Algorithms for Algorithms for Wireless Network DesignWireless Network Design

MohammadTaghi HajiAghayiMohammadTaghi HajiAghayi

Labs – ResearchLabs – Research

Purposes of this TalkPurposes of this Talk

Real-world applications with deep algorithmic Real-world applications with deep algorithmic underpinnings and consequencesunderpinnings and consequences

Present several problems motivated by (wireless Present several problems motivated by (wireless sensor) networkssensor) networks

Show how we can tie together theory and practiceShow how we can tie together theory and practice DemonstrateDemonstrate nice intersections nice intersections ofof wireless multi-wireless multi-

hop networkshop networks, , algorithmic graph theoryalgorithmic graph theory, , probability theory, probability theory, computational geometrycomputational geometry, , computational economics and finally computational economics and finally computational complexitycomputational complexity

OutlineOutline Focus on two real-world applicationsFocus on two real-world applications

Power optimization in fault-tolerant Power optimization in fault-tolerant topology control and related problemstopology control and related problems

The low coverage problem and related The low coverage problem and related

problemsproblems

ConclusionConclusion

  Power Optimization in Power Optimization in Fault-Tolerant Topology Fault-Tolerant Topology

ControlControl Wireless multihop networksWireless multihop networks

Simple low-power devicesSimple low-power devices Power is the Power is the main limitationmain limitation Power assignmentPower assignment

A power setting for each deviceA power setting for each device Defines possible communication linksDefines possible communication links

Power versus distance:Power versus distance: It takes power It takes power rrcc to transmit a message to distance to transmit a message to distance rr for some for some power attenuation exponentpower attenuation exponent cc between between 22 and and 44..

GoalGoal:: Minimize power usage while maintaining Minimize power usage while maintaining key network propertieskey network properties

Connectivity:Connectivity: There is a communication path There is a communication path between any pair of nodes between any pair of nodes

k-Fault tolerancek-Fault tolerance: Connectivity is maintained in : Connectivity is maintained in light of at mostlight of at most k-1 k-1 failures failures Device failures Device failures (our focus)(our focus) Communication link failuresCommunication link failures

By By k-Fault tolerancek-Fault tolerance,, we also have k-disjoint we also have k-disjoint paths and thus higher network capacitypaths and thus higher network capacity

Power Optimization in Power Optimization in Fault-Tolerant Topology Fault-Tolerant Topology

ControlControl

ModelModel

A A wireless networkwireless network is modeled as a graph is modeled as a graph G(V,E)G(V,E) with cost functions with cost functions d d and and pp on on EE V V is the set of is the set of mobile devicesmobile devices EE is the set of pairs of devices which can is the set of pairs of devices which can

communicatecommunicate bi-directionally bi-directionally dduvuv is the is the distancedistance between device between device uu and and vv ppuvuv is the is the powerpower needed to transmit between needed to transmit between

device device uu and and v v (usually it is distance to the (usually it is distance to the power attenuation exponent)power attenuation exponent)

ModelModel

Conversely, a subgraph Conversely, a subgraph H=(V,E’) H=(V,E’) of the of the network graph network graph GG defines an defines an assignment of power settingsassignment of power settings: device : device uu transmits at transmits at

p(u) = max p(u) = max {(u,v) in E’}{(u,v) in E’} p puvuv

The The power power used by a wireless network used by a wireless network with power settings defined by with power settings defined by HH is is

P(H) = P(H) = ΣΣ u in Vu in V p(u) p(u)

Problem FormulationProblem Formulation

GivenGiven A wireless networkA wireless network

FindFind An assignment of power settings that An assignment of power settings that

guarantees guarantees kk-fault tolerance while -fault tolerance while minimizing power usageminimizing power usage

Recall Recall kk-fault tolerance-fault tolerance means the network means the network remains connected even when up to remains connected even when up to k-1k-1

devices (or communication links) faildevices (or communication links) fail

Related Results for Power Related Results for Power MinimizationMinimization

ConnectivityConnectivity Cone-based local Cone-based local heuristicsheuristics

[Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02][Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02]

A A 22-approximation based on -approximation based on minimum weight minimum weight spanning treespanning tree

[Kerousis, Kranakis, Krizanc, Pelc ’00][Kerousis, Kranakis, Krizanc, Pelc ’00]

A A 1.691.69-approximation based on -approximation based on minimum minimum weight Steiner treeweight Steiner tree and a more and a more practicalpractical 1.8751.875-approximation-approximation [Calinescu, Mandoiu, Zelikovsky [Calinescu, Mandoiu, Zelikovsky ’02]’02]

Related Results for Related Results for Power MinimizationPower Minimization

22-Fault tolerance -Fault tolerance Heuristic Heuristic to minimize maximum transmit to minimize maximum transmit

power power

[Ramanathan, Rosales-Hain ’00][Ramanathan, Rosales-Hain ’00] (the only (the only previous result)previous result)

Fault tolerance for general Fault tolerance for general kk Pioneered in Pioneered in [Bahramgiri, H., Mirrokni, [Bahramgiri, H., Mirrokni,

WINET] WINET] andand [H., Immorlica, [H., Immorlica, Mirrokni,IEEE/ACM TON]Mirrokni,IEEE/ACM TON]

More than 100 referencesMore than 100 references

Cone-Based HeuristicCone-Based Heuristic Algorithm:

Input: A set of nodes on the plane, with max. power P Each node increases its power until the angle between any

two consecutive neighbors is less than some threshold or it reaches its maximum power P.

