Approximate nearest neighbor search under universal L_p metrics (ANNS-U-L_p) is an important and challenging research problem, as it requires answering queries under all possible p (0<p <= 2) values simultaneously without building an index for each possible p value. The state-of-the-art solution, called MLSH, is a Locality-Sensitive Hashing (LSH)-based ANNS method with barely acceptable query performance. In contrast, graph-based ANNS methods, which offer significantly improved query efficiency on the ANNS-L_p problem (with a fixed p-value), cannot be naively extended to the ANNS-U-$L_p$ problem. In this paper, we propose U-HNSW, the first graph-based method for ANNS-U-L_p. Our scheme uses HNSW graph indexes built on two base metrics ($L_1$ and $L_2$) to generate promising nearest neighbors candidates, and then verifies these candidates with an early-termination strategy that substantially reduces the number of expensive L_p distance computations. Experimental results show that U-HNSW not only achieves up to 2670 times shorter query times than the original MLSH implementation running on a RAM disk (up to 15 times shorter than the idealized MLSH), but also outperforms the original HNSW on the ANNS-L_p problem (with a fixed p-value), except for a few special p values.

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