Lower Bounds for Additive Spanners, Emulators, and More

An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v \in G, we have \delta _H \left( {u,v} \right) = \delta _G \left( {u,v} \right) + d. For every k = O\left( {\frac{{\ln n}} {{\ln \ln n}}} \right), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k - 1 has \\Omega \left( {\frac{1} {k}n^{1 + 1/k} } \right) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdos. We generalize our lower bound in a number of ways. First, we consider graph emulators introduced by Dor, Halperin, and Zwick (FOCS, 1996), where an emulator of an unweighted undirected graph G with distortion d is like an additive spanner except H may be an arbitrary weighted graph such that\delta _G \left( {u,v} \right)\leqslant\delta _H \left( {u,v} \right) \leqslant \delta _G \left( {u,v} \right) + d. We show a lower bound of \Omega \left( {\frac{1} {{k^2 }}n^{1 + 1/k} } \right) edges for distortion-(2k - 1) emulators. These are the first non-trivial bounds for k \ge 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n^{1/k+c} for any c \geqslant0, we prove a bound of \Omega \left( {\frac{1} {k}n^{1 + 1/k - c(1 + 2/(k - 1))} } \right) for additive spanners and \Omega \left( {\frac{1} {{k^2 }}n^{1 + 1/k - c(1 + 2/(k - 1))} } \right) for emulators. For k = 2 these can be improved to \Omega \left( {n^{3/2 - c} } \right). This partially answers a question of Baswana et al (SODA, 2005) for additive spanners. Finally, we continue the study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin (SODA, 2005) by considering their approximate variants and their relaxation to emulators. We prove the first lower bounds for such graphs.