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Petar Maymounkov ’06 MIT. Geometric Routing, Embeddings and Hyperbolic Spaces. 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)

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outline of talk a little for everybody
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 everybody3
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 1
The problem: Take 1

Space

total routing tables size

Routing tables

Labels

Stretch

Routing decisions

the problem take 2 abraham 06
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
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

slide7

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 -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]
slide9

… 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

slide10

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.
slide11

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]

slide12

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 …
slide13

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.

slide14

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 everybody15
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

slide16

Hyperbolic geometry: Construction

  • -dimensional hyperbolic space :
  • Defined on upper-half space via
  • Defined on unit-disc via
slide17

Hyperbolic isometries

Inversion

Dilation

Translation

slide18

Hyperbolic models

Unit-disk model

Half-plane model

Klein model

outline of talk a little for everybody19
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]

slide20

Greedy embeddings results

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

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

slide21

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

slide22

Lower bound continued …

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.

slide23

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

slide24

Upper bound for trees in [M’06]

Caterpillar

decomposition

depth

-axis

-axis

-axis

-axis

slide25

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!

slide26

Except for the trivial .

  • ... 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.

slide27

One more thing ….

The conjectured relationship between and can be described like this:

Embedding

dimension

Euclidean

Hyperbolic

Graph

complexity

On which graphs does

Hyperbolic geometry have advantage?