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Lower Bounds for Additive Spanners, Emulators, and More. David P. Woodruff MIT and Tsinghua University. To appear in 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 Distance queries: what is G (u,v)?
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Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT and Tsinghua University To appear in 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 • Distance queries: what is G(u,v)? • Exact answers for all pairs (u,v) needs Omega(m) space • What about approximate answers?
Spanners • [A, PS] An (a, b)-spanner of G is a subgraph H such that for all u,v in V, H(u,v) · aG(u,v) + b • If b = 0, H is a multiplicative spanner • If a = 1, H is an additive spanner • Challenge: find sparse H
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) · 3G(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 Omega(n1+1/k) edges and girth 2k+2 • Only (2k-1,0)-spanner of G is G itself
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
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.
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
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) · aG(u,v) + b • An (a,b)-spanner is (a,b)-emulator but not vice versa
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
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.
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) · aG(u,v) + b
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
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.
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?
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
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
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
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)
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.
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
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.
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
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.
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
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.
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
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.
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)
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
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).
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) · aG(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
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
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
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
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
Prescribed Minimum Degree n = 16, degree = 4, care about (1,3)-spanners Suppose we insist on minimum degree 8
Prescribed Minimum Degree Left and middle vertices now have degree 8
Prescribed Minimum Degree Add a new level so everyone has degree 8. What happens to the distortion?
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
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)))
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)
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
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
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
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?