CF1918D.Blocking Elements

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题目描述

You are given an array of numbers $a_1, a_2, \ldots, a_n$ . Your task is to block some elements of the array in order to minimize its cost. Suppose you block the elements with indices $1 \leq b_1 < b_2 < \ldots < b_m \leq n$ . Then the cost of the array is calculated as the maximum of:

the sum of the blocked elements, i.e., $a_{b_1} + a_{b_2} + \ldots + a_{b_m}$ .
the maximum sum of the segments into which the array is divided when the blocked elements are removed. That is, the maximum sum of the following ( $m + 1$ ) subarrays: [ $1, b_1 − 1$ ], [ $b_1 + 1, b_2 − 1$ ], [ $\ldots$ ], [ $b_{m−1} + 1, b_m - 1$ ], [ $b_m + 1, n$ ] (the sum of numbers in a subarray of the form [ $x,x − 1$ ] is considered to be $0$ ).

For example, if $n = 6$ , the original array is [ $1, 4, 5, 3, 3, 2$ ], and you block the elements at positions $2$ and $5$ , then the cost of the array will be the maximum of the sum of the blocked elements ( $4 + 3 = 7$ ) and the sums of the subarrays ( $1$ , $5 + 3 = 8$ , $2$ ), which is $\max(7,1,8,2) = 8$ .

You need to output the minimum cost of the array after blocking.

输入格式

The first line of the input contains a single integer $t$ ( $1 \leq t \leq 30\,000$ ) — the number of queries.

Each test case consists of two lines. The first line contains an integer $n$ ( $1 \leq n \leq 10^5$ ) — the length of the array $a$ . The second line contains $n$ elements $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 10^9$ ) — the array $a$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$ .

输出格式

For each test case, output a single number — the minimum cost of blocking the array.

输入输出样例

输入#1

3
6
1 4 5 3 3 2
5
1 2 3 4 5
6
4 1 6 3 10 7

输出#1

7
5
11

说明/提示

The first test case matches with the array from the statement. To obtain a cost of $7$ , you need to block the elements at positions $2$ and $4$ . In this case, the cost of the array is calculated as the maximum of:

the sum of the blocked elements, which is $a_2 + a_4 = 7$ .
the maximum sum of the segments into which the array is divided when the blocked elements are removed, i.e., the maximum of $a_1$ , $a_3$ , $a_5 + a_6 = \max(1,5,5) = 5$ .

So the cost is $\max(7,5) = 7$ .

In the second test case, you can block the elements at positions $1$ and $4$ .

In the third test case, to obtain the answer $11$ , you can block the elements at positions $2$ and $5$ . There are other ways to get this answer, for example, blocking positions $4$ and $6$ .

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