Problem:
﻿Score : 400 points

Problem Statement

In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.

In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.

According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

1 \leq N \leq 100

1 \leq t_i \leq 200

1 \leq v_i \leq 100

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N

Output

Print the maximum possible that a train can cover in the run.

Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Sample Input 1

1
100
30

Sample Output 1

2100.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 30 seconds, it accelerates at a rate of 1m/s^2, covering 450 meters.

In the subsequent 40 seconds, it maintains the velocity of 30m/s, covering 1200 meters.

In the last 30 seconds, it decelerates at the acceleration of -1m/s^2, covering 450 meters.

The total distance covered is 450 + 1200 + 450 = 2100 meters.

Sample Input 2

2
60 50
34 38

Sample Output 2

2632.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 34 seconds, it accelerates at a rate of 1m/s^2, covering 578 meters.

In the subsequent 26 seconds, it maintains the velocity of 34m/s, covering 884 meters.

In the subsequent 4 seconds, it accelerates at a rate of 1m/s^2, covering 144 meters.

In the subsequent 8 seconds, it maintains the velocity of 38m/s, covering 304 meters.

In the last 38 seconds, it decelerates at the acceleration of -1m/s^2, covering 722 meters.

The total distance covered is 578 + 884 + 144 + 304 + 722 = 2632 meters.

Sample Input 3

3
12 14 2
6 2 7

Sample Output 3

76.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 6 seconds, it accelerates at a rate of 1m/s^2, covering 18 meters.

In the subsequent 2 seconds, it maintains the velocity of 6m/s, covering 12 meters.

In the subsequent 4 seconds, it decelerates at the acceleration of -1m/s^2, covering 16 meters.

In the subsequent 14 seconds, it maintains the velocity of 2m/s, covering 28 meters.

In the last 2 seconds, it decelerates at the acceleration of -1m/s^2, covering 2 meters.

The total distance covered is 18 + 12 + 16 + 28 + 2 = 76 meters.

Sample Input 4

1
9
10

Sample Output 4

20.250000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 4.5 seconds, it accelerates at a rate of 1m/s^2, covering 10.125 meters.

In the last 4.5 seconds, it decelerates at the acceleration of -1m/s^2, covering 10.125 meters.

The total distance covered is 10.125 + 10.125 = 20.25 meters.

Sample Input 5

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70

Sample Output 5

20291.000000000000

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;

namespace ABC076_D___AtCoder_Express
{
    class Program
    {
        static void Main(string[] args)
        {
            var N = int.Parse(Console.ReadLine());
            var T = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
            var V = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
            var totalTime = T.Sum() * 2;

            var limits = new SpeedLimit[N + 2];
            limits[0] = new SpeedLimit(0, 0, 0);
            for (int i = 1; i < N + 1; i++) limits[i] = new SpeedLimit(limits[i - 1].End, limits[i - 1].End + T[i - 1] * 2, V[i - 1]);
            limits[N + 1] = new SpeedLimit(totalTime, totalTime, 0);

            var speed = new double[totalTime + 1];
            for (int t = 1; t <= totalTime; t++)
            {
                speed[t] = double.PositiveInfinity;
                foreach (SpeedLimit l in limits)
                {
                    if (l.Start > t)
                    {
                        speed[t] = Math.Min(speed[t], l.Limit + (l.Start - t) * 0.5);
                    }
                    else if (l.End < t)
                    {
                        speed[t] = Math.Min(speed[t], l.Limit + (t - l.End) * 0.5);
                    }
                    else
                    {
                        speed[t] = Math.Min(speed[t], l.Limit);
                    }
                }
            }

            double distance = 0;
            for (int i = 1; i < totalTime + 1; i++)
            {
                if (speed[i] > speed[i - 1])
                {
                    distance += speed[i - 1] * 0.5 + 0.125;
                }
                else if (speed[i] < speed[i - 1])
                {
                    distance += speed[i] * 0.5 + 0.125;
                }
                else
                {
                    distance += speed[i] * 0.5;
                }
            }
            Console.WriteLine(distance);
        }
        struct SpeedLimit
        {
            public double Start;
            public double End;
            public double Limit;
            public SpeedLimit(double _start, double _end, double _limit)
            {
                Start = _start;
                End = _end;
                Limit = _limit;
            }
        }
    }
}