About the Minimum Energy Broadcasting in Wireless Ad Hoc Networks

Alfredo Navarra, PhD Laboratoire Bordelais de Recherche en Iinformatique (LaBRI) University of Bordeaux, France navarra@labri.fr

Minimum Energy Broadcasting Signal Attenuation: A radio station s with transmission

power PS= βr α reaches all the stations at distance at most r. (for some constant β and usually 2≤α≤6 ).

Multi-hop communication: The messages are transferred from the source to the destination using intermediate radio stations.

Model: Given a set of nodes V in the Euclidean d-dimensional space and a source sV,let G=(V,E) be the complete symmetric directed graph obtained from V, in which every edge (x,y) has weight βdist(x,y)α. Choose a subset of edges of G in such a way that every node is

reached from s. (The transmission power associated to each node x is the weight of the longest outgoing edge from x)

Goal: minimize1

i

n

xi

P

Ex.: Power Assignment in 2-dimension with α=2

2i

i

r

S

ii

cost r Minimum Energy Broadcasting is

in general NP-hard

MST-heuristic: Compute the Minimum Spanning Tree over

the graph G. From s, assign to each node the power

equal to the square of the length of the longest “outgoing” edge.

BIP-heuristic: Starting from s, chose the cheapest way to reach a

node either adding a new edge (MST) or “increasing” an old one.

ABC-heuristic: A sort of “backtracking” is added. The idea is still to reach nodes step by step starting

from s and chosing the cheapest way to reach a new node, either adding a new edge (MST,BIP) or increasing an old one (BIP) but also with the possibility of REMOVING old edges if become useless.

At each step the new node is discovered according to the Prim’ MST order.

At each step, two invariants must be valid: Every node is covered by some circle Every node admits an induced path back to s

An example

SS S

MST BIP ABC

Lemma 1. Given a set of nodes V over the Euclidean 2-dimensional space and a source s belonging to V,

cost(ABC(s,V)) ≤ cost(MST(s,V)) ABC is not always better than BIP

S S

ABC BIP

15 random nodes in a 5x5 square

ILP Formulation

i,t i,ti s, d =0 and x =1

50 random nodes in a 5x5 square

Experimental and Analytic Results, Lower bound

MST (Minimum Spanning Tree) 6 BIP (Broadcast Incremental Power) 13/3 ABC (Adaptive Broadcast Consumption) 2 …

Upper bound in the 2-dimensional case MST, 40-approx (Clementi, Crescenzi, Penna Rossi, Vocca – STACS 2001)

MST, 20-approx (Clementi, Crescenzi, Penna Rossi, Vocca – STACS 2001)

MST, 12-approx (Calinescu, Li, Frieder, Wan – INFOCOM 2001)

MST, 12.5-approx (Klasing, Navarra, Papadopoulos, Perennes – Networking 2004)

MST, 7.45-approx (Flammini, Klasing, Navarra, Perennes – DIALM-POMC 2004)

MST, 6.33-approx (Navarra – WiOpt 2005)

α≥d=2 case

Reminding the MST-heuristic

Compute the Minimum Spanning Tree over the complete graph G.

From the source s, assign to each node the power proportional to the weight of the longest “outgoing” edge.

The following McDiarmid et al. (1998) relation was exploited

0

dc 1-c)N(G, MST(G)

r = r1

# CC = 9

r2

8

r3

7

rmax

1

Assume that each node is inside a circle of radius R=1

R

MOREOVER

2max44rMSTAreaTot

2maxr

R

2

0

12

4

maxr

TotArea N r rdr

2

2max

Tot

rArea R

8-Approximation2

maxTot

2max 2

r1πArear

4

πMST

4

π

maxr14MST

8MST

rmax ≤ 1

13 MST d

More in general, for any dimension d and any α≥d :

For the 2-dimensional case the ratio was first reduced to7.45 by means of geometricaltechniques.

Modifying the Shape

• Inside the circle of radius 1 that we call c(Q), the shape remains the same.• Outside, the new shape occupies the same area while the height decreases.

Properties and Approximation 1

For any d1≤1 and d2≤1 such that 1-r ≤ d1≤ d2, h(r,d1) ≤ h(r, d2)

For any d≤1, h(r,d)≤.3638… h(r,d) ≥ 3/5r For any d1≤1 and d2≤1 such that 1-r ≤ d1≤ d2,

(r,d1) ≥ (r,d2)

MST(G) ≤ 4(1+h(.5,1))2 – 1 ≤ 6.4401…< 6.45

A Further Improvement

• If the station z is on the circumference of c(Q), the outside sector is enlarged according to the radii tangent to its associated circle.• Else it is enlarged according to the radii tangent to the circle centered on the circumference of c(Q) having the same intersections of the original shape associated to z.

Enlarging the sector decreases the height

Properties and Approximation 2

For any d1≤1 and d2≤1 such that 1-r ≤ d1≤ d2,

h(r,d1) ≤ h(r,d2) For any d≤1, h(r,d)≤.3527… h(r,d) ≥ 3/5r For any d1≤1 and d2≤1 such that 1-r ≤ d1≤ d2,

(r,d1) ≥ (r,d2)

MST(G) ≤ 4(1+h(.5,1))2 – 1≤ 6.3203…< 6.33

Remarks

Small gap 6 ≤ MST ≤ 6.33. Considering the lower bound

case, the new associated area fullfil the external sector defined by c(Q) and the circle of radius 1+hmax/2. Assuming 6 as the real bound, the loss of .33 with respect to it must be found then in the “holes” inside c(Q).

