Largest Rectangle in a Histogram（最大矩形面积，动态规划思想）

Largest Rectangle in a Histogram

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others) Total Submission(s): 15013    Accepted Submission(s): 4357

Problem Description

A histogram is a polygon composed of a sequence of rectangles aligned at a common base line. The rectangles have equal widths but may have different heights. For example, the figure on the left shows the histogram that consists of rectangles with the heights 2, 1, 4, 5, 1, 3, 3, measured in units where 1 is the width of the rectangles: Usually, histograms are used to represent discrete distributions, e.g., the frequencies of characters in texts. Note that the order of the rectangles, i.e., their heights, is important. Calculate the area of the largest rectangle in a histogram that is aligned at the common base line, too. The figure on the right shows the largest aligned rectangle for the depicted histogram.

 

Input

The input contains several test cases. Each test case describes a histogram and starts with an integer n, denoting the number of rectangles it is composed of. You may assume that 1 <= n <= 100000. Then follow n integers h1, ..., hn, where 0 <= hi <= 1000000000. These numbers denote the heights of the rectangles of the histogram in left-to-right order. The width of each rectangle is 1. A zero follows the input for the last test case.

 

Output

For each test case output on a single line the area of the largest rectangle in the specified histogram. Remember that this rectangle must be aligned at the common base line.

 

Sample Input

7 2 1 4 5 1 3 3 4 1000 1000 1000 1000 0

 

Sample Output

8 4000

题解：让求最大矩形面积，宽为1，暴力超时

可以发现   当第i-1个比第i个高的时候   比第i-1个高的所有也一定比第i个高  

于是可以用到动态规划的思想

令left[i]表示包括i在内比i高的连续序列中最左边一个的编号   right[i] 为最右边一个的编号

那么有   当 h[left[i]-1]>=h[i]]时   left[i]=left[left[i]-1]  从前往后可以递推出left[i]   

同理      当 h[right[i]+1]>=h[i]]时   right[i]=right[right[i]+1]   从后往前可递推出righ[i]

最后答案就等于  max((right[i]-left[i]+1)*h[i]) 了；

代码：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<vector>
using namespace std;
const int INF=0x3f3f3f3f;
#define mem(x,y) memset(x,y,sizeof(x))
#define SI(x) scanf("%d",&x)
#define PI(x) printf("%d",x)
#define SD(x,y) scanf("%lf%lf",&x,&y)
#define P_ printf(" ")
const int MAXN=100010;
typedef long long LL;
LL a[MAXN];
int l[MAXN],r[MAXN];
int main(){
    int N;
     while(scanf("%d",&N),N){
         for(int i=1;i<=N;i++)scanf("%lld",&a[i]),l[i]=i,r[i]=i;
         a[0]=a[N+1]=-1;
         for(int i=1;i<=N;i++){
             while(a[l[i]-1]>=a[i])
                 l[i]=l[l[i]-1];
         } 
        for(int i=N;i>=1;i--){
            while(a[r[i]+1]>=a[i])
                r[i]=r[r[i]+1];    
        }
        LL ans=0;
        for(int i=1;i<=N;i++){
            ans=max(ans,(r[i]-l[i]+1)*a[i]);
        }
        printf("%lld
",ans);
     }
    return 0;
}