OutputOutput: two nodes are connected if both can hear each : two nodes are connected if both can hear each other with the new power assignmentother with the new power assignment

Theorem [BHM’02]: If the network of max. powers is k-connected and the angle between any pair of adjacent neighbors is at most 2π/3k, then the new network is k-connected (2π/3k is almost tight)

Main disadvantage:Main disadvantage: The algorithm is local and The algorithm is local and thus does not give any bound on the global goal of thus does not give any bound on the global goal of minimizing sum of the powers (or the average minimizing sum of the powers (or the average power)power)

Approximating ConnectivityApproximating Connectivity

Recall the Recall the powerpower P(H)P(H) of subgraph of subgraph HH is is

P(H) = P(H) = ΣΣ u in Vu in V p(u) p(u)

where where p(u) = max p(u) = max {(u,v) in H(E)}{(u,v) in H(E)} p puvuv

Define the Define the weightweight W(H)W(H) of subgraph H of subgraph H asas

W(H) = W(H) = ΣΣ (u,v) in H(E)(u,v) in H(E) p puvuv

Approximating ConnectivityApproximating Connectivity TheoremTheorem [KKKP ’00]:[KKKP ’00]: The minimum weight The minimum weight

spanning tree spanning tree MSTMST of of GG uses at most twice uses at most twice as much power as the minimum power as much power as the minimum power connected subgraph connected subgraph OPTOPT of of GG..

Lemma 1Lemma 1: For any graph : For any graph GG, , P(G) ≤ 2W(G)P(G) ≤ 2W(G).. Lemma 2Lemma 2: For any tree : For any tree TT, , W(T) ≤ P(T)W(T) ≤ P(T).. Lemma 3Lemma 3: : OPTOPT is a tree is a tree

Proof (of Thm)Proof (of Thm): From the above lemmas,: From the above lemmas,P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT)P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT)..

Approximating k-Approximating k-ConnectivityConnectivity

TheoremTheorem:: The minimum weight The minimum weight k-connected subgraph of G is a k-connected subgraph of G is a 2k-approximation to minimum power 2k-approximation to minimum power k-connected subgraphk-connected subgraph Proof sketchProof sketch: use Mader’s theorem to : use Mader’s theorem to

decompose a graph into k forests and then decompose a graph into k forests and then use the previous results for forests (trees) use the previous results for forests (trees)

Minimum weight k-connected Minimum weight k-connected subgraphsubgraph: : an an LP-based algorithmLP-based algorithm gives a gives a solution of weight at most solution of weight at most O(log k)O(log k) times times optimal weight (optimal weight (nn is at least is at least 6k6k22) ) [Cheriyan, [Cheriyan, Vempala, Vetta, STOC’02], Vempala, Vetta, STOC’02], [Kortsarz, Nutov, [Kortsarz, Nutov, STOC’04STOC’04]]

Approximating k-Approximating k-ConnectivityConnectivity

Thus there is an Thus there is an O(k log k)O(k log k) approximation approximation for minimum power k-connected for minimum power k-connected subgraphsubgraph

A more complicated combinatorial A more complicated combinatorial algorithm gives an algorithm gives an O(k)O(k) approximation approximation First we find a 2-approximation to the First we find a 2-approximation to the

minimum weight k-outconnected sub-graph minimum weight k-outconnected sub-graph using using matroid matchingmatroid matching

We can add We can add k-2k-2 disjoint paths to this graph disjoint paths to this graph in order to make it k-connected via a in order to make it k-connected via a min-min-cost k-flow algorithmcost k-flow algorithm

We pay a factor We pay a factor O(k)O(k) in each of the above in each of the above two steps to go from weight to powertwo steps to go from weight to power

Approximating k-Approximating k-ConnectivityConnectivity

Furthermore, a more subtle approach gives Furthermore, a more subtle approach gives O(logO(log44n)n) approximation, an improvement when k approximation, an improvement when k is large is large [H., Kortsarz, Mirrokni, Nutov, IPCO’05, also [H., Kortsarz, Mirrokni, Nutov, IPCO’05, also Math. Prog.’07]Math. Prog.’07]

Theorem:Theorem: If there are If there are c-approximation for min. weight k-connected sub.c-approximation for min. weight k-connected sub. d-approximation for min. power d-approximation for min. power k-edge coverk-edge cover

Then we have 2c+d-approximation for min powerThen we have 2c+d-approximation for min power k-edge cover: k-edge cover: sub-graph with min. degree ≥ ksub-graph with min. degree ≥ k c is in c is in O(log n)O(log n) by previous results by previous results d is in d is in O(logO(log44n)n) by a new involved algorithm by a new involved algorithm There are some hardness results for the There are some hardness results for the k-edge-k-edge-

connectivityconnectivity case (the best factor is in O(√n)) case (the best factor is in O(√n))

