Interference and Topology Interference and Topology ControlControl

Does Topology Control Reduce Interference ?Does Topology Control Reduce Interference ?

Feb 15, 2010Feb 15, 2010University of FreiburgUniversity of Freiburg

H. K. Al-HasaniH. K. Al-Hasani

Seminar Ad Hoc NetzwerkeSeminar Ad Hoc NetzwerkeProf. Dr. Christian SchindelhauerProf. Dr. Christian Schindelhauer

1.1.MotivationMotivation-Interference-Interference

2.2.Goals desired to achieveGoals desired to achieve-Spanner graph-Spanner graph

3.3.Problems and study Problems and study casescases

4.4.AlgorithmsAlgorithms-LIFE-LIFE-LISE-LISE

-LLISE-LLISE

5.5.ConclusionConclusion

Interference and Topology Interference and Topology ControlControl

MotivationMotivation

How to construct the network so that :How to construct the network so that :How to construct the network so that :How to construct the network so that :

• All vertices are connected.All vertices are connected.• Minimum energy consuming, toMinimum energy consuming, to• Extend network lifetime.Extend network lifetime.• Collision handling.Collision handling.• Low Interference.Low Interference.

• All vertices are connected.All vertices are connected.• Minimum energy consuming, toMinimum energy consuming, to• Extend network lifetime.Extend network lifetime.• Collision handling.Collision handling.• Low Interference.Low Interference.

x y

How the communication is establishedHow the communication is establishedHow the communication is establishedHow the communication is established

InterferenceInterference

Scenario:Scenario:Scenario:Scenario:

x y

• |x,|x,yy| refers to the transmission discs radii of x,y .| refers to the transmission discs radii of x,y .• A connection will be established if both radii were A connection will be established if both radii were

identical.identical.• Two nodes x,y are connected if their edge is symmetric.Two nodes x,y are connected if their edge is symmetric.• In this scenario 9 nodes are influenced by this In this scenario 9 nodes are influenced by this

communication including x and y.communication including x and y.• Interference of a node is the number of cycles in which Interference of a node is the number of cycles in which

this node is located.this node is located.

• |x,|x,yy| refers to the transmission discs radii of x,y .| refers to the transmission discs radii of x,y .• A connection will be established if both radii were A connection will be established if both radii were

identical.identical.• Two nodes x,y are connected if their edge is symmetric.Two nodes x,y are connected if their edge is symmetric.• In this scenario 9 nodes are influenced by this In this scenario 9 nodes are influenced by this

communication including x and y.communication including x and y.• Interference of a node is the number of cycles in which Interference of a node is the number of cycles in which

this node is located.this node is located.

Cov (e) = 9Cov (e) = 9

InterferenceInterference

Definition:Definition:Definition:Definition:

x y

Given a graph G (V,E)Given a graph G (V,E)

I (G) = max (Cov (e)) , e I (G) = max (Cov (e)) , e ϵ E E

Lowering a graphs interference is excluding all possible Lowering a graphs interference is excluding all possible edges with high coverages without loosing connectivity.edges with high coverages without loosing connectivity.

Given a graph G (V,E)Given a graph G (V,E)

I (G) = max (Cov (e)) , e I (G) = max (Cov (e)) , e ϵ E E

Lowering a graphs interference is excluding all possible Lowering a graphs interference is excluding all possible edges with high coverages without loosing connectivity.edges with high coverages without loosing connectivity.

What kind of graph can guarantee that What kind of graph can guarantee that ??

GoalsGoals

Desired graphDesired graph::Desired graphDesired graph::

• Low interferenceLow interference• ConnectivityConnectivity• Spanner graphSpanner graph• Planar graphPlanar graph

• Low interferenceLow interference• ConnectivityConnectivity• Spanner graphSpanner graph• Planar graphPlanar graph

Spanner graphSpanner graph

Given a graph G(V,E)Given a graph G(V,E)G* would be a t-spanner for V ifG* would be a t-spanner for V if

|x,y| |x,y| PG* PG* ≤≤ t.t.|x,y| |x,y| PG where t is the stretch factorPG where t is the stretch factor

The path between these pair of nodes in The path between these pair of nodes in G* should be equal to at most the G* should be equal to at most the shortest path between them in the shortest path between them in the original graph G time the stretch factor t.original graph G time the stretch factor t.

