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## PowerPoint Slideshow about 'Geometric Routing, Embeddings and Hyperbolic Spaces' - minty

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Presentation Transcript

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]

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

The problem: Take 2 [Abraham, …’06]

Header rewriting allows for loops!

Routing table at= encoding of

Space = size of encoding of

Routing decision is

Rewritten header!

Header

In-port

Current node

Out-port

The problem: Take 2

Header rewriting allows for loops!

Routing table at = encoding of

Space = size of encoding of

This formulation conceals (or suggests) a lower-bound proof approach by Kolmogorov Complexity!

Routing decision is

Rewritten header!

Current node

Out-port

Header

In-port

- 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 -space on some -vertex graph
- 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]

… but modern applications are more demanding!

- Large distributed systems
- Social networks, P2P systems, adhoc wireless, radio and censor networks
- Can have forced connectivity patterns, hence requiring routing schemes
- So what else do we want?
- Real-world stretch must be constant (think scale-free, high-conductance)
- 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:
- 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?
- … let’s look inside the lower bound

Routing brings security!

Algorithmic?

Name-independence does the job!

Existential?

Graceful space distribution

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.

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.

A harder argument, but with the same connotation, based on Kolmogorov Complexity avoids the girth conjecture!

[Abraham,…’06]

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 …

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!
- 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 …

Note that our motivation of greedy embeddings is purely theoretical, whereas [Papadimitriou, …’05] are motivated by empirical observations.

And b.t.w. recent talks by Karp and Papadimitriou fit right here

- Here’s where things are going:
- Existing routing algorithms … [Thorup,…01], etc.
- … almost invariably look like this:
- A tree-cover is obtained, such that all vertices belong to few trees
- 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]
- Prior work on greedy embeddings … [R. Kleinberg’07]
- … reveals that hyperbolic spaces are perfect for routing on trees
- But this is “almost obvious” anyway, so what’s the catch?
- 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’…]

Marriage between topology and ordinal embeddings?

Because they

require -space!

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

Hyperbolic geometry: Construction

- -dimensional hyperbolic space :
- Defined on upper-half space via
- Defined on unit-disc via

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]

OPEN: Iterative algorithm for any of the upper bounds (rubber bands?)

OPEN: Upper bound for arbitrary graphs in (using SDP duality?)

Lower bound via “hard-crossroad” graph [M’06]

Definition: is a greedy embedding if

Geometric constraints:

Analytic constraints:

Geometric set system among points and

bisecting hyper-planes!

Euclidean

Geometric constraints

(under-specified set system)

Greedy embedding

(fully-specified set system)

Homeomorphisms

Preserve

Set systems

Homeomorphic map to

(same set system)

Hyperbolic

Disc model

-vertices

all

configurations

-vertices

Klein model

Graph with hard crossroads

Use Linear Algebra to show that set system realized in .

Argument generalizes to many geodesic metric spaces.

Upper bound for trees in [R. Kleinberg’06]

Dual of tiling is a greedy embedding

of the infinite 3-ary tree

These are generators of

Problem:

Needs -bits per vertex coordinate

Upper bound for trees in [M’06]

… in the limit we get:

A repulsing force (with rigid edges) straightens paths

Not rubber bands[LLW’88], but repulsing springs will probably get the job done!

- ... What is left to do?
- We don’t know upper bound on -dimension for arbitrary graphs!
- We don’t know upper bound on -dimension for arbitrary graphs, however
- We know how to compute best -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.

The conjectured relationship between and can be described like this:

Embedding

dimension

Euclidean

Hyperbolic

Graph

complexity

On which graphs does

Hyperbolic geometry have advantage?

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