Transcript Lower Bounds for Additive Spanners, Emulators, and More

Lower Bounds for Additive
Spanners, Emulators, and
More
David P. Woodruff
MIT
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) · 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
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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) · 3G(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)
u
v
• 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
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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) · aG(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) · aG(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 instancedependent 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 speciallychosen 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?
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