David S. L. Wei

Dept of Computer and Information SciencesFordham University

Bronx, New York

Szu-Chi Wang and Sy-Yen Kuo

Dept of Electrical EngineeringNational Taiwan University

Taipei, Taiwan

Joint work with

Introduction

Wireless ad hoc networks are - Characterized by scarce resources - Prone to topology changes - Lack of physical infrastructure

The flexibility and mobility of wireless ad hoc networks make them suitable for applications such as automated battlefields and disaster rescues

Introduction (cont.)

A wireless ad hoc network can be modeled by an undirected/directed graph G = (V, E)

Power conservation has been widely used as a primary control parameter in the design of protocols for wireless ad hoc networks

A

B

C

D E

F

A C

F

B

D E

Introduction (cont.)

A C

F

B

D EA

B

C

D E

F

A C

FB

D EA

B

C

D E

F

Motivations

Each node in a wireless ad hoc network can potentially change the network topology by adjusting its transmission range

The primary goal of topology control is to design power-efficient algorithms that

- Maintain network connectivity - Optimize performance metrics (network lifetime, throughput,…)

Preliminaries

In the most common power-attenuation model

- The transmission power between node u and v is denoted as

- All receivers have the same power threshold for signal detection

We assume that - Each mobile host has a low-power GPS receiver - Initially all the nodes are operated at full transmitter power ─ the resulted graph G is a unit-disk graph (denoted as UDG (V))

t ||uv|| + rp(u,v)

transmitter power

receiver power

Preliminaries (cont.)

Hereinafter we use G to present a wireless ad hoc network - The edge weight is defined as w(u, v) = t||uv|| + rp(u,v) - We also call G the transmission graph

A path from node u to node v is denoted as

The total transmission power of this path is

(u, v) = v0v1…vh-1vh where u = v0 and v = vh

h

iii vvwvup

11 ) ,()),((

Transmission Power Assignment

A transmission power assignment on the vertices is a function f from V into real numbers

Given a graph H = (V’, E’) , the transmission power assignment f is induced by H if for each node v V’,

The total transmission power of f is defined as

A transmission power assignment f is complete if the associated graph Gf is strongly connected

),(max)(),(

uvwvfEuv

Vv ii

vf )(

Minimum-Energy Path

Given a communication graph H G, the minimum-energy path between node u and node v, denoted by H

min(u, v), is a path

whose total transmission power is the minimum among all paths that connect (u, v) in H

Let pH(u, v) stand for p(Hmin(u, v)), the power stretch factor of

H with respect to G is defined as

),(

),(max)(

,, vup

vupGpsf

G

H

vuVvuH

Related Works

Our major work is to develop a localized topology control algorithm where each node makes a decision about its transmission power based on only its local information

The two widely used energy conservation approaches in literature are to - Reduce the total transmission power - Reduce the power stretch factor

However, these two approaches may offset each other

The problem of finding a complete f whose total transmission power is the minimum among all of the complete assignments is called the min-total assignment problem

The min-total assignment problem is NP-hard when the nodes are deployed in a d-dimensional space, d 2

The general structure of the minimum-power topology for rp 0 is still unknown

Related Works (cont.)

Basic Ideas

The proposed algorithm is based on the following ideas - First construct a connected subgraph H = (V, E’) - Assure that the power stretch factor of H is bounded - The total transmission power is then minimized as much as possible

We use the local information of each node to excise some links of G while still keeps the power stretch factor being bounded by a predetermined value cb

Our Localized Topology Control Algorithm

The proposed algorithm consists of two phases - Phase I: Local shortest tree construction - Phase II: Path search replacement

A simple illustrative example is shown below

p1

p2

p3

p4

p5

p6

p7

p1

p2

p3

p4

p5

p6

p7

p1

p2

p3

p4

p5

p6

p7

Phase I of the Proposed Algorithm

Definition 1 (Local Topology View) The local topology view of node u, LTV (u, k) = (V’, E’)), is a subgraph of G such that (1) vi V’ if the hop distance between vi and u is no more than k (2) (vi, vj) E’ if both vi and vj belong to V’

Suppose that a subgraph of G is associated with a transmission power assignment f

- For each node u, if link (u, v) satisfies w (u, v) = f (u) then (u, v) is called a critical link of u

Phase I of the Proposed Algorithm (cont.)

