Transcript Geometric Routing, Embeddings and Hyperbolic Spaces

Petar Maymounkov ’06
MIT
Outline of talk – a little for everybody
Compact routing (a small industry)
The problem & summary of prior work
New applications = new open problems [M’06]
Hyperbolic geometry (crash course)
Greedy (ordinal) embeddings
Prior and related work
Lower bounds (on Minkowski and hyperbolic dimension) [M’06]
Techniques: duality and topology
Upper bounds for trees [M’06]
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Outline of talk – a little for everybody
Compact routing (a small industry)
The problem & summary of prior work
New applications = new open problems
Hyperbolic geometry (crash course)
Greedy (ordinal) embeddings
Prior and related work
Lower bounds (on Minkowski and hyperbolic dimension)
Techniques: duality and topology
Upper bounds for trees
3
The problem: Take 1
Input: (un)directed weighted graph
Output: “routing scheme”
Labels
Routing decisions
Space
total routing
tables size
Routing tables
Stretch
4
rewriting
The problem: Take 2 [Abraham,Header
…’06]
allows for loops!
Routing table at
= encoding of
Space = size of encoding of
Routing decision is
Current node
Header
In-port
Out-port
Rewritten
header!
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The problem: Take 2
Routing table at
= encoding of
Header rewriting
allows for loops!
Space = size of encoding of
Routing decision is
This formulation conceals (or suggests) a lower-bound
proof approach by Kolmogorov Complexity!
Current node
Header
In-port
Out-port
Rewritten
header!
6
Problem variations
Name-dependent vs. name-independent routing
– Vertex labels are assigned (in the output) by the routing scheme, vs.
– Vertex labels are pre-assigned (in the input) adversarially, respectively
Directed vs. undirected graphs
– Every
-stretch directed scheme requires
vertex graph
-space on some -
Prior work
Algorithms …
– … run in polynomial, usually quadratic, time
– They are heavily combinatorial and are not parallelizable
Lower bounds
– Interested in: Lower bounds on space for given stretch
– Just keep in mind: Average-case lower bounds are essentially identical to
worst-case bounds [Abraham, …’06]
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Known bounds are not bad
Stretch
(Un)directed graphs
Name-(in)dependent
General graph
Name-dependent
General graph
Name-independent
Trees
Name-dependent
Trees
Name-independent
Space
Lower-bound
Space
Upper-bound
Reference
Obvious
[Thorup,…’01]
[Thorup,…’01]
[Eilam,…’02]
[Abraham,…’06]
[Abraham,…’06]
[Abraham,…’06]
[Thorup,…’01]
[Thorup,…’01],
[Fraigniaud, …’02]
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… but modern applications are more demanding!
Large distributed systems
Routing brings security!
– Social networks, P2P systems, adhoc wireless, radio and censor networks
– Can have forced connectivity patterns, hence requiring routing schemes
Name-independence
So
what else do we want?
1.
2.
3.
4.
Algorithmic?
does
the job! stretch must be constant (think scale-free, high-conductance)
Real-world
Scheme computation must be distributed and furthermore incremental
Vertex labels must be stable over time, despite dynamic graph changes
Economic considerations require that table sizes be small:
Existential?
– A matter of taste dictates that
is too big
– Either, table sizes proportional to vertex degrees
– Or, all table sizes
when conductance is high
For starters: Is existence unreasonable, given the known bounds?
Graceful space
distribution
… let’s look inside the lower bound
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Lower-bound by girth conjecture (Erdös’63)
Theorem [TZ’01]: Every routing scheme for weighted directed
graphs with stretch
uses
space.
Proof:
– Let be an -vertex graph of girth
and
edges (the
conjecture)
– There are sub-graphs of
– Every two of them differ on at least one edge
– Hence routes between and must differ on the two sub-graphs
– Therefore we need
space-per-graph to even differentiate all
corresponding schemes
… the proof really says:
Connotation: Every routing scheme for weighted directed graphs
with stretch
uses
space.
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Lower-bound by girth conjecture (Erdös’63)
Theorem [TZ’01]: Every routing scheme for weighted directed
graphs with stretch
uses
space.
Proof:
– Let be an -vertex graph of girth
and
edges (the
A harder argument, but with the same connotation, based
conjecture)
– Thereon
areKolmogorov
sub-graphsComplexity
of
avoids the girth conjecture!
– Every two of them differ on[Abraham,…’06]
at least one edge
– Hence routes between and must differ on the two sub-graphs
– Therefore we need
space-per-graph to even differentiate all
corresponding schemes
… the proof really says:
Connotation: Every routing scheme for weighted directed graphs
with stretch
uses
space.
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So existence is plausible … what about algorithmics?
Incremental algorithms …
– … are usually force-simulation (e.g. rubber-bands [Linial, …’86], sphere
packings [Nurmela’97], unfolding rigid links [Demaine], etc.)
– or Markov iteration (e.g. PageRank)
– Geometry fits the bill for force-simulation!
Greedy embeddings and greedy routing
– Idea: embed vertices in some nice geometric space and route greedily w.r.t.
geometric locations
– A greedy embedding is
such that
it holds
that has a neighbor with
– Good news: graceful space distribution comes almost [M’06] for free
– Each vertex stores the
-bit coordinates of its neighbors
amounting to a routing table of size
OK, so the plan is …
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So existence is plausible … what about algorithmics?
