Three Sequences

题意翻译

给定一个序列$a_1,a_2,\ldots ,a_n$，长度为$n$的序列$b,c$符合以下条件 1. $a_i=b_i+c_i$ 2. $b$为不下降序列，$c$为不上升序列，即$b_i\geq b_{i-1}, c_i\leq c_{i-1}$ 有$q$个操作，每次操作可以把$[l,r]$的数增加$x$。求每次操作后最小的$\max(b_i,c_i)$

题目描述

You are given a sequence of $ n $ integers $ a_1, a_2, \ldots, a_n $ . You have to construct two sequences of integers $ b $ and $ c $ with length $ n $ that satisfy: - for every $ i $ ( $ 1\leq i\leq n $ ) $ b_i+c_i=a_i $ - $ b $ is non-decreasing, which means that for every $ 1<i\leq n $ , $ b_i\geq b_{i-1} $ must hold - $ c $ is non-increasing, which means that for every $ 1<i\leq n $ , $ c_i\leq c_{i-1} $ must hold You have to minimize $ \max(b_i,c_i) $ . In other words, you have to minimize the maximum number in sequences $ b $ and $ c $ . Also there will be $ q $ changes, the $ i $ -th change is described by three integers $ l,r,x $ . You should add $ x $ to $ a_l,a_{l+1}, \ldots, a_r $ . You have to find the minimum possible value of $ \max(b_i,c_i) $ for the initial sequence and for sequence after each change.

输入输出格式

输入格式

The first line contains an integer $ n $ ( $ 1\leq n\leq 10^5 $ ). The secound line contains $ n $ integers $ a_1,a_2,\ldots,a_n $ ( $ 1\leq i\leq n $ , $ -10^9\leq a_i\leq 10^9 $ ). The third line contains an integer $ q $ ( $ 1\leq q\leq 10^5 $ ). Each of the next $ q $ lines contains three integers $ l,r,x $ ( $ 1\leq l\leq r\leq n,-10^9\leq x\leq 10^9 $ ), desribing the next change.

输出格式

Print $ q+1 $ lines. On the $ i $ -th ( $ 1 \leq i \leq q+1 $ ) line, print the answer to the problem for the sequence after $ i-1 $ changes.

输入输出样例

输入样例 #1

4
2 -1 7 3
2
2 4 -3
3 4 2

输出样例 #1

5
5
6

输入样例 #2

6
-9 -10 -9 -6 -5 4
3
2 6 -9
1 2 -10
4 6 -3

输出样例 #2

3
3
3
1

输入样例 #3

1
0
2
1 1 -1
1 1 -1

输出样例 #3

0
0
-1

说明

In the first test: - The initial sequence $ a = (2, -1, 7, 3) $ . Two sequences $ b=(-3,-3,5,5),c=(5,2,2,-2) $ is a possible choice. - After the first change $ a = (2, -4, 4, 0) $ . Two sequences $ b=(-3,-3,5,5),c=(5,-1,-1,-5) $ is a possible choice. - After the second change $ a = (2, -4, 6, 2) $ . Two sequences $ b=(-4,-4,6,6),c=(6,0,0,-4) $ is a possible choice.

题意翻译

给定一个序列$a_1,a_2,\ldots ,a_n$，长度为$n$的序列$b,c$符合以下条件 1. $a_i=b_i+c_i$ 2. $b$为不下降序列，$c$为不上升序列，即$b_i\geq b_{i-1}, c_i\leq c_{i-1}$ 有$q$个操作，每次操作可以把$[l,r]$的数增加$x$。求每次操作后最小的$\max(b_i,c_i)$