Each node u individually applies Dijkstra’s algorithm to get the shortest-paths from the source u to the other nodes in LTV (u, 1)

The local shortest path tree of node u (denoted by LSPT (u)) can thus be obtained

DC (u) = {v V’ | h (LSPT (u), v) = 1}, where h (LSPT (u), v) is the height of a child node v in LSPT (u)

Node u then deletes the edges {(u, w) | w DC (u)}

The topology generated is denoted as GI

Phase II of the Proposed Algorithm

For each node u, its transmission power could be further reduced by trying to eliminate the critical links that are replaceable with alternative paths

For each critical link (u, v), node u tries to search another path that reaches node v based on LTV (u, k)

- We call such path the replacing path of (u, v) - The entire replacing paths of node u is denoted as RP (u)

The searching procedure is applying Dijkstra’s algorithm again on LTV (u, k)

Phase II of the Proposed Algorithm (cont.)

If no such path exists or no replacing path has transmission power cb w (u, v)

- The search process is ended - RP (u) is set as an empty list - ps (u) is set to 0

The priority of node u is a pair pri(u) = <ps(u), ID(u)>

- pri(v1) = (ps(v1), ID1), pr (v2) = (ps(v2), ID2) - pri(v1) > pri(v2) if ps(v1) > ps(v2) (ps(v1) = ps(v2) ID1 < ID2)

The above procedure for deciding RP (u) starts if node u has the highest priority in its k-hop neighborhood

Phase II of the Proposed Algorithm (cont.)

After Phase I, each node deletes its uni-directional links and the resulting topology is denoted as GII

The constructed topology after Phase II is denoted as GIII

A simple heuristic for further decreasing the total transmission power is also proposed

Important Properties

The minimum-energy path between any two nodes in G is preserved in GI

The minimum-energy path between the two end nodes of each deleted link in GI is preserved in GII

GII preserves the network connectivity of G

is bounded by cb

GIII preserves the network connectivity of G and has a bounded power stretch factor cb

)(III GpsfG

Dealing with Mobility

We also consider the case of modest movement of the nodes

It would be extremely difficult for a topology control algorithm to even effectively guarantee network connectivity if network topology changes too fast

As mentioned in previous works, node movement can be viewed as two events, namely node addition and node deletion

In our case, however, at the beginning of each beacon interval each node u should check if there is a change in transmission radius after deciding the new logical links

Performance Comparisons

We compared the performance via extensive simulations

We observe the following metrics of each constructed topology H

- Total transmission power (denoted by tpc) - Power stretch mean (denoted by psm)

- The maximum power stretch factor (denoted by max psf) - The variance of transmission power (denoted by var tp) - Average node degree (denoted by avg nd) - The maximum node degree (denoted by max nd)

Simulation ResultsThe Performance Measurements with s = 1 The Performance Measurements with s = 3

