Topology Control in Heterogeneous Wireless Networks: Problem and Solution

1
Topology Control in Heterogeneous Wireless
Networks Problem and Solution

• Ning Li and Jennifer C.Hou
• Department of Computer Science
• University of Illinous at Urbana-Champaign

08.03.29 System Software Laboratory Myung-Ho Kim
Team Dae-Woong JO
2
Contents

• Introduction
• Network Model
• Related Work and Why They Cannot Be Directly
  Applied To Heterogeneous Networks
• DRNG and DLMST
• Properties Of DRNG and DLMST
• Simulation Study
• Conclusions
• References

3
Introduction

• Energy efficiency and Network capacity
• Reducing Energy consumption and improving network
  capacity
• Two localized topology control algorithms
• DRNG
• Directed Relative Neighborhood Graph
• DLMST
• Directed Local Minimum Spanning Tree
• Be able to prove
• Derived under both DRNG and DLMST
• DLMST is bounded, DRNG may be unbounded.
• DRNG and DLMST preservers network
  bi-directionality

4
Introduction (cont.)

• Simulation results indicate
• Compared with the other known topology control
  algorithms
• Have smaller average node degree (both logical
  and physical)
• Have smaller average link length.
• In Section 2
• Network model
• In Section 3
• Summarize previous work on topology control
• In Section 4
• DRNG and DLMST algorithms
• In Section 5
• Prove several of their useful properties
• In Section 6
• Evaluate the performance of the proposed
  algorithms
• In Section 7
• conclude

5
Network Model

• V v1, v2, . . . , vn, random distrivuted in
  the 2-D plane.
• Let rvi
• Maximal transmission range of vi
• Heterogeneous network
• All nodes may not be the same.
• rmin minv?V rv
• rmax maxv?V rv
• d(u,v) is distance between node u and node v

6
Network Model (cont.)

• Simple directed graph G (V(G),E(G))
• V(G) randomly distributed in the 2-D plane
• E(G) (u,v) d(u,v) lt ru, u,v ? V(G)
• Definition 1
• Reachable Neighborhood
• Definition 2 (weight function)
• w(u1, v1) gt w(u2, v2)
• ? d(u1, v1) gt d(u2, v2)
• or (d(u1, v1) d(u2, v2)
• maxid(u1), id(v1) gt maxid(u2),
  id(v2))
• or (d(u1, v1) d(u2, v2)
• maxid(u1), id(v1) maxid(u2), id(v2)
• minid(u1), id(v1) gt minid(u2),
  id(v2)).
• Definition 3 (Neighbor Set )
• Algorithm A, denoted u A v
• NA(u) v ? V (G) u A v .

7
Network Model (cont.)

• Definition 4
• Topology
• Directed graph GA (E(GA),V(GA))
• Where V (GA) V (G), E(GA) (u, v) u A
  v , u, v ? V (GA).
• Definition 5
• Radius
• The radius, ru, of node u is defined
• Definition 6
• Connectivity
• Topology generated by an algorithm A
• Node u is connected to node v (denoted u ? v)
• If there exists a path(p0 u, p1,,pm-1,pm v)
• It follows that u gt v if u gt p and p gt v for
  some p ? V(GA)
• Definition 7
• Bi-Directionality
• Any two nodes u,v ? V (GA), u ? NA(v) implies v
  ? NA(u).

8
Network Model (cont.)

• Definition 8
• Bi-Directional Connectivity
• Bi-directionally connected to node v (denoted u ?
  v )
• If there exists a path p0 u, p1,pm-1, pm v)
• It follows that u ? u if u ? p and p ? v for some
  p ? V(GA)
• Definition 9
• Addition and Removal
• Addition operation
• extra edge (v, u) ? E(GA)
• Removal operation
• delete any edge (u, v) ? E(GA)

9
RELATED WORK AND WHY THEY CANNOT BEDIRECTLY
APPLIED TO HETEROGENEOUS NETWORKS

• Ramanathan et al. 5
• Two distrubuted heuristics for mobile networks
• Require global information
• Cannot be directly deployed
• Borbash and Jennings 8
• Proposed to use RNG
• (Relative Neighborhood Graph)
• Topology initialization of wireless networks
• Good overall performance
• Low interference, and reliablity

10
RELATED WORK AND WHY THEY CANNOT BEDIRECTLY
APPLIED TO HETEROGENEOUS NETWORKS

• Definition 10 ( Neighbor Relation in RNG)
• u RNG v if and only if there does not
  exist a third node p such that w(u, p) lt w(u, v)
  and w(p, v) lt w(u, v).
• Or equivalently, there is no node
• inside the shaded area

11
RELATED WORK AND WHY THEY CANNOT BEDIRECTLY
APPLIED TO HETEROGENEOUS NETWORKS (cont.)

• CBTC(a) 6
• Proved to preserve network connectivity
• In 10
• Proposed LMST(Local Minimum Spanning Tree)
• Topology control in homogeneous wireless
  multihop- networks
• Proved that
• LMST preserves the network connectivity
• The node degree of any node
• Can be transformed into one with bi-directional
  links

12
RELATED WORK AND WHY THEY CANNOT BEDIRECTLY
APPLIED TO HETEROGENEOUS NETWORKS (cont.)
13
DRNG and DLMST

• Propose two localized topology control algorithms
• DRNG (Directed Relative Neighborhood Grpah)
• DLMST (Directed Local Minimum Spanning Tree)
• Both algorithms are composed of three pahses
• Information Collection
• Topology Construction
• Construction of Topology with only Bi-Directional
  Links

14
DRNG and DLMST (cont.)

• Definition 12
• Neighbor Relation in DRNG
• u DRNG v if and only if d(u, v) ru and
  there does not exist a third node p
• such that w(u, p) lt w(u, v) and w(p, v) lt
  w(u, v), d(p, v) rp

