Lower Bounds for Additive Spanners, Emulators, and More

1
Lower Bounds for Additive Spanners, Emulators,
and More

• David P. Woodruff
• MIT

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
• 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(n11/k) edges
• Assuming a girth conjecture of Erdos, cannot do
  better than ?(n11/k)
• Girth conjecture there exist graphs G with
  ?(n11/k) edges and girth 2k2
• 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 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.

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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 ?(n11/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 ?(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.

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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 ?(S2 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)-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.

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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 instance-dependent
  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/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)
• 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 specially-chosen
  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/kk k1
• Edges only connect adjacent levels i,i1, and can
  change the ith coordinate arbitrarily
• (a1, a2, , ai, , ak, i) connects to (a1, a2, ,
  ai, , ak, i1)
• Unique shortest path from vertices in level 1 to
  vertices in level k1.

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Additive Spanner Lower Bound

• If subgraph H has less than n11/k edges, use
  the probabilistic method to show there are
  vertices v1, vk1 for which every edge edge along
  canonical path is missing.
• Butterfly network implies in this case, that
• ?G(v1, vk1) k, but ?H(v1, vk1) 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 ?(n11/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?