Additive Spanners in Nearly
Quadratic Time
David Woodruff
IBM Almaden
Talk Outline
1.
Spanners
1.
2.
3.
Definition
Applications
Previous work
2.
Our Results
3.
Our Techniques for Additive-6 spanner
•
•
4.
New existence proof
Efficiency
Conclusion
Spanners
• G = (V, E) undirected unweighted graph, n vertices,
m edges
•
G (u,v) shortest path length from u to v in G
• [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, and do it quickly
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)
• Algorithmic tool: replace complicated dense graph with
spanner
– Spanner should be sparse
– Algorithm should be efficient
Multiplicative Spanners
• [PS, ADDJS] For every k, can find a
(2k-1, 0)-spanner with O(n1+1/k) edges in
O(m) time
• This is optimal assuming a girth conjecture
of Erdos
Surprise, Surprise
• [ACIM, DHZ]: Construct a (1,2)-spanner H
with O(n3/2) edges in O~(n2) time
• 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
• Sparsity Upper Bounds:
– (1,2)-spanner: O(n3/2) edges [ACIM, DHZ]
– (1,6)-spanner: O(n4/3) edges [BKMP]
– For any constant c > 6, best (1,c)-spanner known is
O(n4/3)
– For graphs with few edges on short cycles, can get
better bounds [P]
• Time Bounds:
– A (1,2)-spanner can be found in O~(n2) time [DHZ]
– A (1,6)-spanner can be found in O(mn2/3) time
[BKMP, Elkin]
Talk Outline
1.
Spanners
1.
2.
3.
Definition
Applications
Previous Work
2.
Our Results
3.
Our Techniques for Additive-6 spanner
•
•
4.
New existence proof
Efficiency
Conclusion
Main Contribution
• We construct an additive-6 spanner with O~(n4/3) edges in
O~(n2) time
– Improves previous O(mn2/3) time for all m, n
– Based on a new “path-hitting” framework
Corollary: Graphs with Large Girth
• We construct additive spanners for input graphs G with
large girth in O~(n2) time
• E.g., if G has girth > 4, we construct an additive 4-spanner
with O~(n4/3) edges in O~(n2) time
– Improves previous O(mn) time
• E.g., if G has girth > 4, we construct an additive 8-spanner
with O~(n5/4) edges in O~(n2) time
– Improves previous O(mn3/4) time
• Generalizes to graphs with few edges on short cycles
Corollary: Source-wise Preservers
• Given a subset S of O(n2/3) vertices of G, we compute a
subgraph H of O~(n4/3) edges in O~(n2) time so that
– For all u,v in S, δH(u,v) · δG(u,v) + 2
• Improves previous O(mn2/3) time
Talk Outline
1.
Spanners
1.
2.
3.
Definition
Applications
Previous Work
2.
Our Results
3.
Our Techniques for Additive-6 spanner
•
•
4.
New existence proof
Efficiency
Conclusion
Include Light Edges
• Include all edges incident to degree < n1/3 vertices in spanner
• Call these edges light
• At most O(n4/3) edges included
Path-Hitting
u
v
• Consider any shortest path. Suppose it goes from u to v.
• Done if all edges on path are light
• Otherwise there are heavy edges
• Each heavy edge e is adjacent to a set Se of > n1/3
vertices
• For heavy edges e and e’, Se and Se’ are disjoint
Path-Hitting
Wasn’t heavy
u
r
z
P
P’
v
s
• Randomly sample
vertices. Call these representatives
••Connect
Patheach
from
each
u tovertex
v in the
toeach
spanner:
one representative
possible)
• Connect
path-hitter
to
representative (if
using
an almost (+0,
+1, +2) shortest path P with at most x heavy edges
• W.h.p. every
• traverse
degree
light> edges
n1/3 vertex
+ representative
has an adjacent
edgerepresentative
to get to r
• Only pay for heavy edges along P
• Suppose
• take
thereP are
to get
x heavy
to z edges
• At most x heavy edges along P
• Then there
• take
areP’>tox get
* n1/3
to distinct
s
vertices at distance one from the path
• # of edges included is O~(n2/3)*O~(n2/3/x)*x = O~(n4/3)
• Randomly
• traverse
samplerepresentative
O~(n2/3/x) vertices.
edge Call
+ light
these
edges
path-hitters
to get to v
O~(n2/3)
• W.h.p.
• Bysample
triangleainequality
vertex adjacent
can show
to the
additive
path distortion is 6
Recap
Algorithm:
1. Randomly sample O~(n2/3) representatives
2. Randomly sample O~(n2/3/x) path-hitters
3. Connect each representative to each path-hitter on an
almost (+0, +1, +2) shortest path using O(x) heavy
edges
This works w.h.p. for all shortest paths containing between x and 2x
heavy edges
To make it work for all shortest paths, vary x in powers of 2 and take
the union of the edge-sets
Theorem: there exists an additive-6 spanner with O~(n4/3) edges
Talk Outline
1.
Spanners and related objects
1.
2.
Definition
Applications
2.
Previous Work
3.
Our Results
4.
Our Techniques for Additive-6 spanner
•
•
5.
New existence proof
Efficiency
Conclusion
Efficiency
• Algorithm:
1. Randomly sample O~(n2/3) representatives
2. Randomly sample O~(n2/3/x) path-hitters
3. Connect each representative to each path-hitter on an
almost (+0, +1, +2) shortest path using O(x) heavy edges
• Only one time consuming step
• Bicreteria problem:
– Almost (+0, +1, +2) shortest path AND O(x) heavy edges
• Simple breadth-first-search-like procedure
– Time complexity proportional to # of edges in the graph
– Only get quadratic time if # of edges is O(n4/3x)

Deleting High Degree Vertices
• If maximum degree · n1/3x, we get quadratic time
• Delete all degree > n1/3x vertices and incident
edges, run algorithm on subgraph
• Oops, some shortest paths are disconnected
u
v
Dominating Sets
z
u
Maximum degree = d
v
•
This only happens if there is a degree d > n1/3x vertex on the path
•
Delete all degree > d vertices in graph and incident edges
•
Choose a dominating set of O~(n/d) vertices for degree d vertices
•
Connect vertices in dominating set to representatives via an almost shortest
path with O(x) heavy edges
•
# of edges is O~(n/d * n2/3 * x) = O~(n4/3). Time is O~(n/d * nd) = O~(n2).
•
Vary both d and x in powers of 2. Take the union of the edge-sets found.
Talk Outline
1.
Spanners and related objects
1.
2.
Definition
Applications
2.
Previous Work
3.
Our Results
4.
Our Techniques for Additive-6 spanner
•
•
5.
New existence proof
Efficiency
Conclusion
Conclusion
• New path-hitting framework improves running times of
spanner problems to O~(n2)
– Additive-6 spanner with O~(n4/3) edges
– Further sparsifies inputs with large girth
– Approximate source-wise preservers
• Other graph problems for which technique may apply
– Constructing emulators more efficiently
– Constructing near-additive spanners more efficiently,
i.e., for all pairs u, v,
δH(u,v) · (1+ε)δG(u,v) + 4, where ε > 0 is arbitrary