Distributed (Local) Distributed (Local) ApproximationApproximation

Algorithm: [H., Immorlica, Mirrokni, MOBICOM’03]Algorithm: [H., Immorlica, Mirrokni, MOBICOM’03] Construct minimum weight spanning tree with Construct minimum weight spanning tree with O(n log n + O(n log n +

m)m) messages messages [Gallager, Humbler, Spira, ’83][Gallager, Humbler, Spira, ’83] Use Use local augmentationlocal augmentation to create a to create a

k-connected sub-graph with k-connected sub-graph with O(n)O(n) messages messages

TheoremTheorem: If : If ppuvuv= (d= (duvuv))cc for all pairs of nodes and we for all pairs of nodes and we have metricity of distances, then the algorithm is an have metricity of distances, then the algorithm is an O(1)O(1)-approximation when k is constant.-approximation when k is constant.

Comparing via Comparing via simulationssimulations on random or real inputson random or real inputs Some other variants Some other variants [Bredin, Demaine, H., Rus, [Bredin, Demaine, H., Rus,

MobiHoc’05], [Demaine, H., Mahini, Oveis, Sayedi, MobiHoc’05], [Demaine, H., Mahini, Oveis, Sayedi, Zadi,SODA’07]Zadi,SODA’07]

A Related Problem: A Related Problem: Repairing Fault-ToleranceRepairing Fault-Tolerance

Problem FormulationProblem Formulation: : given an initial given an initial placement of nodes placement of nodes (the unit disk graph (the unit disk graph model)model)

Objective:Objective: Add minimum number of nodes Add minimum number of nodes to obtain to obtain

connectivityconnectivity Or minimize maximum/average movement Or minimize maximum/average movement of of

the current nodes without adding any new nodethe current nodes without adding any new node More generallyMore generally obtaining k-fault tolerance or obtaining k-fault tolerance or

k-connectivity of the whole networkk-connectivity of the whole network

Approximation Algorithms for Approximation Algorithms for Repairing the NetworkRepairing the Network

Minimum number of added nodesMinimum number of added nodes to obtain to obtain connectivity: 5/2-approximation connectivity: 5/2-approximation [Du, Wang, Xu, 2001][Du, Wang, Xu, 2001]

More generallyMore generally obtaining k-fault tolerance or k- obtaining k-fault tolerance or k-connectivity: connectivity: O(kO(k44)-)-approximation (approximation (SimpleSimple Algorithm, Algorithm, ComplicatedComplicated Analysis) Analysis)

[Bredin, Demaine, H., Rus, MobiHoc’05][Bredin, Demaine, H., Rus, MobiHoc’05] More generallyMore generally minimizing movement to obtain a new minimizing movement to obtain a new

configuration with a property configuration with a property PP (e.g. being connected (e.g. being connected being independent, having a perfect matching, etc.)being independent, having a perfect matching, etc.)

More formallyMore formally the goal of movement problem is to the goal of movement problem is to move the agents into a configuration containing at move the agents into a configuration containing at most h vertices that contain all k agents and induce a most h vertices that contain all k agents and induce a “good” target patterns, i.e., an induced graph, in the “good” target patterns, i.e., an induced graph, in the given set given set GG. Agents can have even different colors.. Agents can have even different colors.

Minimizing Movement:Minimizing Movement:Fixed-Parameter Tractability Fixed-Parameter Tractability

Fixed-parameter algorithms:Fixed-parameter algorithms:Parameterize problem by parameter PParameterize problem by parameter P

(typically, the cost of the optimal solution)(typically, the cost of the optimal solution)

and aim for f(P) nand aim for f(P) nO(1)O(1) (or even f(P) + n (or even f(P) + nO(1)O(1))) There is an FPT algorithm for Vertex There is an FPT algorithm for Vertex

CoverCover There is no FPT algorithm for Dominating There is no FPT algorithm for Dominating

SetSet

Minimizing Movement: FPT Minimizing Movement: FPT (cont’d)(cont’d)

CONNECTIVITY. Move the pebbles (agents) so that they are connected and on distinct vertices

GRID. Move the k2 pebbles so that they form a k × k grid.

s-t CONNECTIVITY. Move the pebbles to form a path of pebbled vertices between fixed vertices s and t.

STEINER CONNECTIVITY. Connect the red pebbles by moving the blue pebbles to form a Steiner tree.

Minimizing Movement: FPT Minimizing Movement: FPT (cont’d)(cont’d)

2-CONNECTIVITY. Move the pebbles so that they induce a 2-connected graph and the pebbles are on distinct vertices.

s-t d-CONNECTIVITY. Move the pebbles so that there are d vertex-disjoint paths using pebbled vertices between two fixed vertices s and t.

s-t d-EDGE-CONNECTIVITY. Move the pebbles so that there are d edge-disjoint paths of pebbled vertices between s and t.

