Multi-Exchange Low-Latency Routing Optimization

An ultra-low-latency order routing protocol at Citadel transmits data packets through $N$ sequential exchange routing hubs, indexed from $0$ to $N$. Moving from hub $i$ to hub $i+1$ can be accomplished via two transmission modes:

1. Standard Fiber Link: Incurs latency standard[i].
2. Express Microwave Link: Incurs latency express[i].

Mode Switching Invariant:

• The packet begins transmission at hub $0$ in Standard mode with zero initial latency.
• Switching from Standard mode onto Express mode incurs an immediate hardware switching delay switch_cost.
• Once in Express mode, remaining in Express mode incurs no additional switching cost.
• Switching from Express mode back to Standard mode incurs zero switching penalty.

Your Task

Implement min_latency_routing(standard: list[int], express: list[int], switch_cost: int) -> list[int] that calculates an array of size $N+1$, where index $i$ represents the minimum total latency to reach hub $i$ from hub $0$.

Constraints

• $1 \le N \le 10^5$
• $\text{len}(standard) == \text{len}(express) == N$
• $1 \le standard[i], express[i], switch\_cost \le 10^6$

Examples

standard = [2, 3, 4]
express = [3, 1, 1]
switch_cost = 2
# Hub 0: 0
# Hub 1: min(standard 0+2=2, express 0+2+3=5) = 2
# Hub 2: reach via express = 2 + 2(switch) + 1 = 5, or standard = 2 + 3 = 5
# Hub 3: stay on express = 5 + 1 = 6
# Output: [0, 2, 5, 6]