Lower Bounds for Additive Spanners, Emulators, and

More

David P. WoodruffMIT 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

Overview of Techniques

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)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(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

H has < n3/2 = 8 edges, e1,2 = 3, e2,3 = 4

(1,1,1)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(2,2,3)

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:

1. Choose v1 in level 1 uniformly at random.

2. Choose v2 to be a random neighbor of v1 in level 2.

3. Choose v3 to be a random neighbor of v2 in level 3.

Example: n = 4

(1,1,1)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(2,2,3)

V1

V2

V3

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

(v1, v2) and (v2, v3) are missing from H

(1,1,1)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(2,2,3)

V1

V3V2

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)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(2,2,3)

V1

V3V2

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)

(2,1,1)

(1,2,1)

(2,2,1)

(1,1,3)

(2,1,3)

(1,2,3)

(2,2,3)

V1

V3V2

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

Graph for n= 8:Example 1: |S| =4, |H| must be at least 6

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 SFor 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 choiceThis is a bad choiceThere 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)-spannersSuppose 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?

Modify middle edges so there is a unique edge connecting the clustersChoose a random vertex v1 in level 1

v1 v2

Choose a random v2 amongst first 2 neighbors of v1

v3

v3 is determinedv4 is a random neighbor of v3

v4

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

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 dominating if 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

u v

Shortest path from u to v in G

a

a is in the dominating setPath a, w, x, y, z, v is shortest from a to v in G

w x y

Path a, w, x, y, z, v occurs in BFS(a), so it is in H

z

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)

By triangle inequality, G(a,v) · G(u,v) + 1

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?