Given a graph G(V,E)Given a graph G(V,E)G* would be a t-spanner for V ifG* would be a t-spanner for V if

|x,y| |x,y| PG* PG* ≤≤ t.t.|x,y| |x,y| PG where t is the stretch factorPG where t is the stretch factor

The path between these pair of nodes in The path between these pair of nodes in G* should be equal to at most the G* should be equal to at most the shortest path between them in the shortest path between them in the original graph G time the stretch factor t.original graph G time the stretch factor t.

Besides spanner graph, what else we Besides spanner graph, what else we need to consider need to consider ??

• SparsenessSparseness• Chain of nodesChain of nodes• Worst case graphWorst case graph• Planar graphPlanar graph

Study casesStudy cases

SparsenessSparseness

If the graphs degree could be lowered, If the graphs degree could be lowered, then the interference will be lowered as then the interference will be lowered as wellwell

...?...?

If the graphs degree could be lowered, If the graphs degree could be lowered, then the interference will be lowered as then the interference will be lowered as wellwell

...?...?

Claim :Claim :Claim :Claim :

Low degree graphLow degree graph

SparsenessSparseness

If the graphs degree could be lowered, then the If the graphs degree could be lowered, then the interference will be lowered as wellinterference will be lowered as well

The claim doesn‘t hold, the interference of The claim doesn‘t hold, the interference of this graph is O(n).this graph is O(n).

If the graphs degree could be lowered, then the If the graphs degree could be lowered, then the interference will be lowered as wellinterference will be lowered as well

The claim doesn‘t hold, the interference of The claim doesn‘t hold, the interference of this graph is O(n).this graph is O(n).

Claim :Claim :Claim :Claim :

Chain of nodesChain of nodes

Set of n vertices V, in between the distance grows Set of n vertices V, in between the distance grows exponentially.exponentially.|x|xii,x,xi+1i+1|= 2|= 2i i where I {1,…,n} where I {1,…,n}

|x|xii,x,xi+1i+1| is the radius of these nodes discs :| is the radius of these nodes discs :

When xWhen xii,x,xi+1i+1 communicate, they interfere all other communicate, they interfere all other

nodes in range (1,…i) nodes in range (1,…i)

Set of n vertices V, in between the distance grows Set of n vertices V, in between the distance grows exponentially.exponentially.|x|xii,x,xi+1i+1|= 2|= 2i i where I {1,…,n} where I {1,…,n}

|x|xii,x,xi+1i+1| is the radius of these nodes discs :| is the radius of these nodes discs :

When xWhen xii,x,xi+1i+1 communicate, they interfere all other communicate, they interfere all other

nodes in range (1,…i) nodes in range (1,…i)

ii i+i+11

i+i+22

i+i+33

22ii

Some approaches will choose the greedy way, Some approaches will choose the greedy way, i.e. connect the nearest neighbor.i.e. connect the nearest neighbor.

Minimum Spanning Tree, or Related Neighbor Minimum Spanning Tree, or Related Neighbor Graph will push the interference of such a graph Graph will push the interference of such a graph to to ΩΩ(n).(n).

Some approaches will choose the greedy way, Some approaches will choose the greedy way, i.e. connect the nearest neighbor.i.e. connect the nearest neighbor.

Minimum Spanning Tree, or Related Neighbor Minimum Spanning Tree, or Related Neighbor Graph will push the interference of such a graph Graph will push the interference of such a graph to to ΩΩ(n).(n).

ii i+i+11

i+i+22

i+i+33

Current solution :Current solution :Current solution :Current solution :

Chain of nodesChain of nodes

Ask for help from the nodes in similar situation!Ask for help from the nodes in similar situation!

|x|xii,x,xi+1i+1|= 2|= 2i i where I {1,…,n} where I {1,…,n}

|u|uii,u,ui+1i+1|= 2|= 2ii where I {1,…,n} where I {1,…,n}

Pick a helping node v so that:Pick a helping node v so that:|x|xii,u,uii|< |x|< |xii,v| < |x,v| < |xii,u,uii||

