The best replacement for euclidean is manhattan distance.

The best alternatives to euclidean are rated as following:
★★★★★  manhattan distance
★★★★★  non euclidean
★★★★  cartesian
★★★  spherical geometry
★★★  geodesic
★  hyperbolic
★  bray curtis
★  euclidean geometry
★  non-euclidean
★  euclidean
★  mahalanobis distance
  geometry

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euclidean
manhattan distance
non euclidean