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Lower Bounds for Additive Spanners, Emulators, and

More

David P. WoodruffMIT

FOCS, 2006

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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?

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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

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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)

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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

u v

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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

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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.

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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

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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

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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

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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.

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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

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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

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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.

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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?

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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

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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

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Lower Bound Graphs

• All of our lower bounds are derived from variations of the butterfly network:

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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.

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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.

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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).

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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

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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?