Largest Rectangle in a Histogram

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Time Limit: 2 Seconds      

Memory Limit: 65536 KB

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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

 

动态规划  如果不使用路径压缩的话就会超时

1 # include
      
        2 # define maxn 100010 3 int main() 4 { 5     int n,i,j,left[maxn],right[maxn]; 6     long long int a[maxn]; 7     long long int ans,max; 8     while(scanf("%d",&n) && n) 9     {10         for(i=1; i<=n; i++)11         {12             scanf("%lld",&a[i]);13             left[i] = right[i] = i;14         }15         max=0;16         a[0] = a[n+1] =-1;17         for(i=2; i<=n; i++)18         {19             j = i;20             while(a[i]<= a[left[j]-1])21                 j = left[j]-1;22             left[i] = left[j];23         }24         for(i=n-1; i>=1; i--)25         {26             j=i;27             while(a[i]<=a[right[j]+1])28                 j=right[j]+1;29                 right[i]=right[j];30         }31         for(i=1; i<=n; i++)32         {33             ans = (right[i] - left[i] + 1) *a[i];34             if(ans>max)35                 max=ans;36         }37         printf("%lld\n",max);38     }39     return 0;40 }