Maximum Concurrent Active Orders via Sweep-Line

High-frequency matching engine monitors at Two Sigma, HRT, and Citadel track active resting order memory allocations. You are given a list of $N$ orders, where each order is defined by its lifespan interval $[start_i, end_i)$ in nanoseconds (half-open interval: active from $start_i$ inclusive up to $end_i$ exclusive).

Your Task

Implement max_active_orders(intervals: list[list[int]]) -> int that determines the maximum number of simultaneously active orders present in the matching engine at any nanosecond in $\mathcal{O}(N \log N)$ time.

Constraints

• $1 \le \text{len}(intervals) \le 10^5$
• $0 \le start_i < end_i \le 10^9$

Examples

intervals = [[0, 30], [5, 10], [15, 20]]
# At t=7: orders [0, 30] and [5, 10] are active (2 orders).
# At t=17: orders [0, 30] and [15, 20] are active (2 orders).
# Output: 2

intervals = [[7, 10], [2, 4]]
# No overlap. Max active = 1.
# Output: 1