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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?