What problem should I solve?“

BCTCS, 23 March 2016 1

Magnús M. Halldórsson Tigran Tonoyan

Reykjavik University Iceland

"What problem should I solve?“ or

Efficiency in Wireless Networks?


This talk:

Wireless Scheduling

Which problem to

solve?


Which problem to solve?

• Plausibly doable (by me)

• Challenging enough (for me)

• Gets me going (hours on end)

• Scientifically important (enough)

• The ‚right‘ problem (out of all the zillions of formulations)


Networking: Separation of concerns

• Higher layers What messages do I want to send?

• Network layer:

Decide who to send what; routing

• Data Link/MAC layer Decide when to schedule individual transmissions


Capacity: Maximizing Wireless Thruput

• Given: Set of communication requests (“links”) • Find: Max feasible subset of links


Scheduling: Minimize latency � Partition links

• Given: Set of communication requests (“links”) • Find: Partition links into fewest feasible sets


Which problem Æ Which model?

Models

Realism

Simplicity

Computational Complexity

Generality


Interference model

But interference is not a binary relationship!

Disc Graphs


Interference is:

• Cumulative, not binary • What matters:

Is the received signal strength sufficiently large compared with the interference+noise?

• Î „Feasibility“ is a complicated independence system.


Disc Graphs Fail

Length of link i = 2i [Moscibroda, Wattenhofer 2006]

Feasible set, but forms a clique in any disc graph


Which problem Æ Which model?

Models

Realism

Simplicity

Computational Complexity

Generality


SINR model

1. Affectance (=Relative interference) is additive

= Interference strength / Strength of the (intended) signal

2. Affectance has a threshold

3. Signal strength decreases polynomially with distance

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Feasibility in the SINR model

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A set S is feasible iff the weighted in-degree of every link in 𝐺(𝐿) is small (< 1)

Given set 𝐿 of links, form an edge-weighted digraph 𝐺(𝐿). Weight of edge 𝑖𝑗 = Relative interference of link 𝑖 on link 𝑗

Here: Feasible = there exists a power assignments that

allows all links in S to successfully communicate

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Capacity: Maximizing Wireless Thruput

• Given: Set of communication requests (“links”) • Find: Max feasible subset of links

• 𝑂(1) -approximations known [Kesselheim, SODA’11]


Scheduling: Minimize latency � Partition links

• Given: Set of communication requests (“links”) • Find: Partition links into fewest feasible sets [Moscibroda, Wattenhofer 2006]

• Only 𝑂(log 𝑛)-approximations known


Rethinking graphs for representing interference

• Graphs are preferable to working directly with SINR – Less conceptual complexity – Simplifies description – Lots of theory already established

• How well can graphs work?

– Disc graphs fail, but what about other graphs?

• What does it mean to „represent SINR relationship“?


Abstracting, solving, mapping back


Price of abstraction

• Price of abstraction : How much you lose by solving the abstracted problem

(rather than solving directly)


Examples of abstractions of complex phenomena with simpler ones Reducing size of instance • graph sparsification Simplifying the features • dimensionality reductions • embeddings • graph augmentations and sandwiching properties “Simpler” abstraction • Sketches, adjacency labelings

• Other: curve fitting; generalized Fourier series;

discrepancy theory; PAC learning.


Hierarchies of abstraction

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Wireless „ground truth“

SINR model

Unweighted graphs


Representing link scheduling with a graph

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Î

When should there be an edge?


Requirement: Independent sets in G are feasible

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Independent sets should be feasible

valid coloring of G � valid scheduling


Requirement II:

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Feasible linksets should be „nearly independent“ in G

S feasible � F(GS) small

S GS


Graphs representing SINR

• Want: Schema to form a graph GL on link set L s.t. 1. (Feasibility) S is an independent set in G

Æ S is a feasible subset of links in L 2. (Low cost) S is a feasible set of links Æ G[S] has low chromatic number, k = F(GS) Cost of schema : largest k = F(GS) (over all S) Price of graph abstraction : Minimum cost of a schema


Possible graphs schemas (that fail)

• Primary conflicts – 𝑑 𝑢, 𝑣 ≤ 𝑐 ∙ min 𝑢 , 𝑣 – Too relaxed (fail feasibility)

• Disc graphs

– 𝑑 𝑢, 𝑣 ≤ 𝑐 ∙ max 𝑢 , 𝑣 – Too conservative (high cost) – One of the links will always be

infeasible

• Solution: Interpolate?

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Conflict graph representations [H,Tonoyan, STOC’15]

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Adjacency predicate: 𝑑 𝑢, 𝑤 ≤ 𝑓 𝑤

𝑢|𝑢|,

(𝑓 monotone)

𝑓 linear : disc graphs 𝑓 const : pairwise SINR

All such graphs have O(1) inductive independence. Coloring and WIS are O(1)-approximable

(𝑤 is longer than 𝑣)


Conflict graph representations [H,Tonoyan, STOC’15]

u

w

d(u,w)

Adjacency predicate: 𝑑 𝑢, 𝑤 ≤ 𝑓 𝑤

𝑢|𝑢|,

(𝑓 monotone)

𝑓 linear : disc graphs 𝑓 const : pairwise SINR

Feasibility holds for 𝑓 𝑥 = Ω(log 𝑥)

Cost of abstraction is 𝑓∗ 𝑥 , the iterated application of 𝑓

For 𝑓 = log, the cost is log∗ ∆

∆ = Diversity in link lengths log∗ ∆ is always less than 4 (!)


