Lower Bounds for Additive Spanners, Emulators, and More

1. Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT and Tsinghua University To appear in 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 • Distance queries: what is G(u,v)? • Exact answers for all pairs (u,v) needs Omega(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. 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 Omega(n1+1/k) edges and girth 2k+2 • Only (2k-1,0)-spanner of G is G itself

6. Surprise, Surprise • [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: (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) • Tight for (1,2)-spanners and (1,4)-emulators

17. Overview of Techniques

18. Additive Spanners • 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

19. Lower Bound for (1,3)-spanners • Identify vertices v as points (a,b,i) in [n1/2] £ [n1/2] £ [3] • We call the last coordinate the level • Edges connect vertices in level i to level i+1 which differ only in the ith coordinate: (a,b,1) connected to (a’,b,2) for all a,a’,b (a,b,2) connected to (a,b’,3) for all a,b,b’ • # vertices = 3n. # edges = 2n3/2

20. Example: n = 4 (1,1,1) (1,1,3) (2,1,1) (2,1,3) (1,2,1) (1,2,3) (2,2,1) (2,2,3)

21. Lower Bound for (1,3)-spanners • Recall #vertices = 3n, #edges = 2n3/2 • Consider arbitrary subgraph H with < n3/2 edges • Let e1,2 = # edges in H from level 1 to 2 • Let e2,3 = # edges in H from level 2 to 3 • Then H has e1,2 + e2,3 < n3/2 edges.

22. Example: n = 4 (1,1,1) (1,1,3) (2,1,1) (2,1,3) (1,2,1) (1,2,3) (2,2,1) (2,2,3) H has < n3/2 = 8 edges, e1,2 = 3, e2,3 = 4

23. Lower Bound for (1,3)-spanners Fix the subgraph H. Choose a path v1, v2, v3 in G with vi in level i as follows: • Choose v1 in level 1 uniformly at random. • Choose v2 to be a random neighbor of v1 in level 2. • Choose v3 to be a random neighbor of v2 in level 3.

24. Example: n = 4 V1 (1,1,1) (1,1,3) V3 (2,1,1) (2,1,3) V2 (1,2,1) (1,2,3) (2,2,1) (2,2,3) Red lines are edges in H

25. Lower Bound for (1,3)-spanners Pr[(v1, v2) and (v2, v3) in G \ H] ¸ 1 - Pr[(v1, v2) in H] – Pr[(v2, v3) in H] ¸ 1 - e1,2/n3/2 - e2,3/n3/2 > 0. So, there exist v1, v2, v3 such that (v1, v2) and (v2, v3) are missing from H.

26. Example: n = 4 (1,1,1) (1,1,3) (2,1,1) (2,1,3) (1,2,1) (1,2,3) V1 V3 (2,2,1) (2,2,3) V2 (v1, v2) and (v2, v3) are missing from H

27. Lower Bound for (1,3)-spanners • G(v1, v3) = 2. • Claim: H(v1, v3) ¸ 6. • Proof: • Construction ensures all paths from v1 to v3 in G have an odd # of edges in both levels. • Pigeonhole principle: if H(v1, v3) < 6, some level in any shortest path has only 1 edge.

28. Example: n = 4 (1,1,1) (1,1,3) (2,1,1) (2,1,3) (1,2,1) (1,2,3) V1 V3 (2,2,1) V2 (2,2,3) G(v1, v3) = 2 but H(v1, v3) = 6

29. Lower Bound for (1,3)-spanners • Suppose w.l.o.g., only 1 edge e = (a,b) in level 1 • Path from v1 to v3 in H starts with a level 1 edge e. So, e = (v1, b). • Edges in level i can only change the ith coordinate of a vertex. So, • The 1st coordinate of b and v3 are the same • The 2nd coordinate of b and v1 are the same • So, b = v2 and e = (v1, v2). But (v1, v2) is missing from H. Contradiction.