FACILITY LOCATION. Move client and facility pebbles so that each client pebble is within distance at most d from at least one facility pebble.

MATCHING. Move the pebbles so that the pebbles are on distinct vertices and there is a perfect matching in the graph induced by the pebbles.

Minimizing MovementMinimizing Movement Essentially all these problems are polynomialy

hard to approximate or at least there is no algorithm better than √n-√n-approximatoin (for max connectivity and max path there are such algorithnms) [Demaine, H., Mahini, [Demaine, H., Mahini, Oveis, Sayedi, Zadi, SODA’07]Oveis, Sayedi, Zadi, SODA’07]

But all of them except GIRD, 2-CONNECTIVITY, and FACILITY LOCATION with unbounded number of clients have FPT algorithms which are quite surprising and interesting for practical purposes [Demaine, H.,Marx, [Demaine, H.,Marx, Submitted’07]Submitted’07]

OutlineOutline Focus on two real-world applicationsFocus on two real-world applications

Power optimization in fault-tolerant Power optimization in fault-tolerant topology control and related problemstopology control and related problems

The low coverage problem and related The low coverage problem and related

problemsproblems

ConclusionConclusion

Interference ProblemInterference Problem

The problem is a bit simplified formulation of a The problem is a bit simplified formulation of a real-world application in Bell-Labsreal-world application in Bell-Labs

Input:Input: Decomposition of the plane into regions with various Decomposition of the plane into regions with various

client densitiesclient densities nn base station locations, each with a set of options for base station locations, each with a set of options for

power/directional cones/etc and a cost for each onepower/directional cones/etc and a cost for each one Budget limiting the total cost of optionsBudget limiting the total cost of options Satisfaction sSatisfaction skk for covering a region by for covering a region by k> 0 k> 0 base base

stations where sstations where s11≥ s≥ s2 2 ≥ s≥ s3 3 ≥ . . . ≥ 0 (simplified version ≥ . . . ≥ 0 (simplified version ss11=1 and s=1 and s22=s=s33=. . .=0 is called =. . .=0 is called budgeted unique budgeted unique covercover.).)

Goal:Goal: Find a set of base stations and options Find a set of base stations and options within the total budget which maximizes the within the total budget which maximizes the total satisfaction weighted by client densitiestotal satisfaction weighted by client densities

The Low Coverage Problem, The Low Coverage Problem, MotivationMotivation

[Demaine, H., Feige, Salavatipur, SODA’06, [Demaine, H., Feige, Salavatipur, SODA’06, SICOMP]SICOMP]

The Unique Coverage Problem:The Unique Coverage Problem: Given a universe Given a universe U U ofof n n elements, and elements, and Given a collection Given a collection S S of subsets ofof subsets of U. U. Find a sub-collection Find a sub-collection S’ , S’ , a subset ofa subset of S, S, which which

maximizes the number of elements that are maximizes the number of elements that are uniquely uniquely coveredcovered, i.e., appear in exactly one set of , i.e., appear in exactly one set of S’S’

The Budgeted Unique Coverage Problem:The Budgeted Unique Coverage Problem: Given profits for elements and costs for subsets, andGiven profits for elements and costs for subsets, and Given also a budgetGiven also a budget B B Find a sub-collection Find a sub-collection S’, S’, a subset ofa subset of S, S, whose total whose total

cost is at mostcost is at most B B and maximizes the total profit of and maximizes the total profit of elements that are elements that are uniquely covereduniquely covered

The Unique Coverage Problem The Unique Coverage Problem [Demaine, H., Feige, Salavatipur, SODA’06][Demaine, H., Feige, Salavatipur, SODA’06]

The Unique Coverage Problem The Unique Coverage Problem

The budgeted case is almost equivalent to the The budgeted case is almost equivalent to the wireless network problemwireless network problem

It has the same flavor of the It has the same flavor of the maximum maximum coveragecoverage problem (has problem (has e/(e-1)e/(e-1)-approximation)-approximation)

Unique coverage is a generalization ofUnique coverage is a generalization of MAX-CUTMAX-CUT It has very close connections to the It has very close connections to the radio radio

broadcastbroadcast problem (considered extensively) problem (considered extensively) For simplicity, we focus on the (un-budgeted) For simplicity, we focus on the (un-budgeted)

unique coverage problem in the rest of the talk unique coverage problem in the rest of the talk

The Unique Coverage Problem The Unique Coverage Problem Simple Simple O(log n)O(log n) approximation approximation

algorithmalgorithm Partition the elements into Partition the elements into log nlog n classes classes

according to their according to their degrees, degrees, i.e., the number of i.e., the number of sets that cover an elementsets that cover an element

Let Let i i be the class of maximum cardinalitybe the class of maximum cardinality Choose a set in Choose a set in SS to be in to be in S’S’ with prob. with prob. 1/2 1/2 ii

