# Largest Rectangle in a Histogram Implement `largest_histogram_rectangle(heights: list[int]) -> int`. Each integer is the height of a unit-width histogram bar. Return the largest rectangular area that can be formed from one or more consecutive bars, with the rectangle's base on the histogram baseline. ### Input Domain - `0 <= len(heights) <= 200,000`. - `0 <= heights[i] <= 10^9`. ### Output Rules - A rectangle spanning indices `left` through `right` has height equal to the minimum bar height in that interval and width `right - left + 1`. - Return `0` for an empty input or when all heights are zero. - Return only the exact maximum area; no coordinates are required. ### Constraints - Use signed 64-bit arithmetic for areas. - Target time is `O(n)` and additional space is `O(n)`. ### Examples #### Example 1 Input: `heights = [2,1,5,6,2,3]` Output: `10` #### Example 2 Input: `heights = [2,4]` Output: `4` ```hint Determine when a bar's right boundary becomes known A monotonic structure can retain candidate bars until a shorter height proves that they cannot extend farther right. ```