The Performance Measurements with s = 5 The Performance Measurements with s = 7

tpc psm

s = 1

max psf avg nd max nd

1.0

0.000328

1.0

1.0

0.000123 1.03482

0.000666 1.0

0.000099 1.1967

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99171

1.0

9.70118

96.5312

2.6996

2.1356

3.5992

1.98

99

6

5

8

4

0.000178 1.0ESPT1 1.0 2.4252 5

0.000137 1.016ESPT2 , cb=1.5 1.49991 2.2036 4

0.0

1.20E+15

1.27E+14

4.68E+15

9.30E+13

2.43E+14

1.52E+14

var tp

0.000253 1.01491AMST 2.51846 2.3386 58.76E+14

tpc psm

s = 1

max psf avg nd max nd

1.0

0.000328

1.0

1.0

0.000123 1.03482

0.000666 1.0

0.000099 1.1967

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99171

1.0

9.70118

96.5312

2.6996

2.1356

3.5992

1.98

99

6

5

8

4

0.000178 1.0ESPT1 1.0 2.4252 5

0.000137 1.016ESPT2 , cb=1.5 1.49991 2.2036 4

0.0

1.20E+15

1.27E+14

4.68E+15

9.30E+13

2.43E+14

1.52E+14

var tp

0.000253 1.01491AMST 2.51846 2.3386 58.76E+14

tpc psm

s = 3

max psf avg nd max nd

1.0

0.026874

1.0

1.0

0.010102 1.03485

0.05339 1.0

0.010074 1.08511

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99906

1.0

6.31301

25.5408

2.7016

2.1372

3.59

2.0258

53

6

5

8

4

0.01495 1.0ESPT1 1.0 2.4284 5

0.011188 1.01516ESPT2 , cb=1.5 1.49855 2.2044 4

0.0

8.10E+18

8.56E+17

2.82E+19

2.01E+18

2.60E+18

9.95E+17

var tp

0.020503 1.01413AMST 2.46733 2.3388 45.71E+18

tpc psm

s = 3

max psf avg nd max nd

1.0

0.026874

1.0

1.0

0.010102 1.03485

0.05339 1.0

0.010074 1.08511

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99906

1.0

6.31301

25.5408

2.7016

2.1372

3.59

2.0258

53

6

5

8

4

0.01495 1.0ESPT1 1.0 2.4284 5

0.011188 1.01516ESPT2 , cb=1.5 1.49855 2.2044 4

0.0

8.10E+18

8.56E+17

2.82E+19

2.01E+18

2.60E+18

9.95E+17

var tp

0.020503 1.01413AMST 2.46733 2.3388 45.71E+18

tpc psm

s = 5

max psf avg nd max nd

1.0

0.159539

1.0

1.0

0.086227 1.03671

0.245597 1.0

0.10403 1.03698

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.9971

1.0

4.38695

10.4232

2.6596

2.1564

3.4496

2.135

27

5

5

8

4

0.122629 1.0ESPT1 1.0 2.45 5

0.09287 1.01372ESPT2 , cb=1.5 1.49999 2.2236 5

0.0

1.47E+20

6.51E+19

2.29E+20

9.96E+19

1.08E+20

6.87E+19

var tp

0.128633 1.01305AMST 2.49099 2.3204 41.15E+20

tpc psm

s = 5

max psf avg nd max nd

1.0

0.159539

1.0

1.0

0.086227 1.03671

0.245597 1.0

0.10403 1.03698

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.9971

1.0

4.38695

10.4232

2.6596

2.1564

3.4496

2.135

27

5

5

8

4

0.122629 1.0ESPT1 1.0 2.45 5

0.09287 1.01372ESPT2 , cb=1.5 1.49999 2.2236 5

0.0

1.47E+20

6.51E+19

2.29E+20

9.96E+19

1.08E+20

6.87E+19

var tp

0.128633 1.01305AMST 2.49099 2.3204 41.15E+20

tpc psm

s = 7

max psf avg nd max nd

1.0

0.299838

1.0

1.0

0.228194 1.03382

0.380877 1.0

0.254784 1.0228

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99931

1.0

3.14811

5.6392

2.44

2.1096

2.9548

2.1376

14

6

5

8

4

0.27593 1.0ESPT1 1.0 2.3468 5

0.237019 1.0135ESPT2 , cb=1.5 1.4938 2.158 5

0.0

3.01E+20

2.32E+20

3.46E+20

2.71E+20

2.86E+20

2.40E+20

var tp

0.261474 1.01752AMST 2.8558 2.1856 42.66E+20

tpc psm

s = 7

max psf avg nd max nd

1.0

0.299838

1.0

1.0

0.228194 1.03382

0.380877 1.0

0.254784 1.0228

UDG

SMECN

ESPT2 , cb=2.0

GG

LMST

1.0

1.0

1.99931

1.0

3.14811

5.6392

2.44

2.1096

2.9548

2.1376

14

6

5

8

4

0.27593 1.0ESPT1 1.0 2.3468 5

0.237019 1.0135ESPT2 , cb=1.5 1.4938 2.158 5

0.0

3.01E+20

2.32E+20

3.46E+20

2.71E+20

2.86E+20

2.40E+20

var tp

0.261474 1.01752AMST 2.8558 2.1856 42.66E+20

Network topologies constructed by various algorithms I

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by UDG

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by GG

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by AMST

UDG SMEN

GG AMST

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by SMECN

Network topologies constructed by various algorithms II

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by ESPT2, cb = 1.5

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by ESPT2, cb = 2.0

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by ESPT1LMST

ESPT2, cb = 1.5

ESPT1

ESPT2, cb = 2.0

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Topology by LMST

Conclusions

In this paper we develop a localized algorithm that requires only local information for constructing a logical topology on a given unit disk graph

The topology constructed by our algorithm has several desired features such as bounded power stretch factor, low total power consumption, and small variance of transmission power

The simulation results show that our algorithm outperforms others in terms of various important metrics

Future Research

• Power-aware topology control

• Topology control of ad-hoc networks in three-dimensional space

• Secure topology control algorithm

• Applications in overlay control for P2P communications

Thanks for Your AttentionThanks for Your Attention