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 ∈ G, we have δH(u, v) ≤ δG(u, v) + d. For every k = O( ln n ln ln n ), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k − 1 has Ω(1 k n1+1/k ) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erd¨os. 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 δG(u, v) ≤ δH(u, v) ≤ δG(u, v) + d. We show a lower bound of Ω( 1 k2 n1+1/k ) edges for distortion-(2k − 1) emulators. These are the first non-trivial bounds for k > 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimu...

• Type: Conference
• Year: 2006
• Where: FOCS
• Authors: David P. Woodruff
• Keywords: Additive Spanners, FOCS 2006, Lower Bounds, Theoretical Computer Science, Unweighted Undirected Graph