Order Book Depth Snapshot Histogram Liquidity

During an auction clearing phase at Jump Trading and HRT, an aggregated Limit Order Book snapshot provides available resting liquidity across $N$ discrete price levels. The available volume at price tick $i$ is represented by a non-negative integer heights[i].

An institutional block trading algorithm seeks to clear a uniform block of liquidity spanning a contiguous sequence of price levels. The volume cleared across contiguous ticks $[i..j]$ at a uniform depth $H \le \min(heights[i..j])$ is given by the rectangular area: $\text{Liquidity Block Area} = H \times (j - i + 1)$

Your Task

Implement largest_liquidity_rectangle(heights: list[int]) -> int that determines the maximum possible liquidity area that can be cleared in strictly $\mathcal{O}(N)$ time using a monotonic stack.

Constraints

• $1 \le \text{len}(heights) \le 10^5$
• $0 \le heights[i] \le 10^9$

Examples

heights = [2, 1, 5, 6, 2, 3]
# Contiguous rectangle spanning indices 2 and 3 (heights 5 and 6) at depth 5 has area 5 * 2 = 10.
# Output: 10

heights = [2, 4]
# Rectangle spanning index 1 has area 4 * 1 = 4. Rectangle spanning indices 0..1 has min height 2 * 2 = 4.
# Output: 4