Bubble Sort Graph

题意翻译

给定一个 $1 \sim n$ 的排列 $a_{1 \sim n}$，基于此排列构建一张 $n$ 个点的无向图：对于每对逆序对 $a_i > a_j$（$i < j$），在 $a_i, a_j$ 两点间连一条无向边。试求在这个无向图中的最大独立集。

题目描述

Iahub recently has learned Bubble Sort, an algorithm that is used to sort a permutation with $ n $ elements $ a_{1} $ , $ a_{2} $ , ..., $ a_{n} $ in ascending order. He is bored of this so simple algorithm, so he invents his own graph. The graph (let's call it $ G $ ) initially has $ n $ vertices and 0 edges. During Bubble Sort execution, edges appear as described in the following algorithm (pseudocode). `<br></br>procedure bubbleSortGraph()<br></br> build a graph G with n vertices and 0 edges<br></br> repeat<br></br> swapped = false<br></br> for i = 1 to n - 1 inclusive do:<br></br> if a[i] > a[i + 1] then<br></br> add an undirected edge in G between a[i] and a[i + 1]<br></br> swap( a[i], a[i + 1] )<br></br> swapped = true<br></br> end if<br></br> end for<br></br> until not swapped <br></br> /* repeat the algorithm as long as swapped value is true. */ <br></br>end procedure<br></br>`For a graph, an independent set is a set of vertices in a graph, no two of which are adjacent (so there are no edges between vertices of an independent set). A maximum independent set is an independent set which has maximum cardinality. Given the permutation, find the size of the maximum independent set of graph $ G $ , if we use such permutation as the premutation $ a $ in procedure bubbleSortGraph.

输入输出格式

输入格式

The first line of the input contains an integer $ n $ ( $ 2<=n<=10^{5} $ ). The next line contains $ n $ distinct integers $ a_{1} $ , $ a_{2} $ , ..., $ a_{n} $ $ (1<=a_{i}<=n) $ .

输出格式

Output a single integer — the answer to the problem.

输入输出样例

输入样例 #1

3
3 1 2

输出样例 #1

2

说明

Consider the first example. Bubble sort swaps elements 3 and 1. We add edge (1, 3). Permutation is now \[1, 3, 2\]. Then bubble sort swaps elements 3 and 2. We add edge (2, 3). Permutation is now sorted. We have a graph with 3 vertices and 2 edges (1, 3) and (2, 3). Its maximal independent set is \[1, 2\].

题意翻译

给定一个 $1 \sim n$ 的排列 $a_{1 \sim n}$，基于此排列构建一张 $n$ 个点的无向图：对于每对逆序对 $a_i > a_j$（$i < j$），在 $a_i, a_j$ 两点间连一条无向边。试求在这个无向图中的最大独立集。