30. Example: n = 4 (1,1,1) (1,1,3) (2,1,1) (2,1,3) (1,2,1) (1,2,3) V1 V3 (2,2,1) V2 (2,2,3) Every path in G with G(v1, v3) < 6 contains (v1, v2) or (v2, v3)

31. Extension to General k • Lower bound for (1,2k-1)-spanners same: • Vertices are points in [n1/k]k£ [k+1] • Edges only connect adjacent levels i,i+1, and can change the ith coordinate arbitrarily • If subgraph H has less than n1+1/k edges, there are vertices v1, vk+1 for which G(v1, vk+1) = k, but H(v1, vk+1) ¸ 3k

32. 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’. • Our graphs have diameter 2k = O(1), so H’ has at most 2rk edges. Thus, r = (n1+1/k).

33. Extension to Preservers • An (a,b)-approximate source-wise preserver of a graph G 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 • Use same lower bound graph • Restrict to subgraph case. Can apply “diameter argument” • Choose a “hard’’ set S of vertices, based on |S|, whose distances to preserve

34. Lower Bound for (1,5)-approximate source-wise preserver Example 1: |S| =4, |H| must be at least 6 Graph for n= 8: Red lines indicate edges on shortest paths to and from S

35. Lower Bound for (1,5)-approximate source-wise preserver Example 2: |S| =8, our technique implies |H| ¸ 8 Red lines indicate edges on shortest paths to and from S For n = 8, can improve bound on |H|, but not asymptotically

36. Lower Bound for (1,5)-approximate source-wise preserver Intuition: “Spread out” source S This is a good choice This is a bad choice There is a small H

37. Other Extensions • For (1,2k-1)-approximate source-wise preservers, we achieve (|S|min(|S|, n1/k)) • Prescribed minimum degree d • Insert Kd,ds to ensure the minimum degree constraint is satisfied, while preserving the distortion property

38. Prescribed Minimum Degree n = 16, degree = 4, care about (1,3)-spanners Suppose we insist on minimum degree 8

39. Prescribed Minimum Degree Left and middle vertices now have degree 8

40. Prescribed Minimum Degree Add a new level so everyone has degree 8. What happens to the distortion?

41. v1 v2 v3 v4 Modify middle edges so there is a unique edge connecting the clusters Any sparse subgraph H is likely not to contain (v1, v2) and (v3, v4) G(v1, v4) = 3, but H(v1, v4) = 7, so H is not a (1,3)-spanner Choose a random v2 amongst first 2 neighbors of v1 Choose a random vertex v1 in level 1 v4 is a random neighbor of v3 v3 is determined

42. Prescribed Minimum Degree • (1,2)-spanners require (n3/2 – c) edges if the minimum degree is n1/2 + c • Corresponding O(n3/2-c log n) upper bound • General result: if min degree is n1/k+c, any (1,2k-1)-emulator has size (n1+1/k-c(1+2/(k-1)))

43. Upper Bound for (1,2)-spanners • A set S is dominatingif for all vertices v 2 V, there is an s 2 S such that (s,v) is an edge in G • If minimum degree n1/2+c , then there is a dominating S of size O(n1/2 –c log n) • For v 2 V, BFS(v) denotes the shortest-path tree in G rooted at v • H = [v in S BFS(v). Then |H| = O(n3/2 – c log n)

44. Upper Bound for (1,2)-spanners a w x y z v u Path u, a, w, x, y, z, v in H H(u,v) · 1+ H(a,v) = 1 + G(a,v) · 2 + G(u,v) Path a, w, x, y, z, v is shortest from a to v in G By triangle inequality, G(a,v) ·G(u,v) + 1 Shortest path from u to v in G a is in the dominating set Path a, w, x, y, z, v occurs in BFS(a), so it is in H

45. Upper Bound Recap • If minimum degree n1/2+c , then there is a dominating S of size O(n1/2 –c log n) • H = [v in S BFS(v). • |H| = O(n3/2 – c log n) • H is a (1,2)-spanner

46. 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. In some cases the bounds are tight

47. Future Directions • Moral: • One can show the equivalence of the girth conjecture to lower bounds for multiplicative spanners, • However, for additive spanners are 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?