Ask for help from the nodes in similar situation!Ask for help from the nodes in similar situation!

|x|xii,x,xi+1i+1|= 2|= 2i i where I {1,…,n} where I {1,…,n}

|u|uii,u,ui+1i+1|= 2|= 2ii where I {1,…,n} where I {1,…,n}

Pick a helping node v so that:Pick a helping node v so that:|x|xii,u,uii|< |x|< |xii,v| < |x,v| < |xii,u,uii||

ii i+i+11

i+i+22

i+i+33

Suggested solution :Suggested solution :Suggested solution :Suggested solution :

Chain of nodesChain of nodes

Constant interferenceConstant interferenceConstant interferenceConstant interference

uui+i+

11

xxi+1i+1xxii

uuii

vv

Worst case Worst case graphgraph

Given a graph G (V,E)Given a graph G (V,E)The vertices are located equally to each other.The vertices are located equally to each other.Except two nodes x,y :Except two nodes x,y :|x,y| >|x,y| > |u,v| where { u,v |u,v| where { u,v ϵϵ V } V }For connectivity reason, this edge must be set.For connectivity reason, this edge must be set.The edge The edge |x,y| increases the interference.|x,y| increases the interference.

Given a graph G (V,E)Given a graph G (V,E)The vertices are located equally to each other.The vertices are located equally to each other.Except two nodes x,y :Except two nodes x,y :|x,y| >|x,y| > |u,v| where { u,v |u,v| where { u,v ϵϵ V } V }For connectivity reason, this edge must be set.For connectivity reason, this edge must be set.The edge The edge |x,y| increases the interference.|x,y| increases the interference.

xxyy aa

Planar graphPlanar graph

A Planar graph, is a graph A Planar graph, is a graph in which no two edges in which no two edges intersect.intersect.

Planarity doesn‘t have to Planarity doesn‘t have to increase / decrease graphs increase / decrease graphs interference.interference.

Our desired graph might Our desired graph might be a planar one.be a planar one.

A Planar graph, is a graph A Planar graph, is a graph in which no two edges in which no two edges intersect.intersect.

Planarity doesn‘t have to Planarity doesn‘t have to increase / decrease graphs increase / decrease graphs interference.interference.

Our desired graph might Our desired graph might be a planar one.be a planar one.

aa

bb

aa

bb

a,b are small groups of nodesa,b are small groups of nodesa,b are small groups of nodesa,b are small groups of nodes

Then how can the optimal graph be Then how can the optimal graph be constructed constructed ??

Algorithms Algorithms

• LIFELIFE• LISELISE• LLISELLISE

AlgorithmsAlgorithms

LIFELIFE

Low Interference Forest EstablisherLow Interference Forest EstablisherLow Interference Forest EstablisherLow Interference Forest Establisher

Hence the algorithm constructs Minimum Hence the algorithm constructs Minimum Spanning Trees, the constructed graph is a Spanning Trees, the constructed graph is a forest.forest.If two nodes in range of each other, pick the If two nodes in range of each other, pick the edge with minimum coverage.edge with minimum coverage.

Input :Input :Set of vertices, no more requirementsSet of vertices, no more requirements..Running time : O (nRunning time : O (n22 log n) log n)

Hence the algorithm constructs Minimum Hence the algorithm constructs Minimum Spanning Trees, the constructed graph is a Spanning Trees, the constructed graph is a forest.forest.If two nodes in range of each other, pick the If two nodes in range of each other, pick the edge with minimum coverage.edge with minimum coverage.

Input :Input :Set of vertices, no more requirementsSet of vertices, no more requirements..Running time : O (nRunning time : O (n22 log n) log n)

cc

bb ee

aa

ff

ggddaa 33bb 22cc 22dd 33ee 44ff 33gg 55

Simulating LIFESimulating LIFESimulating LIFESimulating LIFE

Given a set of vertices and their Given a set of vertices and their maximum transmission radius:maximum transmission radius:

ee

ggcc

bb

aa

ff

dd

The connectivity is violated.The connectivity is violated.

