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BCTCS, 23 March 2016 1
Magnús M. Halldórsson Tigran Tonoyan
Reykjavik University Iceland
"What problem should I solve?“ or
Efficiency in Wireless Networks?
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BCTCS, 23 March 2016 2
This talk:
Wireless Scheduling
Which problem to
solve?
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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)
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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
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Capacity: Maximizing Wireless Thruput
• Given: Set of communication requests (“links”) • Find: Max feasible subset of links
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Scheduling: Minimize latency � Partition links
• Given: Set of communication requests (“links”) • Find: Partition links into fewest feasible sets
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Which problem Æ Which model?
Models
Realism
Simplicity
Computational Complexity
Generality
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Interference model
But interference is not a binary relationship!
Disc Graphs
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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.
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Disc Graphs Fail
Length of link i = 2i [Moscibroda, Wattenhofer 2006]
Feasible set, but forms a clique in any disc graph
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Which problem Æ Which model?
Models
Realism
Simplicity
Computational Complexity
Generality
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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
56.03/14/1)( 2
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| wau
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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]
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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
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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“?
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Abstracting, solving, mapping back
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Price of abstraction
• Price of abstraction : How much you lose by solving the abstracted problem
(rather than solving directly)
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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.
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Hierarchies of abstraction
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Wireless „ground truth“
SINR model
Unweighted graphs
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Representing link scheduling with a graph
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Î
When should there be an edge?
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Requirement: Independent sets in G are feasible
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Independent sets should be feasible
valid coloring of G � valid scheduling
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Requirement II:
Î
Feasible linksets should be „nearly independent“ in G
S feasible � F(GS) small
S GS
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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
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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)
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Conflict graph representations [H,Tonoyan, STOC’15]
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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 (!)
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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
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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?
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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
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v u L
Every conflict graph schema is sandwich by formulations
𝑑 𝑢, 𝑤 ≤ 𝑓 𝑤𝑢
|𝑢|, where 𝑓 is a monotone function
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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
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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)
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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
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Modeling Reality
reflection
scattering
diffraction shadowing
Non-omnidirectional antennas
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Two-ray model
Slope = 2
Slope = 4
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Testbeds
Classroom (TB-20) Basement (TB-40)
[Gudmundsdottir, Asgeirsson, Bodlaender, Foley, H, Mitra, Vigfusson, MSWiM 2014]
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Do the SINR axioms hold (within reasonable errors)?
Additivity Thresholding
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The headache: Geometric pathloss
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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
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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 ?
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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)
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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.
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Collaborators
• Tigran Tonoyan
• Marijke Bodlaender
• Eyjólfur Ásgeirsson
Experimental group: • Helga Gudmundsdottir • Ýmir Vigfusson • Joe Foley
Alumni: • Pradipta Mitra
Roger Wattenhofer At ETH, Zurich:
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