1

Minimum-energy broadcasting in multi-hop wireless networks using a single broadcast tree

Department of Computer Science and Information Engineering National Cheng Kung University, Taiwan R.O.C.

Authors: Ioannis Papadimitriou and Leomidas Georgiadis

Publisher: Mobile Networks and Applications 11, 361–375, 2006

Present: Min-Yuan Tsai (蔡旻原 )

Date: October, 16, 2007

2

Outline

1. Introduction 2. Definitions and problem description 3. Broadcasting using a single

broadcast tree 4. Numerical results 5. Conclusion

3

Introduction

In this paper, the authors focus on the problem of energy-efficient broadcasting in wireless networks where omni-directional antennas are used and there is flexibility of power adjustment.

Common solution depends on the source node that initiates the broadcast request. (Every time a node needs to broadcast some information to all other nodes in the network, the algorithm for the broadcast tree construction is executed for the specific source node.)• Each node in the network has to keep track of n broadcast

trees, one for each of the possible source nodes (n is the number of nodes in the network).

• large memory space

4

Introduction (cont.)

The authors proposed minimum-energy broadcasting using a single broadcast tree(SBT) to simplified the above situation .

Advantages of the proposed scheme

• General networks

• Independence of the source node – considerable simplification, scaling

• Approximation ratio close to best achievable bound in polynomial time

5

Outline

1. Introduction 2. Definitions and problem

description 3. Broadcasting using a single

broadcast tree 4. Numerical results 5. Conclusion

6

Model for Wireless Broadcasting

Definition:

• G(N , L): connected undirected graph

• cl > 0: power needed for successful transmission over link l=(i , j)• If node i transmits with power p, it can reach any node j f

or which c (i , j) ≤ p

• Ts=(N , LTs): s-rooted directed spanning tree that induced by undirected tree T

• node s transmits with power

7

Model for Wireless Broadcasting (cont.)

Example:• T : {(A,B) , (A,C) , (B,D)} (undirected)

• TA and TD (directed) are induced by T

• , C & D are leaves node

in TA

• , (D,B) is outgoing

link of D in TD

8

The Minimum-Energy Broadcast Problem : total power consumed for broadcasting

from source node s In general, for different source nodes, the trees that mi

nimize the sum of node powers are different (each node has to keep track of |N| broadcast trees, one for every possible source)

To simplify the above situation we have to “Find a single (undirected) spanning tree T to be used by all nodes for broadcasting, such that the sum of consumed node powers P(Ts) is minimized for any source node s”.• Each node needs to store only a small set of links that belon

g to tree T• Processing of broadcast information is minimal (scaling to lar

ger networks)

Ni

TisspTP )(

9

The Minimum-Energy Broadcast Problem (cont.) Two open issues: If all broadcasts take place on the

same tree, then• Issue 1 : Certain broadcasts may need much more total

power than others, depending on the source node (widely varying total power consumption for different source nodes).

• Issue 2 : If one attempts to find a tree for which the total powers consumed for broadcasting initiated by different source nodes are approximately the same, then, for a given source node, the resulting total power may be far away from the optimal.

10

Outline

1. Introduction 2. Definitions and problem description 3. Broadcasting using a single

broadcast tree 4. Numerical results 5. Conclusion

11

Broadcasting using a single broadcast tree Addressing Issue 1 : We use the following Corollary.

• If the same spanning tree T is used for broadcasting by all nodes, then the total broadcast power consumption for source node s is at most twice the total broadcast power consumption for any other source node s΄, P( Ts ) ≤ 2P( Ts΄ ).

Addressing Issue 2 : We propose a polynomial time approximation algorithm for the construction of a single broadcast tree .• For any source node s, the total power consumed for broad

casting using tree has an approximation ratio 2H(n-1) with respect to the optimal power.• Approximation ratio close to the best achievable bound in poly

nomial time (n=|N| is the number of nodes in the network and H(n) is the harmonic function)

12

Broadcasting using a single broadcast tree (cont.) Single Broadcast Tree (SBT) algorithm:

• At each iteration, SBT maintains a forest of trees in the network, such that each node belongs to a forest tree.

• Initially, each node constitutes a forest tree.• The forest is expanded by joining trees through

nodes, so that the “incremental power consumed per joined tree” is minimal.

• This is achieved by examining the adjacent links of every node i in the network that terminate outside the tree to which node i belongs.

• The algorithm terminates when the forest consists of a single (undirected) spanning tree.

13

Broadcasting using a single broadcast tree (cont.) Example of SBT algorithm :

• Node imin is selected to be joined with the forest tress TF1 and TF2.

• Link lmin joins tree TFmin with TF1 .

• Only one of the links (imin , m) , (imin , n) must be selected to join tree TFmin with TF2 to avoid the creation of cycle.

14

Broadcasting using a single broadcast tree (cont.) A minimum spanning tree (MST) of G is a spanning t

ree whose sum of link costs is minimal.• For any source node s, the total power consumed

for broadcasting using a minimum spanning tree, is at most Δ times the optimal power, where Δ is the maximum node degree in the network.

• For any source node s, the total power consumed for broadcasting using a single broadcast tree has an approximation ratio 2H(n-1) with respect to the optimal power where H(n) is the harmonic function.

MST may be a good candidate for broadcasting in sparse networks.

15

Outline

1. Introduction 2. Definitions and problem description 3. Broadcasting using a single

broadcast tree 4. Numerical results 5. Conclusion

16

Numerical results Algorithms compared:

• 1) Broadcast Incremental Power algorithm (“BIP”) • 2) Single Broadcast Tree (“SBT”)• 3) Minimum Spanning Tree (“MST”)

Networks: • 1) With a specified number of nodes (20,40,…,100) in a

rectangular grid of 100×100 points (The power needed

for successful transmission over link (i, j ) depends on

the distance d(i,j) between the two nodes and it is given by c(i,j) = da(i,j ), where a is the propagation loss exponent.)• 2) “Special” nodes added to the grid – 3-dimensional

network Performance metric: Average total broadcast power con

sumption

17

Numerical results (cont.)

a = 2 , complete networks a = 4 , complete networks BIP determines a different broadcast tree for every possible so

urce node, while SBT algorithm constructs a single tree used by all nodes for broadcasting.

Average tree power of SBT is slightly larger than that of BIP. The difference in performance of the algorithms vanishes for l

arger values of a. The “penalty” of using longer links increases and all algorithms converge to MST.

18

Numerical results (cont.)

The power of a link between the “special” node and any other node on the grid at distance d is f d2, where f is a factor 0 < f ≤ 1.

There is a range of values of f for which SBT outperforms significantly BIP.

SBT succeeds in selecting links of “special” node when they are more cost efficient.

Ratio of avg. tree power of SBT to BIP, a = 2, 100-node sparse networks + 1 “special” node

a = 2, 1 “special” node added to the sparse networks, factor f = 0.1

19

Numerical results SBT algorithm performs fairly well, compared to BIP

algorithm, for networks represented by unit disk graphs, while using a single broadcast tree.

There are interesting instances of general networks, for which SBT algorithm significantly outperforms BIP and MST.

MST algorithm performs worse for most of the network instances considered.

SBT algorithm presents a good compromise between simplicity and achieved performance.

20

Outline

1. Introduction 2. Definitions and problem description 3. Broadcasting using a single

broadcast tree 4. Numerical results 5. Conclusion

21

Conclusion

The main contribution of this paper is that we do not have to determine a different broadcast tree every time a source node initiates a broadcast request.

Some further study:• Network environments with high mobility and frequent

topological changes.

• Energy-limited and resource-limited environment, Lifetime maximization.

• Dynamic power assignments