15
DRNG and DLMST (cont.)

• Definition 13
• Neighbor Relation in DLMST
• Directed Local Minimum Spanning Tree Graph
  (DLMST)
• u DLMST v if and only if (u, v) ? E(Tu),
  where Tu is the directed local MST rooted at u
  that spans NRu .
• each node u computes a directed MST that spans
  NRu and takes on-tree
• nodes that are one hop away as its neighbors

16
Properties of DRNG and DLMST

• Discuss the connectivity, bi-directionality and
  degree bound of DLMST and DRNG
• Connectivity
• Theorem 1 (Connectivity of DLMST)
• If G is strongly connected, then G DLMST is also
  strongly connected.
• Proof
• For any two nodes u, v ? V (G), there exists a
  unique global MST T rooted at u since G is
  strongly connected. Since E(T) ? E(GDLMST ) by
  Lemma 2, there is a path from u to v in GDLMST .
• Lemma 2 Let T be the global directed MST of G
  rooted at any node w ? V(G) ,then E(T) ? E(Gdlmst)

17
Properties of DRNG and DLMST

• Theorem 2 (Connectivity of DRNG)
• If G is strongly connected, then G DRNG is also
  strongly connected.
• Proof
• For any two nodes u, v ? V (G), since G is
  strongly connected, there exists a path (p0 u,
  p1, p2, . . . , pm-1, pm v) from u to v, such
  that (pi, pi1) ? E(G), i 0, 1, . . .,m - 1.
  Thus pi ? pi1 in GDRNG by Lemma 3. Therefore, u
  ? v in GDRNG. Hence we can conclude that GDRNG is
  strongly connected.
• Lemma 3 For any edge (u,v) ? E(G), we have u gt
  v in Gdrng

18
Properties of DRNG and DLMST (Cont.)

• Bi-directionality
• Theorem 3
• If the original topology G is strongly connected
  and bi-directional, then G DLMST and G DRNG are
  also strongly connected and bi-directional
• Proof
• For any two nodes u, v ? V (G), there exists at
  least one path p (w0 u,w1, w2, , wm-1,
  wm v) from u to v, where (wi, wi1) ? E(G), i
  0, 1, ,m - 1. Since wi ? wi1 in GDLMST by
  Lemma 5, we have u ? v in GDLMST . Therefore, wi
  ? wi1 in GDRNG, which means u ? v in GDRNG. The
  same results still hold after Addition or Removal
• Lemma 5 If the original topology G is strongly
  connected and bi-directional, then any edge (u,v)
  ? E(G) satisfies that u ? v in Gdlmst

19
Properties of DRNG and DLMST (Cont.)

• Degree Bound
• Theorem 4
• For any node u ? V (GDLMST ), the number of
  neighbors in GDLMST that are inside Disk(u, rmin)
  is at most 6.
• Theorem 5
• The out degree of node in GDLMST is bounded by a
  constant that depends only on rmax and rmin.

20
Simulation Study

• Evaluate the performance
• RM, DRNG and DLMST by simulations.
• Preserve network connectvity in heterogeneous
  networks
• First simulation
• 50 nodes are uniformly distributed
• 1000m x 1000m region
• RM, DRNG and LMST all reduce
• Average node degree, while maintaining network
  connectivity
• DRNG and DLMST outperforms RM

21
Simulation Study (cont.)
22
Simulation Study (cont.)

• Second simulation
• Vary the number of nodes in the region
• 80 to 300
• Average of 100 simulation runs
• Each data point
• Nodes are uniformly distributed in 10m,250m
• Average radius and the average edge length
• NONE(no topology control)
• RM, DRNG, and DLMST
• DLMST outperforms the others
• Better spatial reuse and use less energy

23
Simulation Study (cont.)

• Compare the out degree
• The topologies by different algorithms
• The result of NONE is not shown
• Langer than under RM, DRNG, DLMST
• Out degrees increase linearly
• Shows the average logical/physical
• Derived by RM, DRNG, DLMST
• Under RM and DRNG increase
• Under DLMST actually decrease

24
Simulation Study (cont.)
25
Conclusions

• Proposed two local topology control algorithms
• DRNG, DLMST
• Heterogeneous wireless multi-hop networks
• Have different transmission ranges
• Show that
• Most existing topology control algorithms
• Have different transmission ranges
• Disconnected network topology
• Directly applied to heterogeneous networks.

26
Conclusions (cont.)

• DRNG and DLMST prove
• Preserve network connectivity
• Preserve network bi-directionality
• Bounded in the topology under DLMST ,Unbounded
  under DRNG
• Future research
• Different maximal transmission power
• Density of nodes, distribution of the
  transmission ranges
• MAC-level interference affect network
• Connectivity and bi-directionality

27
References

• 3 S. Narayanaswamy, V. Kawadia, R. S.
  Sreenivas, and P. R. Kumar,
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• IEEE J. Select. Areas Commun., vol. 17,
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• Aug. 1999.
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• 264273.

28
References (cont.)

• 8 S. A. Borbash and E. H. Jennings,
  Distributed topology control algorithm for
  multihop wireless networks, in Proc. 2002 World
  Congress on
• Computational Intelligence (WCCI 2002),
  Honolulu, Hawaii, US, May 2002.
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  Distributed construction of planar
• spanner and routing for ad hoc networks,
  in Proc. IEEE INFOCOM
• 2002, New York, New York, US, June 2002.
• 10 N. Li, J. C. Hou, and L. Sha, Design and
  analysis of an MSTbased
• topology control algorithm, in Proc. IEEE
  INFOCOM 2003, San
• Francisco, California, US, Apr. 2003.