs = growing_costs.dup
  end

  def calc_total_cost
    total_cost = 0

    @fields_productions = Array.new(@number_of_houses, 0)
    @left_moves = Array.new(@number_of_houses-1, 0)
    @right_moves = Array.new(@number_of_houses-1, 0)

    # order houses index by growing cost
    houses = (0...@number_of_houses).zip(@growing_costs).sort {|a, b| a[1] <=> b[1]}.map {|t| t[0]}

    min_index = 0
    houses.each do |house|
      min_new_crop_value = @growing_costs[houses[min_index/2]]

      min_index += 1

      local_cost = @growing_costs[house]
      left_cost, left_crop_house = calc_left_crop_cost(consumer_house: house,
                                                       local_crop_price: local_cost,
                                                       min_new_crop_value: min_new_crop_value)
      right_cost, right_crop_house = calc_right_crop_cost(consumer_house: house,
                                                          local_crop_price: local_cost,
                                                          min_new_crop_value: min_new_crop_value)

      costs = []
      costs << local_cost unless local_cost.nil?
      costs << left_cost unless left_cost.nil?
      costs << right_cost unless right_cost.nil?

      if local_cost == costs.min
        @fields_productions[house] += 1
      elsif left_cost == costs.min
        @fields_productions[left_crop_house] += 1
        change_moves_to_left(from: house, to: left_crop_house)
      elsif right_cost == costs.min
        @fields_productions[right_crop_house] += 1
        change_moves_to_right(from: house, to: right_crop_house)
      end
    end

    sum_prices
  end

  private

  def change_moves_to_left(from:, to:)
    (to...from).each do |road|
      if @right_moves[road] > 0
        @right_moves[road] -= 1
      else
        @left_moves[road] += 1
      end
    end
  end

  def change_moves_to_right(from:, to:)
    (from...to).each do |road|
      if @left_moves[road] > 0
        @left_moves[road] -= 1
      else
        @right_moves[road] += 1
      end
    end
  end

  def sum_prices
    total_cost = 0
    (0...@number_of_houses).each do |house|
      total_cost += @growing_costs[house] * @fields_productions[house]
    end
    @carrying_costs.each_with_index do |cost, index|
      total_cost += @left_moves[index] * cost
      total_cost += @right_moves[index] * cost
    end
    total_cost
  end

  def calc_left_crop_cost(consumer_house:, local_crop_price:, min_new_crop_value:)
    lower_cost = nil
    crop_house = nil

    road_cost = 0
    (0...consumer_house).each do |house|
      aux_house = consumer_house - 1 - house
      if @right_moves[aux_house] > 0
        road_cost -= @carrying_costs[aux_house]
      else
        road_cost += @carrying_costs[aux_house]
      end
      next unless @fields_productions[aux_house] < 2

      house_cost = road_cost + @growing_costs[aux_house]
      if lower_cost.nil? || house_cost < lower_cost
        lower_cost = house_cost
        crop_house = aux_house
      end

      if @right_moves[aux_house] < 0
        break if road_cost < local_crop_price
      end
    end

    [lower_cost, crop_house]
  end

  def calc_right_crop_cost(consumer_house:, local_crop_price:, min_new_crop_value:)
    lower_cost = nil
    crop_house = nil

    road_cost = 0
    ((consumer_house+1)...@number_of_houses).each do |house|
      if @left_moves[house-1] > 0
        road_cost -= @carrying_costs[house-1]
      else
        road_cost += @carrying_costs[house-1]
      end
      next unless @fields_productions[house] < 2

      house_cost = road_cost + @growing_costs[house]
      if lower_cost.nil? || house_cost < lower_cost
        lower_cost = house_cost
        crop_house = house
      end

      if @left_moves[house-1] < 0
        break if road_cost < local_crop_price
      end
    end

    [lower_cost, crop_house]
  end
end

number_of_houses = gets.to_i
carrying_costs = gets.split.map(&:to_i)
growing_costs = gets.split.map(&:to_i)

farm_village = FarmVillage.new(number_of_houses: number_of_houses,
                               carrying_costs: carrying_costs,
                               growing_costs: growing_costs)

puts farm_village.calc_total_cost