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(n11/k) edges
• Assuming a girth conjecture of Erdos, cannot do
  better than ?(n11/k)
• Girth conjecture there exist graphs G with
  Omega(n11/k) edges and girth 2k2
• 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 gt 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 ?(n11/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 ?(n11/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 ?(n11/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 ?(S2 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)-approximat
  e 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 ?(Smin(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/kc. For any constant k, there is
  an infinite family of graphs G such that any
  (1,2k-1)-emulator of G has ?(n11/k-c(12/(k-1)))
  edges.
• If d ?(n1/k) recover our ?(n11/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 i1
  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)
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Lower Bound for (1,3)-spanners

• Recall vertices 3n, edges 2n3/2
• Consider arbitrary subgraph H with lt 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 lt 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 lt 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
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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 gt 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
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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) lt 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)
(2,2,3)
V2
?G(v1, v3) 2 but ?H(v1, v3) 6
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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)
(2,2,3)
V2
Every path in G with ?G(v1, v3) lt 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/kk k1
• Edges only connect adjacent levels i,i1, and can
  change the ith coordinate arbitrarily
• If subgraph H has less than n11/k edges, there
  are vertices v1, vk1 for which
• ?G(v1, vk1) k, but ?H(v1, vk1) 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 ?(n11/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
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
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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
• ?(Smin(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
Modify middle edges so there is a unique edge
connecting the clusters
Choose a random vertex v1 in level 1
Choose a random v2 amongst first 2 neighbors of v1
v3 is determined
v4 is a random neighbor of v3
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
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/kc, any
  (1,2k-1)-emulator has size ?(n11/k-c(12/(k-1)))

43
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 n1/2c , 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
Shortest path from u to v in G
By triangle inequality, ?G(a,v) ?G(u,v) 1
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 n1/2c , 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?