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New applications such as soft real-time data center applications, algorithmic trading, and high-performance computing require extremely low latency (in microseconds) from networks. Network operators today lack sufficient fine-grain measurement tools to detect, localize, and repair delay spikes that cause application service level agreement (SLA) violations. A recently proposed solution called LDA provides a scalable way to obtain latency, but only provides aggregate measurements. However, debugging application-specific problems requires per-flow measurements since different flows may exhibit significantly different characteristics even when they are traversing the same link. To enable fine-grained per-flow measurements in routers, we propose a new scalable architecture called reference latency interpolation (RLI) that is based on our observation that packets potentially belonging to different flows that are closely spaced to each other exhibit similar delay properties. In our evaluation using simulations over real traces, we show that while having small overhead, RLI achieves a median relative error of 12% and one to two orders of magnitude higher accuracy than previous per-flow measurement solutions. We also observe RLI achieves as high accuracy as LDA in aggregate latency estimation, and RLI outperforms LDA in standard deviation estimation.