About the Minimum Energy Broadcasting in Wireless Ad Hoc Networks Alfredo Navarra, PhD Laboratoire...

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Transcript of About the Minimum Energy Broadcasting in Wireless Ad Hoc Networks Alfredo Navarra, PhD Laboratoire...
About the Minimum Energy Broadcasting in Wireless Ad Hoc Networks
Alfredo Navarra, PhD Laboratoire Bordelais de Recherche en Iinformatique (LaBRI) University of Bordeaux, France [email protected]
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 ).
Multihop 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 ddimensional 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 2dimension with α=2
2i
i
r
S
ii
cost r Minimum Energy Broadcasting is
in general NPhard
MSTheuristic: 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.
BIPheuristic: Starting from s, chose the cheapest way to reach a
node either adding a new edge (MST) or “increasing” an old one.
ABCheuristic: 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
Lemma 1. Given a set of nodes V over the Euclidean 2dimensional 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
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 2dimensional case MST, 40approx (Clementi, Crescenzi, Penna Rossi, Vocca – STACS 2001)
MST, 20approx (Clementi, Crescenzi, Penna Rossi, Vocca – STACS 2001)
MST, 12approx (Calinescu, Li, Frieder, Wan – INFOCOM 2001)
MST, 12.5approx (Klasing, Navarra, Papadopoulos, Perennes – Networking 2004)
MST, 7.45approx (Flammini, Klasing, Navarra, Perennes – DIALMPOMC 2004)
MST, 6.33approx (Navarra – WiOpt 2005)
α≥d=2 case
Reminding the MSTheuristic
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 1c)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
8Approximation2
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 2dimensional 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 1r ≤ 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 1r ≤ 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 1r ≤ 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 1r ≤ 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 8Approximation 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=V1; 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 jth 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
High Density Case
62.33
2MST(S)
Theorem: In the 2dimensional Euclidean space, the upper bound on the approximation ratio of the MST heuristic for the Minimum Energy BroadcastRouting problem with highdensity distribution of the nodes is between 3.62 and 4.
2
1MST(S)
4
3 2
3Dimension
Known approximation ratio: 3d1, i.e., 26 The idea to reduce such an approximation
is by adapting the 2dimensional method Suitable calculations must be done
18.8Approximation3
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 4approximation for the highdensity 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 IFIPTC6 International Networking Conference, volume 3042 of Lecture Notes in Computer Science, pp. 866877. 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 (DIALMPOMC), pp. 8591, 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. 313322, 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. 2231, 2005. (To appear in the associated Special Issue of JEA)
[5] A. Navarra. 3D 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. 240253. Springer Verlag, 2006.