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Lower Bounds for Additive Spanners, Emulators, and More

Lower Bounds for Additive Spanners, Emulators, and More. David P. Woodruff MIT. FOCS, 2006. 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)

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Lower Bounds for Additive Spanners, Emulators, and More

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  1. Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT FOCS, 2006

  2. 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?

  3. 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

  4. Spanner Application • 3-approximate distance queries G(u,v) with small space • Construct a (3,0)-spanner H with O(n3/2) edges. [PS, ADDJS] do this efficiently • Query answer: G(u,v) ·H(u,v) · 3G(u,v)

  5. u v Multiplicative Spanners • [PS, ADDJS] For every k, can quickly find a (2k-1, 0)-spanner with O(n1+1/k) edges • Assuming a girth conjecture of Erdos, cannot do better than (n1+1/k) • Girth conjecture: there exist graphs G with (n1+1/k) edges and girth 2k+2 • Only (2k-1,0)-spanner of G is G itself

  6. Surprise: Additive Spanners • [ACIM, DHZ]: Construct a (1,2)-spanner H with O(n3/2) edges! • Remarkable: for all u,v: G(u,v) ·H(u,v) ·G(u,v) + 2 • Query answer is a 3-approximation, but with much stronger guarantees for G(u,v) large

  7. Additive Spanners • Upper Bounds: • (1,2)-spanner: O(n3/2) edges [ACIM, DHZ] • (1,6)-spanner: O(n4/3) edges [BKMP] • For any constant b > 6, best (1,b)-spanner known is O(n4/3) Major open question: can one do better than O(n4/3) edges for constant b? • Lower Bounds: • Girth conjecture: (n1+1/k) edges for (1,2k-1)-spanners. Only resolved for k = 1, 2, 3, 5.

  8. 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 (1,2k-1)-spanner of G requires (n1+1/k) edges • Matches girth conjecture up to constants • Improves weaker unconditional lower bounds by an n(1) factor

  9. 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

  10. Known Results • Focus on (1,2k-1)-emulators • Previous published bounds [DHZ] • (1,2)-emulator: O(n3/2), (n3/2 / polylog n) • (1,4)-emulator: O(n4/3), (n4/3 / polylog n) • Lower bounds follow from bounds on graphs of large girth

  11. 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 (1,2k-1)-emulator of G requires (n1+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.

  12. 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

  13. 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

  14. Our Third Result • Simple lower bound for general (1,2k-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 (1,2k-1)-approximate source-wise preserver of G with source S has (|S|min(|S|, n1/k)) edges. • Lower bound for emulators when |S| = n. • No previous non-trivial lower bounds known.

  15. 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 (n1/k) • What happens when we want instance-dependent lower bounds as a function of d?

  16. Our Fourth Result • A generalization of our lower bound graphs to satisfy the minimum degree d constraint: Suppose d = n1/k+c. For any constant k, there is an infinite family of graphs G such that any (1,2k-1)-emulator of G has (n1+1/k-c(1+2/(k-1))) edges. • If d = (n1/k) recover our (n1+1/k) bound • If k = 2, can improve to (n3/2 – c) • We show tight for (1,2)-spanners and (1,4)-emulators

  17. 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 specially-chosen graphs. This is crucial

  18. Lower Bound Graphs • All of our lower bounds are derived from variations of the butterfly network:

  19. Lower Bound Graphs • Lower bound for (1,2k-1)-spanners: • Vertices are points in [n1/k]k£ [k+1] • Edges only connect adjacent levels i,i+1, and can change the ith coordinate arbitrarily (a1, a2, …, ai, …, ak, i) connects to (a1, a2, …, ai’, …, ak, i+1) • Unique shortest path from vertices in level 1 to vertices in level k+1.

  20. Additive Spanner Lower Bound If subgraph H has less than n1+1/k edges, use the probabilistic method to show there are vertices v1, vk+1 for which every edge edge along canonical path is missing. Butterfly network implies in this case, that G(v1, vk+1) = k, but H(v1, vk+1) ¸ 3k, so get additive distortion 2k.

  21. Extension to Emulators • Recall that a (1,2k-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 2k = O(1), so H’ has at most 2rk edges. Thus, r = (n1+1/k).

  22. 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

  23. 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(n3/2) vs. (n4/3) • Challenge: What is the size of additive spanners?

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