Title: Lower Bounds for Additive Spanners, Emulators, and More
1Lower Bounds for Additive Spanners, Emulators,
and More
- David P. Woodruff
- MIT and Tsinghua University
To appear in FOCS, 2006
2The 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?
3Spanners
- 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
4Spanner 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)
5Multiplicative 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
6Surprise, 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
7Additive 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.
8Our 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
9Emulators
- 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
10Known 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
11Our 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.
12Distance 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
13Known 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
14Our 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.
15Prescribed 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?
16Our 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 18Additive 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
19Lower 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
20Example 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)
21Lower 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.
22Example 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
23Lower 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.
24Example 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
25Lower 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.
26Example 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
27Lower 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.
28Example 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
29Lower 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.
30Example 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)
31Extension 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
32Extension 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).
33Extension 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
34Lower 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
35Lower 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
36Lower 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
37Other 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
38Prescribed Minimum Degree
n 16, degree 4, care about (1,3)-spanners
Suppose we insist on minimum degree 8
39Prescribed Minimum Degree
Left and middle vertices now have degree 8
40Prescribed Minimum Degree
Add a new level so everyone has degree 8. What
happens to the distortion?
41Modify 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
42Prescribed 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)))
43Upper 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)
44Upper 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
45Upper 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
46Summary 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
47Future 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?