Seminar PPT

Algorithms for Algorithms for Wireless Network DesignWireless Network Design

Santhosh Reddy S.V

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

Show how we can tie together theory and Show how we can tie together theory and practicepractice

DemonstrateDemonstrate nice intersections nice intersections ofof wireless wireless multi-hop networksmulti-hop 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

Key WordsKey Words

NetworkNetwork –a collection of autonomous –a collection of autonomous computerscomputers

Ad-hoc NetworksAd-hoc Networks Critical PowerCritical Power -The universal -The universal

minimum power used by all wireless minimum power used by all wireless nodes such that the induced network nodes such that the induced network Topology is connected is called the Topology is connected is called the critical powercritical power..

Fault ToleranceFault Tolerance

Important to wireless networksImportant to wireless networks Prone to link failureProne to link failure Prone to node failureProne to node failure Approach in wired networkApproach in wired network K-vertex/edge connectivity with K-vertex/edge connectivity with

minimum weightminimum weight NP hard problem NP hard problem Interference ProblemInterference Problem

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

ControlControl Wireless multihop networksWireless multihop networks

Simple low-power devicesSimple low-power devicesPower is the main limitationPower is the main limitation

Power assignmentPower assignmentA 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 for some powersome power attenuation exponentattenuation exponent c c between between 22 and and 44..

GoalGoal:: Minimize power usage while maintaining Minimize power usage while maintaining key network properties.key 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 failures.Communication link failures. By By k-Fault tolerancek-Fault tolerance,, we also have k-disjoint we also have k-disjoint

paths and thus higher network capacity.paths and thus higher network capacity.

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

ControlControl

Interference ProblemInterference Problem

Preliminaries and ModelPreliminaries and Model Oriented transmittersOriented transmitters Omni-directionalOmni-directional Multi-hop networksMulti-hop networks

Static and BidirectionalStatic and Bidirectional

ModelModel A A wireless networkwireless network is modeled as a is modeled as a

graph graph G(V,E)G(V,E) with cost functions with cost functions d d and and pp on on EEV V is the set of is the set of mobile devicesmobile devicesEE is the set of pairs of devices which can is the set of pairs of devices which can

communicatecommunicate bi-directionally bi-directionallydduvuv is the is the distancedistance between device between device uu and and vvppuvuv is the is the powerpower needed to transmit needed to transmit

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

DEFINITION 3:DEFINITION 3:An Undirected Minimum Power k-Vertex An Undirected Minimum Power k-Vertex Connected Sub graph (k-UPVCS) of a graph Connected Sub graph (k-UPVCS) of a graph G=(V,E)G=(V,E) is a is a k-vertex connected sub graph k-vertex connected sub graph H=(V,F),F H=(V,F),F E, E, such that such that PP(H) (H) P(H') P(H') for any k-vertex connected sub graph for any k-vertex connected sub graph H'=(V,F'), F'H'=(V,F'), F'E.E.

DEFINITION 4:DEFINITION 4: An Undirected Minimum Cost k-Vertex An Undirected Minimum Cost k-Vertex Connected Sub graph (k-UCVCS) of a graph Connected Sub graph (k-UCVCS) of a graph G= (V, E)G= (V, E) is is a k-vertex connected sub graph a k-vertex connected sub graph H= (V, F), F H= (V, F), F E, E, such such That CThat C(H) (H) C(H') C(H') for any k-vertex connected sub graph for any k-vertex connected sub graph H'=(V,F'), F'H'=(V,F'), F'E.E.

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 k-fault tolerancek-fault tolerance means the means the network remains connected even when network remains connected even when

up to up to k-1k-1 devices (or communication devices (or communication links) faillinks) fail

Power MinimizationPower Minimization

ConnectivityConnectivityCone-based local heuristicsCone-based local heuristics

CBTC AlgorithmCBTC Algorithm

Global approximationGlobal approximation Lemma 1Lemma 1

Lemma 2Lemma 2

Lemma 3Lemma 3

Distributed approximationDistributed approximation

Cone-Based HeuristicCone-Based Heuristic Algorithm:

Input: A set of nodes on the plane, with maxpower 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 : two nodes are connected if both can hear each other with the new power assignmenteach other 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 power)minimizing sum of the powers (or the average power)

Global ApproximationGlobal Approximation

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) ≤ 2C(G)P(G) ≤ 2C(G).. Lemma 2Lemma 2: For any tree : For any tree TT, , C(T) ≤ P(T)C(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)..

