A16396 · Blocking Elements

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.

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 $10^5$ .

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

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