Proof Sketch: Proof Sketch: we uniquely cover we uniquely cover 1/e1/e2 2

fraction of elements of class fraction of elements of class ii in expectation in expectation Can be de-randomized by the standard Can be de-randomized by the standard

method of conditional expectationmethod of conditional expectation Can be generalized for budgets, real-worldCan be generalized for budgets, real-world O(log B) O(log B) where where BB is the max size of a subset is the max size of a subset

(nontrivial)(nontrivial)

Hardness ResultHardness Result

The algorithm seems naïveThe algorithm seems naïve Several other problems have the same solutionSeveral other problems have the same solution Can we do better?Can we do better? It seems combination sometimes can be hardIt seems combination sometimes can be hard TheoremTheorem: This problem is hard to approximate : This problem is hard to approximate

within a factor better than within a factor better than O(logO(logcc n), 0<c<1 n), 0<c<1, , unless NP has a sub-exponential algorithm.unless NP has a sub-exponential algorithm.

O(logO(log1/31/3 n) n) hard or evenhard or even O(log n) O(log n) hard under hard under stronger but plausible complexity assumptionsstronger but plausible complexity assumptions

Proof IdeasProof Ideas A bad instance that we cannot A bad instance that we cannot

uniquely cover > 1/log n fractionuniquely cover > 1/log n fraction

O(log n)O(log n)

BB11

2 2 i i random random subsetssubsets

nn

BBi i : Class: Class i i

nn BBpp

BB: Elements: Elements

AA: Sets: Sets

In this graph, at In this graph, at most most O(n)O(n) elements elements of B can be uniquely of B can be uniquely covered by sets of Acovered by sets of A

Proof IdeasProof Ideas Bipartite Independent Set (BIS) problem:Bipartite Independent Set (BIS) problem:

Given a bipartite graph Given a bipartite graph G(A G(A UUB,E)B,E) where |A|=|B|=n where |A|=|B|=n Find a bipartite sub-graph Find a bipartite sub-graph G’(A’ G’(A’ UUB’, E’)B’, E’) where where |A’|=a, |B’|=b|A’|=a, |B’|=b

and and E’E’ is an empty set. is an empty set. Theorem: Theorem: Unless NP has sub-exponential algorithm, it Unless NP has sub-exponential algorithm, it

is hard to decide between is hard to decide between (n (n cc,n/log,n/logddn)-n)-BIS and BIS and (n (n c’c’ ,n/log,n/logd’d’ n)- n)-BIS where BIS where 0<c’<c ≤10<c’<c ≤1 and and 0 ≤ d<d’ ≤10 ≤ d<d’ ≤1

Now between Now between AA and and BBii we put a random matching and we put a random matching and an instance of BIS where the edges remain with an instance of BIS where the edges remain with probability 1/2probability 1/2i-1 i-1

ForFor Yes Yes instance, we have a unique cover of size instance, we have a unique cover of size ΩΩ(nlog(nlog1-d1-d n) n)

For For NoNo instance, we have a unique cover of size at instance, we have a unique cover of size at most O(nlogmost O(nlog1-d’1-d’n)n)

The inapproximability threshold can be improved under The inapproximability threshold can be improved under other stronger but still plausible complexity other stronger but still plausible complexity assumptionsassumptions

Other Aspects of the ResultsOther Aspects of the Results Unique coverage is the Unique coverage is the firstfirst natural natural maximizationmaximization problem problem

with almost logarithmic-hardness with almost logarithmic-hardness We believe that it can be a central problem for We believe that it can be a central problem for

maximization problems like maximization problems like set coverset cover for minimization for minimization problemsproblems

Toward this end, we obtain the same almost logarithmic Toward this end, we obtain the same almost logarithmic hardness for hardness for envy-free pricingenvy-free pricing, an important problem in , an important problem in computational economics considered by computational economics considered by [Guruswami, [Guruswami, Hartline, Karlin, Kempe, Kenyon, McSherry, SODA’05]Hartline, Karlin, Kempe, Kenyon, McSherry, SODA’05]

Other maximization candidates: deadline TSP Other maximization candidates: deadline TSP [BBCM, [BBCM, STOC’04]STOC’04] and budgeted connected sub-graph and budgeted connected sub-graph [Moss, [Moss, Rabani, STOC’01]Rabani, STOC’01] where both have O(log n) approximation where both have O(log n) approximation but not better so farbut not better so far

Finally, better models of interference for the real-world Finally, better models of interference for the real-world application also has been considered via some primal-dual application also has been considered via some primal-dual schemas and also schemas and also simulationssimulations on real-world inputs on real-world inputs [Bahl, [Bahl, H., Mirrokni, Qui, Saberi,IEEE TMC]H., Mirrokni, Qui, Saberi,IEEE TMC]

Application of Market Equilibrium in Distributed Load Balancing

[Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE [Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE TMC]TMC]

Wireless devicesWireless devices Cell-phones, laptops with WiFi cardsCell-phones, laptops with WiFi cards Referred as clients or users Referred as clients or users

interchangeablyinterchangeably Demand connections to access pointsDemand connections to access points

Uniform for cell-phones (voice connection)Uniform for cell-phones (voice connection) Non-uniform for laptops (application Non-uniform for laptops (application

dependent)dependent)