Incremental algorithms …
– … are usually force-simulation (e.g. unfolding rigid links [Demaine])
– or Markov iteration (e.g. PageRank)
– Geometry fits the bill for force-simulation!
Note
that our motivation
of greedy
Greedy
embeddings
and greedy
routingembeddings is purely
theoretical,
whereas
[Papadimitriou,
…’05]
are
motivated
– Idea:
embed vertices
in some
nice geometric space
and
route
greedily w.r.t.
geometric locations by empirical observations.
– A greedy embedding is
such that
it holds
that has a neighbor with
– Good news: graceful space distribution comes almost [M’06] for free
– Each vertex stores the
-bit coordinates of its neighbors
amounting to a routing table of size
OK, so the plan is …
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And b.t.w. recent talks by Karp
Papadimitriou fit right here
Here’s where things areand
going:
Marriage between
topology and ordinal
– A tree-cover is obtained, such that all vertices belong to few treesembeddings?
– To route, a vertex chooses the best tree (from the cover)
– Then, routes with respect to this tree using optimal routing for trees
Our goal will be to match the last step (in an incremental and graceful manner) [M’06]
But we will also derive a unified lower-bounds for all graphs by topology [M’06]
Unusual geometric and optimization open questions will follow [M’06]
Existing routing algorithms … [Thorup,…01], etc.
– … almost invariably look like this:
–
–
–
Prior work on greedy embeddings … [R. Kleinberg’07]
–
–
–
–
–
… reveals that hyperbolic spaces are perfect for routing on trees
Because they
But this is “almost obvious” anyway, so what’s the catch?
require
-space!
Why not just use ultra-metrics?
The catch is that hyperbolic spaces accommodate concise embeddings [M’06]
And, they have degrees of freedom to allow “rapidly-mixing” force-simulation [M’…]
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Outline of talk – a little for everybody
Compact routing (a small industry)
The problem & summary of prior work
New applications = new open problems
Hyperbolic geometry (crash course)
Greedy (ordinal) embeddings
Prior and related work
Lower bounds (on Minkowski and hyperbolic dimension)
Techniques: duality and topology
Upper bounds for trees
15
Hyperbolic geometry: Construction
-dimensional hyperbolic space
–
Defined on upper-half space
–
Defined on unit-disc
:
via
via
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Hyperbolic isometries
Inversion
Dilation
Translation
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Hyperbolic models
Unit-disk model
Half-plane model
Klein model
Outline of talk – a little for everybody
Compact routing (a small industry)
The problem & summary of prior work
New applications = new open problems [M’06]
Hyperbolic geometry (crash course)
Greedy (ordinal) embeddings
Prior and related work
Lower bounds (on Minkowski and hyperbolic dimension) [M’06]
Techniques: duality and topology
Upper bounds for trees [M’06]
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Greedy embeddings results
Hyperbolic
Trees in :
-bit [R. Kleinberg’06]
Upper bounds
Trees in :
-bit [M’06]
Lower bounds
Hard-crossroad graph:
[M’06]
Euclidean
Trees in
:
-bit [M’06]
Star graph:
[R. Kleinberg’06]
Hard-crossroad graph:
[M’06]
OPEN: Iterative algorithm for any of the upper bounds (rubber bands?)
OPEN: Upper bound for arbitrary graphs in (using SDP duality?)
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Lower bound
Definition:
via “hard-crossroad” graph [M’06]
is a greedy embedding if
Geometric constraints:
Euclidean
Analytic constraints:
Geometric set system among points and
bisecting hyper-planes!
Geometric constraints
(under-specified set system)
Greedy embedding
(fully-specified set system)
Homeomorphisms
Preserve
Set systems
Hyperbolic
Homeomorphic map to
(same set system)
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Disc model
Lower bound continued …
-vertices
all
configurations
Klein model
-vertices
Graph with hard crossroads
Use Linear Algebra to show that
set system realized in
.
Argument generalizes to many
geodesic metric spaces.
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Upper bound for trees in
Dual of
[R. Kleinberg’06]
tiling is a greedy embedding
of the infinite 3-ary tree
Needs
Problem:
-bits per vertex coordinate
These are generators of
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Upper bound for trees in
Caterpillar
decomposition
[M’06]
depth
-axis
-axis
-axis
-axis
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Upper bound for trees in
… in the limit we get:
[M’06]
A repulsing force (with rigid
edges) straightens paths
Not rubber bands [LLW’88], but repulsing springs will probably get the job done!
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... What is left to do?
– We don’t know upper bound on
– We don’t know upper bound on
– We know how to compute best
Except for the trivial
.
-dimension for arbitrary graphs!
-dimension for arbitrary graphs, however
-embedding using an SDP:
… Hence we can look for an
embedding on the unit sphere …
… This can be expressed as an SDP …
… It is thus possible that the dual gives insight to upper bounds …
… and an iterative algorithm will be sure to find best embedding.
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One more thing ….
The conjectured relationship between
Embedding
dimension
and
can be described like this:
Euclidean
Hyperbolic
On which graphs does
Hyperbolic geometry have advantage?
Graph
complexity
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Thank you!
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