Corollaries

• SINR Scheduling with arbitrary power control is log∗ (∆)-approximable

• Our schema implies bounds on every subset of links!

• Obtain easily equivalent results for various extensions: – Weighted Capacity problem – Stochastic Packet Scheduling (w/ power control) – Multi-channel Multi-antennas – Max concurrent flow etc. – Online algorithms (admission control) – Spectrum auctions


How far can we go? Limits of solvability

• No (theoretical) study is complete without exploring the

limits of the doable.

• Can we show that no conflict graph schema can perform better?


Axioms for conflict graph representations

• Defined by pairwise relationship of links

• Independent of position and scale (scale-free)

• Monotonic with increasing distances

• Symmetric w.r.t. sender and receiver

GL

v u L

Every conflict graph schema is sandwich by formulations

𝑑 𝑢, 𝑤 ≤ 𝑓 𝑤𝑢

|𝑢|, where 𝑓 is a monotone function


Limitation results

• A. Any conflict graph representation incurs a Ω(log∗ ∆ ) factor Æ Price of abstraction is Θ(log∗ (∆)) – i) For every monotone 𝑓, there is an instance that is feasible but

whose conflict graph is a clique and requires Ω(𝑓∗(Δ)) colors – Ii) For 𝑓 = 𝑂(log1/𝛼 𝑛), there is an instance whose conflict graph

is independent, but requires Θ(log∗ (∆)) slots to schedule.

• Builds on a construction of [H, Mitra, SODA‘12]

• B. No approximation in terms of n is possible. • C. Requires Euclidean or doubling metrics


Open questions

• Still have not answered the question if purely constant-factor approximation is possible.

• Can we leverage this graph representation further?

• In which other context can we study „the price of graph abstraction“?

• Distributed algorithms?

• New modes of communication (interference alignment)


Is the SINR model really realistic?

1. (Additivity) Interference accumulates – It is not a pairwise property, but aggregate

2. (Thresholding) Transmission is successful if the

received signal-strength is stronger than the accumulated interference

3. (Polynomial decay) Signal decays as an inverse polynomial of distance

𝑑𝛼


Modeling Reality

reflection

scattering

diffraction shadowing

Non-omnidirectional antennas


Two-ray model

Slope = 2

Slope = 4


Testbeds

Classroom (TB-20) Basement (TB-40)

[Gudmundsdottir, Asgeirsson, Bodlaender, Foley, H, Mitra, Vigfusson, MSWiM 2014]


Do the SINR axioms hold (within reasonable errors)?

Additivity Thresholding


The headache: Geometric pathloss


How are distances actually used in the proofs?

Triangular inequality

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New Approach : The reality on the ground

• Idea: Signal decay needs not be a function of distance

• Geometric SINR model: – Nodes know distances d (or can obtain them) – They also know the pathloss constant, 𝛼 – Signal decay (and affectances) is computed based on these

distances

• Decay model – Nodes (typically) measure the signal decay between the nodes – They use these decays, and resulting affectances, directly – The performance guarantees are a function of how „metric-like“

the decay matrix is


Relation of Distance to Signal Strength

𝑑𝑎𝑏

Distance (Predicted) Received Signal Strength

(𝑑𝑎𝑏) 𝛼

(Actual) Received Signal Strength

𝑓𝑎𝑏 (𝑓𝑎𝑏)1/𝛼

? Sort of distance ?


Metricity [Bodlaender, H, PODC‘14]

• 𝑓𝑎𝑏 : The (measured) signal decay from a to b

• The metricity of a matrix 𝑓 is the smallest value ζ such that

(𝑓𝑎𝑐)1/ζ ≤ (𝑓𝑎𝑏)1/ζ + (𝑓𝑏𝑐)1/ζ • For geometric SINR, ζ = 𝛼

• Any result that holds for basic SINR in general metric spaces, holds equally in the Decay model!

• If performance ratio in Geo-SINR was f(𝛼) then the performance ratio in the Decay model is f(ζ)


Take-home message

• Our role as theorists is to elucidate fundamental properties, and discover common threads

• The „model“ matters

• The „right“ model combines fidelity, simplicity, generality, and (good) computational complexity

• All abstractions leak

• Understanding the underlying assumptions is important

• Which problem to solve or not to solve ...

That‘s the question.


Collaborators

• Tigran Tonoyan

• Marijke Bodlaender

• Eyjólfur Ásgeirsson

Experimental group: • Helga Gudmundsdottir • Ýmir Vigfusson • Joe Foley

Alumni: • Pradipta Mitra

Roger Wattenhofer At ETH, Zurich:


Documents

What problem should I solve?“