“…H−e (s, t) ≤ d A (s, cnt(z)) + d H−e (cnt(z), cnt(z ′ )) + d A (cnt(z ′ ), t) ≤ d G−e (s, z) + β − 1 + d G−e (z, z ′ ) + 2 + d G−e (z ′ , t) + β − d G−e (s, t) + 2β.Finally, when e = (cnt(z ′ ), z ′ ), we have:d H−e (s, t) ≤ d A (s, cnt(z)) + d H−e (cnt(z), cnt(z ′ )) + d H−e (cnt(z ′ ), z ′ ) + d A (z ′ , t) ≤ d G−e (s, z) + β − 1 + d G−e (z, z ′ ) + 2 + 1 + d G−e (z ′ , t) + β ≤ d G (s, t) + 2β + 2.This concludes the proof.⊓ ⊔This result can immediately be applied to the 6-additive spanner of size O(n ) in[4], which is clustering-based and uses O(n ) clusters. Using the 5multiplicative EFT spanner M of size O(n 4/3 ) from[10], we obtain: There exists a polynomial time algorithm to compute a 14-additive EFT spanner of size O(n 43 ).…”