【Atcoder - ARC101D/ABC107D】Median of Medians

最新推荐文章于 2022-05-23 19:36:24 发布

Tiw_Air_OAO

最新推荐文章于 2022-05-23 19:36:24 发布

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分类专栏： @分治算法看这里！@ --CDQ分治！-- --二分答案！-- --Atcoder！-- @那些日子做过的网赛！@

本文链接：https://blog.youkuaiyun.com/Tiw_Air_Op1721/article/details/82077501

这是一个关于Atcoder比赛中的ARC101D/ABC107D问题的解析，主要讨论如何在不构建新序列m的情况下，通过二分查找算法确定新序列的中位数。通过将原序列中的数与二分查找的可能中位数进行比较，转换问题为判断序列中大于中位数的元素数量，进而利用前缀和和归并排序（或CDQ分治）解决二维偏序问题，实现O(nlog^2n)时间复杂度的解决方案。

@Median of Medians@

@Median of Medians@

@题目描述 - English@

Time limit : 2sec / Memory limit : 1024MB

Score : 700 points

Problem Statement
We will define the median of a sequence b of length M, as follows:

Let b’ be the sequence obtained by sorting b in non-decreasing order. Then, the value of the (M/2+1)-th element of b’ is the median of b. Here, /is integer division, rounding down.
For example, the median of (10,30,20) is 20; the median of (10,30,20,40) is 30; the median of (10,10,10,20,30) is 10.

Snuke comes up with the following problem.

You are given a sequence a of length N. For each pair (l,r) (1≤l≤r≤N), let ml,r be the median of the contiguous subsequence (al,al+1,…,ar) of a. We will list ml,r for all pairs (l,r) to create a new sequence m. Find the median of m.

Constraints
1≤N≤10^5
ai is an integer.
1≤ai≤10^9

Input
Input is given from Standard Input in the following format:
N
a1 a2 … aN

Output
Print the median of m.

Sample Input 1
3
10 30 20
Sample Output 1
30

The median of each contiguous subsequence of a is as follows:
The median of (10) is 10.
The median of (30) is 30.
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