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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

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

Assume each link has a unit capacity (withoutinterference)

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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

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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

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

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Rate Feasibility

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

r - the query Rate Vector

Problem: Does r ∈ Λ?

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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

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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)

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InterferencePrimary Constraints : Links conflict when theyshare a node.

pi

(a) (b) (c)

pj pkpj

pj

pi pi

pkpk

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

pk

pl

pi

pj

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

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Background . . .

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

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Background . . .

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

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

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Background . . .

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

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

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

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Our Results . . .Bounded density

A randomized approximation algorithmMore generally relevant to membership incomplex convex sets

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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.

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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.

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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.

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Algorithm overview

Partition the graph into:

Regions, small enough to be efficiently solved

Boundaries, thick enough separate them

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Algorithm overview

Partition the graph appropriately.

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Algorithm overview

Partition the graph appropriately.

Solve for feasible schedules in each region

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Algorithm overview

Partition the graph appropriately.

Solve for feasible schedules in each region

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

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Algorithm overview

Partition the graph appropriately.

Solve for feasible schedules in each region

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.)

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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).

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

X

Y

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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

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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]

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Contrast with RateFeasibility

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

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Contrast with RateFeasibility

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

Wireless adjoints do not have unit disk graphstructure.

D1

D2

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Key Observations

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

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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.

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(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.

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(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.

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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.

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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.

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Thank you for your attention!

. . . Questions?

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