Rate Feasibility in Wireless Networksramki-gummadi.github.io/other/feasiblerate_pres.pdf · rate feasibility is tractable for wireless networks, even though the general problem is

Rate Feasibility inWireless Networks

INFOCOM 2008Ramakrishna Gummadi, UIUC (speaker)

Kyomin Jung, MIT

Devavrat Shah, MIT

RS Sreenivas, UIUC

Ramakrishna Gummadi, UIUC – p. 1/35

IntroductionSimple abstraction of a wireless network:Vertices: wireless nodesEdges: possible communication links



Assume each link has a unit capacity (withoutinterference)




But Interference ⇒ Not all links can besimultaneously active




But Interference ⇒ Not all links can besimultaneously active

A fundamental question: Can a given vectorof link demands be satisfied simultaneously?


Rate Feasibility

Λ - set of all Rate Vectors for which thereexists a TDMA schedule.

r - the query Rate Vector

Problem: Does r ∈ Λ?


Rate Feasibility

Λ - set of all Rate Vectors for which thereexists a TDMA schedule.

r - the query Rate Vector

Problem: Does r ∈ Λ?

T - the incidence matrix for the set of allsubsets of non conflicting links

z = min 1Tx

Tx = r,

Then r ∈ Λ iff z ≤ 1


Is r “feasible”?

An answer to whether a given rate vector isfeasible depends on the following:

1. Graph structure (the specific probleminstance)

2. Interference constraints (the problemmodel - critical for the computationalcomplexity)


InterferencePrimary Constraints : Links conflict when theyshare a node.

pi

(a) (b) (c)

pj pkpj

pj

pi pi

pkpk


InterferenceSecondary constraints : Links can conflict evenwhile not sharing a common node.

pk

pl

pi

pj

Links (pi, pj) and (pk, pl) conflict


Background . . .

Primary and Secondary constraints(appropriate for wireless networks): problemis NP-hard [Arikan,’84]


Background . . .


Primary alone: Polynomial schedulingalgorithms exist [Hajek, Sasaki ’88]


Background . . .


Primary alone: Polynomial schedulingalgorithms exist [Hajek, Sasaki ’88]

Question: Are there restricted subclasses ofwireless networks that are tractable?


Our Results . . .Bounded density

A randomized approximation algorithmMore generally relevant to membership incomplex convex sets


Our Results . . .Bounded density

A randomized approximation algorithmMore generally relevant to membership incomplex convex sets

Fixed width slabGeneralize results on fractional coloringunit disk graphs to a more general classDeterministic, Exact solution.


Adjoint Graph

Adjoint Graph:Vertices - Link midpointsEdges - defined by link conflicts (bothprimary and secondary)

Communication (Interference) radius - rC(rI)for wireless nodes ⇒ Structure of its adjoint.

Link Rate-feasibility ≡ Node rate-feasibility inadjoint graph with independent sets.


Bounded density

Let B(v,R) = |{u ∈ V : u 6= v, d(u, v) < R}|Then, G has bounded density D > 0, if for allv ∈ V

B(v,R)

R2≤ D

Bounded density of wireless graph ⇒Bounded density of adjoint

∃R > 0 such that two vertices farther than Rin the adjoint have no edge.


Algorithm overview

Partition the graph into:

Regions, small enough to be efficiently solved

Boundaries, thick enough separate them


Algorithm overview

Partition the graph appropriately.


Algorithm overview


Solve for feasible schedules in each region


Algorithm overview



Merge them to get a global schedule –(schedule satisfies everyone except nodes inboundary)


Algorithm overview



Merge them to get a global schedule –(schedule satisfies everyone except nodes inboundary)

Randomize the boundary and compute anaverage schedule obtained over sufficientlylarge iterations. – (such that its unlikely for anode to fall in the boundary.)


Main Properties

(w.h.p.) Given ǫ > 0:

(1) If r ∈ Λ, algorithm outputs a TDMA scheduleT̂ = (αk, Ik)k≤M (with M = poly(n)), such that:

(1 − ǫ)r ≤∑

k

αkIk,

(2) If (1 − ǫ)r /∈ Λ, it declares NOT FEASIBLE.

Complexity is O

n log n2O

(

R2D

ǫ2

)

ǫ

≈ O(n log n).


Fixed width slabAssume wireless nodes are restricted in onecoordinate. (e.g. Y- dimension bounded,while X can range all over: IVHS)

X

Y


Adjoint Structure

Represent the vertex corresponding to thelink between pi,pj as (pi,pj). Then:

1. There cannot be an edge between vertices(pi,pj) and (pk,pl) if‖(pi,pj) − (pk,pl)‖2 ≥ rc + ri

2. There will be an edge between vertices(pi,pj) and (pk,pl) if‖(pi,pj) − (pk,pl)‖2 < ri − rc


Useful ResultsLet M be the incidence matrix for the set ofall independent sets of a graph G. Fractionalcoloring on G is to solve:

z = min 1Tx

Mx ≥ 1

x ≥ 0

Fractional coloring problem on a Unit DiskGraph has polynomial solution if nodes arewithin a fixed width slab. [Matsui 02]


Contrast with RateFeasibility

Fractional coloring algorithms solve a Nodebased rate feasibility problem with equalrates.


Contrast with RateFeasibility

Fractional coloring algorithms solve a Nodebased rate feasibility problem with equalrates.

Wireless adjoints do not have unit disk graphstructure.

D1

D2


Key Observations

1. The results for unit disk graphs can beextended to a more general class called(dmin, dmax) graphs, which includes ourwireless adjoints.


Key Observations

1. The results for unit disk graphs can beextended to a more general class called(dmin, dmax) graphs, which includes ourwireless adjoints.

2. The equal rates in fractional coloring can begeneralized to arbitrary rates.


(dmin, dmax) - graphs

G(P ) = (P,E) induced by point setP ⊆ {(x, y) ∈ R2} satisfies the followingproperties:

1. Property P1:∀p1,p2 ∈ P, ‖p1 − p2‖ ≥ dmax ⇒ (p1,p2) /∈ E

2. Property P2: ‖p1 − p2‖ < dmin ⇒ (p1,p2) ∈ E

If dmin = dmax, we get Unit Disk Graphs.

Note: Between dmin and dmax we have norestriction.


(dmin, dmax) - graphs

Theorem: Matsui’s algorithm can begeneralized to (dmin, dmax) graphs.

Proof idea: The properties P1 and P2, thoughimplicitly buried together in the specificationof the unit disk graphs, are actually onlyneeded separately.


Rate Feasibility onFixed Width Slabs

Wireless adjoints are (rI − rC , rI + rC)graphs.

Apply the Fractional Coloring algorithm,generalized to arbitrary rates.

Works in polynomial time, due to ourgeneralization from unit disk graphs to(dmin, dmax) graphs.


Conclusion andFuture interests

Have shown two relevant subclasses whererate feasibility is tractable for wirelessnetworks, even though the general problem isNP-hard.

Negatives: These are quite centralizeddescriptions of the algorithms.

Potential interest for practicality: Finding evenlower complexity distributed versions.

Integrate Rate Feasibility checkingseamlessly into cross layer design.


Thank you for your attention!

. . . Questions?


Documents

Rate Feasibility in Wireless Networksramki-gummadi.github.io/other/feasiblerate_pres.pdf · rate feasibility is tractable for wireless networks, even though the general problem is