Lower Bounds for Additive Spanners Emulators and More
Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT FOCS, 2006 1
The Model • G = (V, E) undirected unweighted graph, n vertices, m edges • G (u, v) shortest path length from u to v in G • Want to preserve pairwise distances G(u, v) • Exact answers for all pairs (u, v) needs (m) space • What about approximate answers? 2
Spanners • [A, PS] An (a, b)-spanner of G is a subgraph H such that for all u, v in V, H(u, v) · a G(u, v) + b • If b = 0, H is a multiplicative spanner • If a = 1, H is an additive spanner • Challenge: find sparse H 3
Spanner Application • 3 -approximate distance queries G(u, v) with small space • Construct a (3, 0)-spanner H with O(n 3/2) edges. [PS, ADDJS] do this efficiently • Query answer: G(u, v) · H(u, v) · 3 G(u, v) 4
Multiplicative Spanners • [PS, ADDJS] For every k, can quickly find a (2 k-1, 0)-spanner with O(n 1+1/k) edges • Assuming a girth conjecture of Erdos, cannot do better than (n 1+1/k) u v • Girth conjecture: there exist graphs G with (n 1+1/k) edges and girth 2 k+2 – Only (2 k-1, 0)-spanner of G is G itself 5
Surprise: Additive Spanners • [ACIM, DHZ]: Construct a (1, 2)-spanner H with O(n 3/2) edges! • Remarkable: for all u, v: G(u, v) + 2 G(u, v) · H(u, v) · • Query answer is a 3 -approximation, but with much stronger guarantees for G(u, v) large 6
Additive Spanners • Upper Bounds: – (1, 2)-spanner: O(n 3/2) edges [ACIM, DHZ] – (1, 6)-spanner: O(n 4/3) edges [BKMP] – For any constant b > 6, best (1, b)-spanner known is O(n 4/3) Major open question: can one do better than O(n 4/3) edges for constant b? • Lower Bounds: – Girth conjecture: (n 1+1/k) edges for (1, 2 k-1)spanners. Only resolved for k = 1, 2, 3, 5. 7
Our First Result • Lower Bound for Additive Spanners for any k without using the (unproven) girth conjecture: For every constant k, there exists an infinite family of graphs G such that any (1, 2 k-1)-spanner of G requires (n 1+1/k) edges • Matches girth conjecture up to constants • Improves weaker unconditional lower bounds by an n (1) factor 8
Emulators • In some applications, H must be a subgraph of G, e. g. , if you want to use a small fraction of existing internet links • For distance queries, this is not the case • [DHZ] An (a, b)-emulator of a graph G = (V, E) is an arbitrary weighted graph H on V such that for all u, v G(u, v) · H(u, v) · a G(u, v) + b • An (a, b)-spanner is (a, b)-emulator but not vice versa 9
Known Results • Focus on (1, 2 k-1)-emulators • Previous published bounds [DHZ] – (1, 2)-emulator: O(n 3/2), (n 3/2 / polylog n) – (1, 4)-emulator: O(n 4/3), (n 4/3 / polylog n) • Lower bounds follow from bounds on graphs of large girth 10
Our Second Result • Lower Bound for Emulators for any k without using graphs of large girth: For every constant k, there exists an infinite family of graphs G such that any (1, 2 k-1)-emulator of G requires (n 1+1/k) edges. • All existing proofs start with a graph of large girth. Without resolving the girth conjecture, they are necessarily n (1) weaker for general k. 11
Distance Preservers • [CE] In some applications, only need to preserve distances between vertices u, v in a strict subset S of all vertices V • An (a, b)-approximate source-wise preserver of a graph G = (V, E) with source S ½ V, is an arbitrary weighted graph H such that for all u, v in S, G(u, v) · H(u, v) · a G(u, v) + b 12
Known Results • Only existing bounds are for exact preservers, i. e. , H(u, v) = G(u, v) for all u, v in S • Bounds only hold when H is a subgraph of G • In this case, lower bounds have form (|S|2 + n) for |S| in a wide range [CE] • Lower bound graphs are complex – look at lattices in high dimensional spheres 13
Our Third Result • Simple lower bound for general (1, 2 k-1)approximate source-wise preservers for any k and for any |S|: For every constant k, there is an infinite family of graphs G and sets S such that any (1, 2 k-1)approximate source-wise preserver of G with source S has (|S|min(|S|, n 1/k)) edges. • Lower bound for emulators when |S| = n. • No previous non-trivial lower bounds known. 14
Prescribed Minimum Degree • In some applications, the minimum degree d of the underlying graph is large, and so our lower bounds are not applicable • In our graphs minimum degree is (n 1/k) • What happens when we want instancedependent lower bounds as a function of d? 15
Our Fourth Result • A generalization of our lower bound graphs to satisfy the minimum degree d constraint: Suppose d = n 1/k+c. For any constant k, there is an infinite family of graphs G such that any (1, 2 k-1)emulator of G has (n 1+1/k-c(1+2/(k-1))) edges. • If d = (n 1/k) recover our (n 1+1/k) bound • If k = 2, can improve to (n 3/2 – c) • We show tight for (1, 2)-spanners and (1, 4)emulators 16
Techniques • All previous methods looked at deleting one edge in graphs of high girth • Thus, these methods were generic, and also held for multiplicative spanners • We instead look at long paths in speciallychosen graphs. This is crucial 17
Lower Bound Graphs • All of our lower bounds are derived from variations of the butterfly network: 18
Lower Bound Graphs • Lower bound for (1, 2 k-1)-spanners: • Vertices are points in [n 1/k]k £ [k+1] • Edges only connect adjacent levels i, i+1, and can change the ith coordinate arbitrarily (a 1, a 2, …, ai, …, ak, i) connects to (a 1, a 2, …, ai’, …, ak, i+1) • Unique shortest path from vertices in level 1 to vertices in level k+1. 19
Additive Spanner Lower Bound If subgraph H has less than n 1+1/k edges, use the probabilistic method to show there are vertices v 1, vk+1 for which every edge along canonical path is missing. Butterfly network implies in this case, that G(v 1, vk+1) = k, but H(v 1, vk+1) ¸ 3 k, so get additive distortion 2 k. 20
Extension to Emulators • Recall that a (1, 2 k-1)-emulator H is like a spanner except H can be weighted and need not be a subgraph. • Observation: if e=(u, v) is an edge in H, then the weight of e is exactly G(u, v). • Reduction: Given emulator H with less than r edges, can replace each weighted edge in H by a shortest path in G. The result is an additive spanner H’. • Butterfly graphs have diameter 2 k = O(1), so H’ has at most 2 rk edges. Thus, r = (n 1+1/k). 21
Summary of Results • Unconditional lower bounds for additive spanners and emulators beating previous ones by n (1), and matching a 40+ year old conjecture, without proving the conjecture • Many new lower bounds for approximate source-wise preservers and for emulators with prescribed minimum degree. We show in some cases that the bounds are tight 22
Future Directions • Moral: – One can show the equivalence of the girth conjecture to lower bounds for multiplicative spanners, – However, for additive spanners our lower bounds are just as good as those provided by the girth conjecture, so the conjecture is not a bottleneck. • Still a gap, e. g. , (1, 4)-spanners: O(n 3/2) vs. (n 4/3) • Challenge: What is the size of additive spanners? 23
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