Application of Market Equilibrium in Distributed Load Balancing

[Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE [Bahl, H., Jain, Mirrokni, Qui, Saberi,IEEE TMC]TMC]

Access pointsAccess points Cell-towers, Base stations, Wireless routersCell-towers, Base stations, Wireless routers

CapacitiesCapacities Total traffic they can serveTotal traffic they can serve Integer for cell-towersInteger for cell-towers

Variable transmission powerVariable transmission power Capable of operating at various power Capable of operating at various power

levelslevels Assume levels are continuous real numbersAssume levels are continuous real numbers

Clients to APs assignmentClients to APs assignment

Assign clients to APs in an efficient Assign clients to APs in an efficient wayway No over-loading of APsNo over-loading of APs Assigning the maximum number of Assigning the maximum number of

clients and thus satisfying maximum clients and thus satisfying maximum demanddemand

One Heuristic SolutionOne Heuristic Solution

A client connects to the AP with A client connects to the AP with reasonable signal and then the lightest reasonable signal and then the lightest loadload Requires support both from AP and ClientsRequires support both from AP and Clients APs have to communicate their current loadAPs have to communicate their current load Clients have WiFi cards from various Clients have WiFi cards from various

vendors running legacy softwarevendors running legacy software Overall it has limited benefit in practiceOverall it has limited benefit in practice

Ideal CaseIdeal Case

We would like a client connects to the AP We would like a client connects to the AP with the best received signal strengthwith the best received signal strength

If an AP If an AP jj transmitting at power level transmitting at power level PPjj then a client then a client ii at distance d at distance dijij receives receives signal with strengthsignal with strength

PPijij = = a.Pa.Pjj.d.dijij-c-c

where where aa and and cc are constants capturing are constants capturing various models of power attenuation various models of power attenuation

Cell Breathing HeuristicCell Breathing Heuristic

An overloaded AP decreases its An overloaded AP decreases its communication radius by decreasing powercommunication radius by decreasing power

A lightly loaded AP increases its A lightly loaded AP increases its communication radius by increasing powercommunication radius by increasing power

Hopefully an equilibrium would be reached Hopefully an equilibrium would be reached Will show that an equilibrium exist Will show that an equilibrium exist Can be computed in polynomial timeCan be computed in polynomial time Can be reached by a tatonnement processCan be reached by a tatonnement process

Lets start withLets start with economics and game theory economics and game theory

Market Equilibrium – A Market Equilibrium – A distributed load balancing distributed load balancing

mechanism.mechanism. Fisher setting with linear Utilities:Fisher setting with linear Utilities: m buyers (each with budget Bm buyers (each with budget Bii) and n goods for sale ) and n goods for sale

(each with quantity q(each with quantity qjj)) Each buyer has linear utility Each buyer has linear utility uuii, i.e. utility of i is , i.e. utility of i is

sumsumjj u uij ij xxijij where u where uijij>= 0 is the utility of buyer i for >= 0 is the utility of buyer i for good j and xgood j and xijij is the amount of good j bought by i. is the amount of good j bought by i.

A A market equilibriummarket equilibrium or or market clearancemarket clearance is a is a price vector price vector p p that that maximizes utility maximizes utility sumsumjj u uij ij xxijij of buyer i subject to his of buyer i subject to his

budgetbudget sum sumjj p pj j xxijij <= B <= Bii

The demand and supply for each good j are equalThe demand and supply for each good j are equal

sumsumjj x xij ij = q= qj j (and thus the budgets are totally spent).(and thus the budgets are totally spent).

Fisher Setting with Linear Fisher Setting with Linear UtilitiesUtilitiesBuyers Goods

1 1 1 1, j jj

M u u x

, i i ij ijj

M u u x

, n n nj njj

M u u x

1q

jq

mq

ijx

ij iju x

Market Equilibrium – A Market Equilibrium – A distributed load balancing distributed load balancing

mechanism.mechanism. Static supplyStatic supply

corresponding to capacities of APscorresponding to capacities of APs PricesPrices

corresponding to powers at APscorresponding to powers at APs Utilities Utilities

Analogous to received signal strength functionAnalogous to received signal strength function Either all clients are served or all APs are Either all clients are served or all APs are

saturatedsaturated Analogous to the market clearance(equiblirum) Analogous to the market clearance(equiblirum)

conditioncondition Thus our situation is analogous to Fisher setting Thus our situation is analogous to Fisher setting

with linear utilitieswith linear utilities

Clients assignment to APsClients assignment to APsClients APs

1 1 1 1, max j jj

D u P x

, maxi i ij ijj

D u P x

, maxn n nj njj

D u P x

1C

jC

mC

ijx

ij ijP x

Analogousness Is Only Analogousness Is Only InspirationalInspirational

Get inspiration from various Get inspiration from various algorithms for the Fisher setting and algorithms for the Fisher setting and develop algorithms for our settingdevelop algorithms for our setting

Though we do not know any Though we do not know any reduction – in fact there are some reduction – in fact there are some key differenceskey differences