There are three MST, which There are three MST, which are not connected.are not connected.

The connectivity is violated.The connectivity is violated.

There are three MST, which There are three MST, which are not connected.are not connected.

ccbbee

aaffggdd

ee

ggcc

bb

aa

ff

dd

Then the constructed graph Then the constructed graph would look like …would look like …

Simulating LIFESimulating LIFESimulating LIFESimulating LIFE

LISELISE

Low Interference Spanner EstablisherLow Interference Spanner EstablisherLow Interference Spanner EstablisherLow Interference Spanner Establisher

The algorithm constructs Spanner graph.The algorithm constructs Spanner graph.Shortest path is computed separately.Shortest path is computed separately.Low weighted (low coverage) edges are inserted Low weighted (low coverage) edges are inserted to the graph.to the graph.

Input :Input :Set of vertices, Stretch factor (t).Set of vertices, Stretch factor (t).Running time : polynomial.Running time : polynomial.

The algorithm constructs Spanner graph.The algorithm constructs Spanner graph.Shortest path is computed separately.Shortest path is computed separately.Low weighted (low coverage) edges are inserted Low weighted (low coverage) edges are inserted to the graph.to the graph.

Input :Input :Set of vertices, Stretch factor (t).Set of vertices, Stretch factor (t).Running time : polynomial.Running time : polynomial.

LLISELLISE

Local Low Interference Spanner EstablisherLocal Low Interference Spanner EstablisherLocal Low Interference Spanner EstablisherLocal Low Interference Spanner Establisher

The algorithm applied locally for edge e.The algorithm applied locally for edge e.Search within t/2 neighbors for eligible paths.Search within t/2 neighbors for eligible paths.Find the optimal interference path among these Find the optimal interference path among these paths.paths.Inform the neighbors to use this path only.Inform the neighbors to use this path only.

Input :Input :Set of vertices, stretch factor.Set of vertices, stretch factor.

The algorithm applied locally for edge e.The algorithm applied locally for edge e.Search within t/2 neighbors for eligible paths.Search within t/2 neighbors for eligible paths.Find the optimal interference path among these Find the optimal interference path among these paths.paths.Inform the neighbors to use this path only.Inform the neighbors to use this path only.

Input :Input :Set of vertices, stretch factor.Set of vertices, stretch factor.

ConclusionConclusion

1.1. Stretch factorStretch factor1.1. Stretch factorStretch factor

For some kind of graphs, stretch factor is known.Compute the stretch factor considered to be very expensive.

For some kind of graphs, stretch factor is known.Compute the stretch factor considered to be very expensive.

2.2. Networks knowledgeNetworks knowledge2.2. Networks knowledgeNetworks knowledge

Either to find the neighbors, or to find the shortest path.In LIFE it is required to set the edge with minimum coverage.

Either to find the neighbors, or to find the shortest path.In LIFE it is required to set the edge with minimum coverage.

LISE and LLISE are a good start although certain LISE and LLISE are a good start although certain information is required :information is required :LISE and LLISE are a good start although certain LISE and LLISE are a good start although certain information is required :information is required :

ConclusionConclusion

3.3. Shortest path computationShortest path computation3.3. Shortest path computationShortest path computation

The running time of the algorithms depends on the implementation of this point.

The running time of the algorithms depends on the implementation of this point.

The proofs trigger many arguments!The proofs trigger many arguments!The proofs trigger many arguments!The proofs trigger many arguments!

Algorithms performance is close to the optimal if the Algorithms performance is close to the optimal if the stretch factor has large value. Nevertheless for stretch factor has large value. Nevertheless for small values the algorithms performance doesn’t small values the algorithms performance doesn’t differ from Related neighbor Graphs.differ from Related neighbor Graphs.

Algorithms performance is close to the optimal if the Algorithms performance is close to the optimal if the stretch factor has large value. Nevertheless for stretch factor has large value. Nevertheless for small values the algorithms performance doesn’t small values the algorithms performance doesn’t differ from Related neighbor Graphs.differ from Related neighbor Graphs.

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Thank you!Thank you!