Approximating Gromov-Hausdorff distance in Euclidean space

The Gromov-Hausdorff distance (dGH) proves to be a useful measure between shapes. In order approximate dGH for X,Y⊂Rd, we look into its relationship with dH,iso, the infimum Hausdorff under Euclidean isometries. As already known dimension d≥2, dH,iso cannot bounded above by constant factor times dGH. For d=1, however, prove that dH,iso≤54dGH. We also show bound is tight. effect, X,Y⊂R at most n points, this gives rise an O(nlog⁡n)-time algorithm dGH(X,Y) approximation of (1+14).