Differences from the Market Differences from the Market Equilibrium settingEquilibrium setting

DemandDemand Price dependent in Market equilibrium settingPrice dependent in Market equilibrium setting Power independent in our settingPower independent in our setting

Is demand splittable?Is demand splittable? Yes for the Market equilibrium settingYes for the Market equilibrium setting No for our settingNo for our setting

Market equilibrium clears both sides but our Market equilibrium clears both sides but our solution requires clearance on either client side solution requires clearance on either client side or AP sideor AP side This also means two separate linear This also means two separate linear

programs for these two separate casesprograms for these two separate cases

Three Approaches for Three Approaches for Market EquilibriumMarket Equilibrium

Convex Programming BasedConvex Programming Based Eisenberg, Gale 1957Eisenberg, Gale 1957

Primal-Dual BasedPrimal-Dual Based Devanur, Papadimitriou, Saberi, Vazirani Devanur, Papadimitriou, Saberi, Vazirani

20042004 Auction BasedAuction Based

Garg, Kapoor 2003Garg, Kapoor 2003

Three Approaches for Three Approaches for Load BalancingLoad Balancing

Linear ProgrammingLinear Programming Minimum weight complete matchingMinimum weight complete matching

Primal-DualPrimal-Dual Uses properties of bipartite graph Uses properties of bipartite graph

matchingmatching AuctionAuction

Useful in dynamically changing situationUseful in dynamically changing situation

Linear Programming Based Linear Programming Based Solution Solution

Create a complete bipartite graphCreate a complete bipartite graph One side is the set of all clientsOne side is the set of all clients The other side is the set of all APs, The other side is the set of all APs,

conceptually each AP is repeated as conceptually each AP is repeated as many times as its capacity (unit many times as its capacity (unit demand)demand)

The weight between client The weight between client i i and AP and AP jj is is wwijij = = c.c.ln(ln(ddijij) – ln() – ln(aa)) Find the minimum weight complete Find the minimum weight complete

matchingmatching

TheoremTheorem

Minimum weight matching is supported Minimum weight matching is supported by a power assignment to APsby a power assignment to APs

Power assignment are the dual Power assignment are the dual variablesvariables

Two cases for the primal program Two cases for the primal program which is known at the beginningwhich is known at the beginning Solution can satisfy all clientsSolution can satisfy all clients Solution can saturate all APsSolution can saturate all APs

Case 1 – Complete Case 1 – Complete matching covers all clientsmatching covers all clients

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

ij ji C

ij

w x

x

j A x C

i C j A x

Case 1 – Pick Dual VariablesCase 1 – Pick Dual Variables

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

i

ij j ji C

ij

w x

x

j A x C

i C j A x

Write Dual ProgramWrite Dual Program

maximize

subject to

,

0

i j ji C j A

i j ij

j

C

i C j A w

j A

xxijij

Optimize the dual programOptimize the dual program

Choose Choose PPjj = e = e ππjj

Using the complementary slackness Using the complementary slackness condition we will show that the condition we will show that the minimum weight complete matching minimum weight complete matching is supported by these power levelsis supported by these power levels

ProofProof

Dual feasibility gives:Dual feasibility gives:

--λλi i ≥ ≥ ππjj – w – wijij= = ln(ln(PPjj) – ) – c.c.ln(ln(ddijij) + ln() + ln(aa) = ln() = ln(a.Pa.Pjj.d.dijij--

cc)) Complementary slackness gives:Complementary slackness gives:

xxijij=1=1 implies implies --λλi i = ln(= ln(a.Pa.Pjj.d.dijij-c-c))

(Remember if an AP (Remember if an AP jj transmitting at power level transmitting at power level PPjj then then a client a client ii at distance d at distance dijij receives signal with strength receives signal with strength PPijij = = a.Pa.Pjj.d.dijij

-c)-c)

Together they imply that Together they imply that ii is connected to the AP is connected to the AP with the strongest received signal strengthwith the strongest received signal strength

Case 2 – Complete Case 2 – Complete matching saturates all APsmatching saturates all APs

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

ij ji C

ij

w x

x

j A x C

i C j A x

Case 2 – The rest of the proof Case 2 – The rest of the proof is similaris similar

Unsplittable DemandUnsplittable Demand

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

i ij ji C

ij

w x

x

j A D x C

i C j A x

Unsplittable DemandUnsplittable Demand

The integer program is APX-hard in The integer program is APX-hard in general (because of knapsack)general (because of knapsack)

Assuming that the number of clients is Assuming that the number of clients is much larger than the number of APs, a much larger than the number of APs, a realistic assumption, we can obtain a realistic assumption, we can obtain a nice approximation heuristic.nice approximation heuristic.