21 maxh

Indeed…

The MST approximation bound is already reduced to the optimal one, that is, 6.C. Ambuhl, in Proceedings of ICALP 2005The analysis is based on Delauney

triangulation We had our glory… let’s go back to

simulations and higher dimensions!!!

Reminding about the 8-Approximation and more…

maxr14MST rmax ≤ 1

MST ≤ 8

For rmax → 0 MST ≤ 4

2

maxTot

2max 2

r1πArear

4

πMST

4

π

Worst case for the MST heuristic

5 1.301 2.875

7 1.479 2.479

10 1.802 3.123

15 1.887 2.669

20 1.854 2.618

30 1.825 2,232

50 1.812 1.972

100 1.683 1.883

How was it found? Probably not randomly!!Throwing 6 nodes randomly at uniform in a circle of radius 1 and considering as source the center of such a circle, it is really “lucky” to happen that a similar high cost instance appears.

n Av. | Max

GOAL

Investigate more carefully the possible input instances in order to better understand this phenomenon.The Idea is to start from random instances and then

increase the cost of the MST heuristic by slight movements of the nodes.

Given an edge of the MST, increasing the distance between its endpoints “usually” increases the total cost.

Allowed Movements

More in general for any v≠s, let N(v)={v1…vk} be its neighborood, we compute the median point p=(x,y):

A further way to increase the cost of the MST is to try to delete a node. We choose as candidate the node with highest degree.

Augmenting Algorithm (sketch)flagi1= 1; flag2=1; N=|V|-1; i=1; j=1;Compute MST over G2(V), save its cost in cost1;While flag2≤N do While flag1≤N do if vi is not on the circumference then let v’i be a point inside C1 on the line passing through vi and p in such a way that |vi,p| < |v’i,p|≤(1+ε•rand)|vi,p|; else let v’i be a point on the circumference further from p with respect to

vi such that the arc joining vi and v’i has length ε•rand; Compute the MST over G2((V \ vi)U v’I), save its cost in cost2; if cost2>cost1 then ... let vj be the j-th highest degree node of the current MST, compute the MST over G2(V \ vj), save its cost in cost2; if cost2>cost1 then ...

Experimental Results

5 1.301 2.875 3.645 4 3.627 4

7 1.479 2.479 4,545 5.738 4.56 5.88

10 1.802 3.123 5.285 5.785 5.353 5.918

15 1.887 2.669 4.865 5.48 4.777 5.773

20 1.854 2.618 4.281 5.09 4.131 5.122

30 1.825 2,232 4.137 4.45 3.991 4.182

50 1.812 1.972 3,732 3.89 3.633 3.76

100 1.683 1.883 3.567 3.722 3.49 3.812

n Random

Av. | Max

Augm. ε=.5

Av. | Max

Augm. ε=.1

Av. | Max

Instance of 100 nodes

Surprisengly the obtained “bad” instances look like regular grids!

High Density Case

62.33

2MST(S)

Theorem: In the 2-dimensional Euclidean space, the upper bound on the approximation ratio of the MST heuristic for the Minimum Energy BroadcastRouting problem with high-density distribution of the nodes is between 3.62 and 4.

2

1MST(S)

4

3 2

3-Dimension

Known approximation ratio: 3d-1, i.e., 26 The idea to reduce such an approximation

is by adapting the 2-dimensional method Suitable calculations must be done

Rotation of the shape

18.8-Approximation3

maxTot

3max 1,

2

r1π

3

4Volr

6

πMST

6

π

h

3max

3

max r1,2

r18MST

h

8.802...1MST

rmax ≤ 1

Conclusion We closely examined the MEBR problem by extensive

experiments and analytical results Behind the mere numerical results of 7.45, 6.33 and 6, very

interesting are the applied techniques The 6.33 is in fact also scalable to higher dimensions The almost tight 4-approximation for the high-density case

introduces an interesting approach for studying problems The case of α<d is open A better performing heuristic or the analysis of the real

approximation factor of heuristics like BIP, ABC, … remains a challenging problem

Many variations of the original model are studied like changing the pattern of communication or the function to minimize

References[1] R. Klasing, A. Navarra, A. Papadopoulos, and S. Perennes. Adaptive Broadcast Consumption (ABC), a new heuristic and new bounds for the minimum energybroadcast routing problem. In Proceedings of the 3rd IFIP-TC6 International Networking Conference, volume 3042 of Lecture Notes in Computer Science, pp. 866-877. Springer Verlag, 2004.

[2] M. Flammini, R. Klasing, A. Navarra, and S. Perennes. Improved approximation results for the Minimum Energy Broadcasting Problem. In Proceedings of ACM Joint Workshop on Foundations of Mobile Computing (DIALM-POMC), pp. 85-91, 2004. (To appear in the associated Special Issue of Algorithmica)

[3] A. Navarra. Tighter bounds for the Minimum Energy Broadcasting problem. In Proceedings of the 3rd International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), pp. 313-322, 2005. (To appear in the associated Special Issue of WiNet)

[4] M. Flammini, A. Navarra and S. Perennes. The “Real” approximation factor of the MST heuristic for the Minimum Energy Broadcasting. In Proceedings of the 4th International Workshop on Efficient and Experimental Algorithms (WEA). Lecture Notes in ComputerScience, vol. 3503. Springer Verlag, pp. 22-31, 2005. (To appear in the associated Special Issue of JEA)

[5] A. Navarra. 3-D Minimum Energy Broadcasting. In Proceedings of the 13th Colloquium on Structural Information and Communication Complexity(SIROCCO), volume 4056 of Lecture Notes in Computer Science, pp. 240-253. Springer Verlag, 2006.