HISTOGRA - Largest Rectangle in a Histogram

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 Specification

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 Specification

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

Sample Output


hide comments
x4fectax: 2020-01-11 19:32:54


We will use a mono-stack to solve this problem, meaning that all values in the stack are non-decreasing. For each bar, we will push it into the stack along with its index as soon as the stack is empty or the top of the stack is not taller than your current bar. While the top of the stack is taller than your current bar, we will pop it, use its height as the tallest height, and update our answer. Specifically, our answer will be max(answer, (top of stack height * (i - next_item_in_stack.index - 1)). For convenience, we can add a bar of height 0 and index 0 to the front of the stack before we start our algorithm, and a bar of height 0 to the stack after we finish our algorithm. Since each bar is pushed and popped at most once, our complexity is O(n). Hope this helps.

Last edit: 2020-01-11 19:33:22
hirbod313: 2019-12-08 14:16:50

Could you please talk about the idea of solving the problem with stack?

Last edit: 2019-12-08 14:17:06
codefresher: 2019-11-23 10:53:48

Nice problem...excellent example of stack...

sa_chin123: 2019-07-14 10:30:41

how to take number of test cases

mgroovyank: 2019-06-13 07:45:31

Use everything as long long int

hetp111: 2019-05-29 17:14:32

how to do it without stacks?

wingman__7: 2019-05-19 08:32:44

If you are using CPP, d&c, segment trees then just use long long everywhere ... you'll get AC

Last edit: 2019-05-19 08:35:26
iceelement: 2019-05-06 13:09:18

sparse table and binary search also works O(nlogn)

stones: 2019-04-16 23:44:00

It can also be done using Union and Find in O(nlogn).

Jumpy: 2019-03-31 19:31:54

I was playing with values in stack, turned out it is dependent on index.

1) Very good implementation of stack, One should try out this.
2) Use this link to learn about this: https://www.geeksforgeeks.org/largest-rectangle-under-histogram/ .

Last edit: 2019-04-02 05:32:14

Added by:Wanderley Guimarăes
Time limit:0.800s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO
Resource:University of Ulm Local Contest 2003