First we compute a basic feasible First we compute a basic feasible solutionsolution

Analysis of Basic Feasible Analysis of Basic Feasible SolutionSolution

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

i ij ji C

ij

w x

x

j A D x C

i C j A x

Approximate SolutionApproximate Solution

All All xxijij’s but a small number of ’s but a small number of xxijij’s are integral’s are integral Theorem: Number of Theorem: Number of xxijij which are integral is which are integral is

at least the number of clients – the number at least the number of clients – the number APsAPs

Most clients are served unsplittablyMost clients are served unsplittably Clients which are served splittably – do not Clients which are served splittably – do not

serve themserve them The algorithm is almost optimalThe algorithm is almost optimal

Discrete Power LevelsDiscrete Power Levels

Over the shelf APs have only fixed number Over the shelf APs have only fixed number of discrete power levelsof discrete power levels

Equilibrium may not existEquilibrium may not exist In fact it is NP-hard to test whether it exists or In fact it is NP-hard to test whether it exists or

notnot If every client has a deterministic tie If every client has a deterministic tie

breaking rule then we can compute the breaking rule then we can compute the equilibrium – if exists under the tie breaking equilibrium – if exists under the tie breaking rulerule

Discrete Power LevelsDiscrete Power Levels

Start with the maximum power levels for Start with the maximum power levels for each APeach AP

Take any overloaded AP and decrease its Take any overloaded AP and decrease its power level by one notchpower level by one notch

If an equilibrium exist then it will be If an equilibrium exist then it will be computed in time computed in time mkmk, where , where mm is the is the number of APs and number of APs and kk is the number of is the number of power levelspower levels

This is a distributed tatonnement This is a distributed tatonnement process!process!

Proof.Proof.

Suppose Suppose PPjj is an equilibrium power level is an equilibrium power level for the for the jjthth AP. AP.

Inductively prove that when Inductively prove that when jj reaches the reaches the power level power level PPjj then it will not be then it will not be overloaded again.overloaded again. Here we use the deterministic tie breaking Here we use the deterministic tie breaking

rule.rule.

Reconstruction ProblemReconstruction Problem Geometric graph structureGeometric graph structure

Given:Given: Information about local Information about local geometrygeometry E.g., approximate distancesE.g., approximate distances

Goal:Goal: Reconstruct global geometry Reconstruct global geometry

Sensor network

Cricket v2

Motivation: Cricket Location Motivation: Cricket Location SystemSystem

[MIT SLAM Project since 2000][MIT SLAM Project since 2000] [Badoiu, Demaine, H., Indyk, SOCG’04, [Badoiu, Demaine, H., Indyk, SOCG’04,

DCG]DCG] Cricket devices “chirp” radio & Cricket devices “chirp” radio & ultrasoundultrasound Radio travels at speed of lightRadio travels at speed of light Ultrasound travels at speed of soundUltrasound travels at speed of sound

MeasureMeasuredistancesdistances

LocalizedLocalized Accuracy ~1cmAccuracy ~1cm

? ?

??

General Embedding ProblemGeneral Embedding Problem

Given graph with edge weightsGiven graph with edge weights Embed into desired metric spaceEmbed into desired metric space

(e.g., 2D or 3D) with edge lengths (e.g., 2D or 3D) with edge lengths matching specified weightsmatching specified weights

Weights may be exact orWeights may be exact orapproximateapproximate

Simple only if we know all Simple only if we know all the distances exactlythe distances exactly

Embedding with Approx. Embedding with Approx. DistancesDistances Exact embedding likely impossibleExact embedding likely impossible

Goal:Goal: Minimize maximum distortion Minimize maximum distortion Additive:Additive: |embed length − true length| |embed length − true length| Multiplicative:Multiplicative: |embed length / true length| |embed length / true length|

Sample results:Sample results: [Badoiu, Demaine, H., Indyk, SOCG’04], [Badoiu, Demaine, H., Indyk, SOCG’04], Embed any Embed any

metric on n points into Euclidean 2D plane with metric on n points into Euclidean 2D plane with multiplicative/additive distortion (by knowing multiplicative/additive distortion (by knowing anglesangles) ) or or quasi-polynomial timequasi-polynomial time if we do not have the if we do not have the anglesangles

[Alon, Badoiu, Demaine, Farach-Colton, H., [Alon, Badoiu, Demaine, Farach-Colton, H., Sidiropoulos, SODA’05], Sidiropoulos, SODA’05], Embed any metric on n Embed any metric on n points into low dimensions such that we preserve points into low dimensions such that we preserve the the orderorder approximately approximately

Approximating best embedding into plane in Approximating best embedding into plane in polynomial time is a very important open polynomial time is a very important open problemproblem

ConclusionsConclusions Introduction and motivationsIntroduction and motivations Focused on two real-world applicationsFocused on two real-world applications

Power optimization in fault-tolerant Power optimization in fault-tolerant topology control and related problemstopology control and related problems

The low coverage problem and related The low coverage problem and related problemsproblems

Several other related areas to wireless Several other related areas to wireless networksnetworks Geometric embedding & Reconstruction Geometric embedding & Reconstruction

ProblemProblem Market equiblirum & Load BalancingMarket equiblirum & Load Balancing Game theory & Low CoverageGame theory & Low Coverage

Questions ?Questions ?

Thanks for your attention…Thanks for your attention…

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