Lemma 1:Lemma 1: For any graph G, P (G) ≤ 2C (G).For any graph G, P (G) ≤ 2C (G).

Proof:Proof: The proof is straightforward from the The proof is straightforward from the following inequalities.following inequalities.

Lemma 2:Lemma 2: For any treeFor any tree T, C (T) ≤P (T).T, C (T) ≤P (T).

Proof:Proof: Root T at an arbitrary vertex r. Note the Root T at an arbitrary vertex r. Note the Power of each node is at least the cost of its Power of each node is at least the cost of its parent edge. The statement follows.parent edge. The statement follows.

Lemma 3: For any graph G which can be written as Lemma 3: For any graph G which can be written as a union of t forests, C (G) ≤ t P (G)a union of t forests, C (G) ≤ t P (G)

Algorithm Global k-UPVCS (G (V, E))Algorithm Global k-UPVCS (G (V, E))// choose arbitrary root r// choose arbitrary root rrrVV//and covering set F' using subroutine A(r,G)//and covering set F' using subroutine A(r,G)H, F' H, F' A(r, G)A(r, G)for for (u,v)(u,v) F' F'//find k vertex disjoint paths Fuv with the cheapest//find k vertex disjoint paths Fuv with the cheapest//(normal) cost from u to v in G//(normal) cost from u to v in GFuvFuvk vertex disjoint paths with cheapest costk vertex disjoint paths with cheapest costendend//replace edges in cover by the sets of//replace edges in cover by the sets of//cheapest k vertex disjoint paths//cheapest k vertex disjoint pathsforfor (u,v) (u,v) F' F'HH H H Fuv Fuvendendoutput Gk=Houtput Gk=HFigure 2: A formal description of Algorithm Global k-UPVCSFigure 2: A formal description of Algorithm Global k-UPVCS

Distributed Approximation Distributed Approximation

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

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)

More generallyMore generally minimizing movement to obtain a minimizing movement to obtain a new configuration with a property new configuration with a property PP (e.g. being (e.g. being connected being independent, having a perfect connected being independent, having a perfect matching, etc.)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 move the agents into a configuration containing at most h vertices that contain all k agents and at most h vertices that contain all k agents and induce a “good” target patterns, i.e., an induced induce a “good” target patterns, i.e., an induced graph, in the given set graph, in the given set GG. Agents can have even . Agents can have even different colors.different colors.

Performance EvaluationPerformance Evaluation

Figure : Cone-Based Topology Control (ο) (low performance), Distributed k-Figure : Cone-Based Topology Control (ο) (low performance), Distributed k-UPVCS (+) (middle performance), and Global k-UPVCS (*) (high UPVCS (+) (middle performance), and Global k-UPVCS (*) (high performance). These graphs depict EER (Expended Energy Ratio) versus performance). These graphs depict EER (Expended Energy Ratio) versus density.density.

ConclusionsConclusions

Previous heuristics and approaches do not give us Previous heuristics and approaches do not give us good approximation factors.good approximation factors.

Distributed algorithm is more suitable for Distributed algorithm is more suitable for static ad-static ad-hoc networks.hoc networks.

Add minimum number of nodes to obtain Add minimum number of nodes to obtain connectivity.connectivity.

Or minimize maximum/average movement of the Or minimize maximum/average movement of the current nodes without adding any new node.current nodes without adding any new node.

More generally obtaining More generally obtaining k-fault tolerance or k-k-fault tolerance or k-connectivityconnectivity of the whole network. of the whole network.

Questions?Questions?