poj3903 Stock Exchange 二分+dp

最新推荐文章于 2018-09-25 21:03:07 发布

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最新推荐文章于 2018-09-25 21:03:07 发布

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原文链接：http://www.cnblogs.com/weeping/p/5782450.html

本文介绍了一种高效算法来解决寻找股票价格序列中最长上升趋势的问题。通过使用动态规划和二分查找技术，该算法能在O(n log n)的时间复杂度内找到最长上升子序列的长度。

题目地址：http://poj.org/problem?id=3903

题目：

Description

The world financial crisis is quite a subject. Some people are more relaxed while others are quite anxious. John is one of them. He is very concerned about the evolution of the stock exchange. He follows stock prices every day looking for rising trends. Given a sequence of numbers p1, p2,...,pn representing stock prices, a rising trend is a subsequence pi1 < pi2 < ... < pik, with i1 < i2 < ... < ik. John’s problem is to find very quickly the longest rising trend.

Input

Each data set in the file stands for a particular set of stock prices. A data set starts with the length L (L ≤ 100000) of the sequence of numbers, followed by the numbers (a number fits a long integer).
White spaces can occur freely in the input. The input data are correct and terminate with an end of file.

Output

The program prints the length of the longest rising trend.
For each set of data the program prints the result to the standard output from the beginning of a line.

Sample Input

6 
5 2 1 4 5 3 
3  
1 1 1 
4 
4 3 2 1

Sample Output

3 
1 
1

Hint

There are three data sets. In the first case, the length L of the sequence is 6. The sequence is 5, 2, 1, 4, 5, 3. The result for the data set is the length of the longest rising trend: 3.

思路：dp入门水题。n^2的算法肯定会T的。所以只能用nlogn的算法；

　　c【i】：表示长度为i的上升子序列的最后一个值（也是序列中的最大值）；

　　从左向右扫描题目所给的数组，然后在c数组中二分查找第一个大于a【i】的位置，然后更新c数组。最后c数组的大小就是最长上升子序列的长度。

　　具体见代码吧，，没看懂的话可以看我dp分类里的另一个上升子序列的题目。讲的更详细。

代码：

#include <cstdlib>
#include <cctype>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <vector>
#include <string>
#include <iostream>
#include <sstream>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <fstream>
#include <numeric>
#include <iomanip>
#include <bitset>
#include <list>
#include <stdexcept>
#include <functional>
#include <utility>
#include <ctime>
#define PB push_back
#define MP make_pair
using namespace std;
typedef long long LL;
typedef pair<int,int> PII;
#define PI acos((double)-1)
#define E exp(double(1))
const int K=100000+9;
int a[K],c[K];
int main(void)
{
    int n,cnt;
    while(cin>>n)
    {
        cnt=0;
        memset(c,0,sizeof(c));
        for(int i=1; i<=n; i++)
            scanf("%d",&a[i]);
        for(int i=1;i<=n;i++)
        {
            int d=lower_bound(c+1,c+1+cnt,a[i])-c;
            c[d]=a[i];
            cnt=max(cnt,d);
        }
        cout<<cnt<<endl;
    }
    return 0;
}