Minimum%20Energy%20Broadcast%20Routing%20Problem

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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_at_labri.fr
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Minimum Energy Broadcasting

• Signal Attenuation A radio station s with
  transmission power PS ßr a reaches all the
  stations at distance at most r. (for some
  constant ß and usually 2a6 ).
• 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 s?V,let G(V,E)
  be the complete symmetric directed graph obtained
  from V, in which every edge (x,y) has weight
  ßdist(x,y)a.
• 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 minimize

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Ex. Power Assignment in 2-dimension with a2
Minimum Energy Broadcasting is in general
NP-hard
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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.

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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

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An example
MST
BIP
ABC
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• 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 ?

ABC
BIP
8
15 random nodes in a 5x5 square
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ILP Formulation
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50 random nodes in a 5x5 square
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ad2 case

• 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)

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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

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Assume that each node is inside a circle of
radius R1
R
r
r1
r2
r3
rmax
9
8
7
1
CC
MOREOVER
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8-Approximation
For the 2-dimensional case the ratio was first
reduced to 7.45 by means of geometrical techniques
.
rmax 1
More in general, for any dimension d and any ad

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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.

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Properties and Approximation 1

• For any d11 and d21 such that 1-r d1 d2,
  h(r,d1) h(r, d2)
• For any d1, h(r,d).3638
• ?h(r,d) 3/5?r
• For any d11 and d21 such that 1-r d1 d2,
• ??(r,d1) ??(r,d2)
• MST(G) 4(1h(.5,1))2 1

6.4401lt 6.45
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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

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Properties and Approximation 2

• For any d11 and d21 such that 1-r d1 d2,
• h(r,d1) h(r,d2)
• For any d1, h(r,d).3527
• ?h(r,d) 3/5?r
• For any d11 and d21 such that 1-r d1 d2,
• ??(r,d1) ??(r,d2)
• MST(G) 4(1h(.5,1))2 1

6.3203lt 6.33
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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
  1hmax/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).

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Indeed

• The MST approximation bound is already reduced to
  the optimal one, that is, 6.
• C. Ambuhl, in Proceedings of ICALP 2005
• The analysis is based on Delauney triangulation
• We had our glory lets go back to simulations
  and higher dimensions!!!

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Reminding about the 8-Approximation and more
rmax 1
MST 8
MST 4
For rmax ? 0
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Worst case for the MST heuristic
n Av. Max
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.
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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.

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Allowed Movements

• More in general for any v?s, let N(v)v1vk 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.

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Augmenting Algorithm (sketch)

• flagi1 1 flag21 NV-1 i1 j1
• Compute MST over G2(V), save its cost in cost1
• While flag2N do
• While flag1N do
• if vi is not on the circumference then
• let vi be a point inside C1 on the line
  passing through vi and p in such
• a way that vi,p lt vi,p(1erand)vi,p
  
• else let vi be a point on the circumference
  further from p with respect to vi such that the
  arc joining vi and vi has length erand
• Compute the MST over G2((V \ vi)U vI), save
  its cost in cost2
• if cost2gtcost1 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 cost2gtcost1 then
• ...

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Experimental Results
n Random Av. Max Augm. e.5 Av. Max Augm. e.1 Av. Max
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
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Instance of 100 nodes
Surprisengly the obtained bad instances look
like regular grids!
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High Density Case
Theorem In the 2-dimensional Euclidean space,
the upper bound on the approximation ratio of
the MST heuristic for the Minimum Energy
Broadcast Routing problem with high-density
distribution of the nodes is between 3.62 and 4.
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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

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Rotation of the shape
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18.8-Approximation
rmax 1
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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 altd 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

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References

• 1 R. Klasing, A. Navarra, A. Papadopoulos, and
  S. Perennes. Adaptive Broadcast
• Consumption (ABC), a new heuristic and new bounds
  for the minimum energy
• broadcast 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 Computer
• Science, vol. 3503. Springer Verlag, pp. 22-31,
  2005. (To appear in the associated Special Issue
  of JEA)