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Overlay networks for wireless ad hoc networks

Overlay networks for wireless ad hoc networks. Christian Scheideler Dept. of Computer Science Johns Hopkins University. Motivation. Wireless ad hoc networks have many important applications: search and rescue missions emergency situations military applications

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Overlay networks for wireless ad hoc networks

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  1. Overlay networks for wireless ad hoc networks Christian Scheideler Dept. of Computer Science Johns Hopkins University

  2. Motivation • Wireless ad hoc networks have many important applications: • search and rescue missions • emergency situations • military applications • Signal processing and MAC: OK • Scheduling: somewhat OK • Routing: HARD!!

  3. Outline • In this talk: • Basic model and goals • Basic approach (graph spanners) • Basic results about spanners • Examples • Extensions and problems • Realistic wireless models and protocols • References

  4. Wireless ad hoc network

  5. Overlay network

  6. Unit disk graph 1

  7. Unit disk graph Basic goals: low energy consumption, high throughput Some links: high energy / low success prob No particular structure Contention can be problem in dense graphs! Strategy: find structured sparsification

  8. Sparsification is not trivial! Every node connects to two nearest neighbors. • < 0.074 log n nearest neighbors: disconnected w.h.p. • > 5.1774 log n nearest neighbors: connected w.h.p. • if nodes distributed uniformly at random in convex reg.

  9. Goals of Sparsification • Guarantee connectivity • Guarantee energy-efficient paths resp.paths with high success probability • Maintain self-routing network(no preprocessing for path selection) • General strategy:graph spanners

  10. Assumptions Node set V distributed in 2-dim Euclidean space e u v Euclidean distance: ||u v|| or ||e|| Power consumption: ||u v||dfor some  > 2 -cost of path p=(e1,…,ek): ||p|| = i=1k ||ei||

  11. Distance G=(V,E) u v dG(u,v): min ||p|| over all paths p from u to v in G Unit disk graph UDG(V): simply d(u,v)

  12. UDG Spanners • Basic idea: S is spanner of graph G for some : subgraph of G with dS(u,v) ¼ dG(u,v) for all u,v 2 V • c>1 constant, G: UDG(V) • Geometric spanner: dS(u,v) < c ¢ d(u,v) • Power spanner: dS(u,v) < c ¢ d(u,v), >2 • Weak spanner: path p from u to v within disk of diameter c ¢ d(u,v) • Topological spanner: dS0(u,v) < c ¢ d0(u,v) S G

  13. UDG Spanners Geometric spanner: Weak spanner: Power spanner:

  14. Spanners Geometric spanner: Weak spanner: Power spanner:

  15. UDG Spanners • Easier: c>1 constant. • Geometric spanner: dS(u,v) < c ¢||u v|| • Power spanner: dS(u,v) < c ¢||u v||, >2 • Weak spanner: path p from u to v within disk of diameter c ¢||u v|| • Constrained spanner: there is a path p satisfying constraint above and ||e|| <= ||u v|| for all e 2 p • E(constrained spanner) Å UDG(V): spanner of UDG(V) • We want: c>1 constant • Geometric spanner: dS(u,v) < c ¢d(u,v) • Power spanner: dS(u,v) < c ¢d(u,v),>2 • Weak spanner: path p from u to v within disk of diameter c ¢d(u,v)

  16. Spanner Properties • Geometric spanner ) power spanner • Geometric spanner ) weak spanner • Weak spanner ) geometric spanner • Power spanner ) weak spanner • Weak spanner ) power spanner geometric ) weak ) power spanner geometric ) power ) weak spanner / /

  17. Spanners Geometric spanner: Weak spanner: Power spanner:

  18. Proximity graphs G=(V,E) is proximity graph of V if 8 u,w 2 V: • either (u,w) 2 E • or (u,v) 2 E for some v: ||v w|| < ||u w|| v w u u w

  19. Example v

  20. Proximity graphs Every proximity graph of V is a weak 2-spanner. Proof: by induction on distance min distance: v induction w w u u

  21. Relative neighborhood graphs G=(V,E) is a RNG of V if for all u,w 2 V: • either (u,w) 2 E • or (u,v) 2 E for some v: ||u v|| <= ||u w|| and ||v w|| < ||u w|| v RNG: constrained proximity graph u w

  22. Relative neighborhood graphs Local control rule: v Gives weak 2-spanner of UDG(V).

  23. Relative neighborhood graphs Problem: Minimal RNGs have outdegree at most 6 but indegree can be large.

  24. Routing in RNGs ??? v u What if nodes have GPS? Better: establish/maintain paths Naïve approach: flooding

  25. Sector-based spanners Yao graph or -graph (special RNG): Rule: Connect to nearest node in each sector

  26. Sector based spanners -graphs with >6 sectors are geometric spanners Routing: destination Go to node in sector of destination.Requires dense node distribution, GPS!

  27. Sector based graphs Distribution not dense: v s t ???

  28. Delaunay-based spanners Idea: use variants of Delaunay graph Delaunay graph Del(V) of V: contains all {u,v}: 9 w 2 V where O(u,v,w) does not contain any node of V

  29. Delaunay-based spanners Gabriel graph GG(V) of V: contains all {u,w} with no v 2 V s.t. ||u v||2 + ||v w||2 < ||u w||2 u w GG(V) ½ Del(V)

  30. Relative neighborhood graph G=(V,E) is a RNG of V if for all u,w 2 V: • either (u,w) 2 E • or (u,v) 2 E for some v: ||u v|| <= ||u w|| and ||v w|| < ||u w|| v RNG: constrained proximity graph u w

  31. RNG for path loss The GG satisfies for all u,w 2 V: • either (u,w) 2 E • or (u,v) 2 E for some v: c(u,v) <= c(u,w) and c(v,w) < c(u,w) v GG: constrained proximity graph for path loss with =2 u w

  32. Delaunay-based spanners GG(V) is an optimal power spanner. Proof: Let p be energy-optimal path for {u,v}. Consider any edge {x,y} in p. {x,y} 2 GG(V): {x,y} 2 GG(V): / Contradiction!

  33. Delaunay-based spanners Problem with Gabriel graphs: can have high degree, can be poor geometric spanner Alternative: k-localized Delaunay graphs LDel(k)(V) v No node in k-neighborhood of u,v,w in UDG(V) u w

  34. Delaunay-based spanners • LDel(1)(V): not planar • LDel(2)(V): planar and geometric spanner Planarity important for routing!

  35. Routing in planar graphs Face routing: s t

  36. That’s nice, but do these strategies work in practice???

  37. Problem: Unit Disk Model In reality, hard to maintain planar overlay network.

  38. Reality u • Cost function c and constant >0 s.t. for any two points v and w • c(v,w) 2 [(1-) ¢ ||v w||, (1+) ¢ ||v w||] • c(v,w) = c(w,v)

  39. Realistic wireless model • When using cost function c: • Proximity graphs: OK • Relative neighborhood graphs: OK • -graphs: < arccos (2+1/2)/(1-2),  < 1/2 • Gabriel graphs, localized Delaunay graphs:not planar but other spanner results OK • Face routing extendable to more realistic model? • Is GPS necessary?

  40. Problem: GPS GPS not necessary for topology control, butcan we avoid GPS for routing? Possible solution: use force approach

  41. Problem: Cost Model Power spanner prefers over Energy consumption:

  42. Solution Only allow long edges (except last).

  43. Problem: Contention Still too much contention because every node participates in routing. Better to have highway system (i.e., only few nodes act as relay nodes).

  44. Solution Construct dominating set: every haswithin its transmission range

  45. Problem: Mobility Solution: view mobility as another dimension

  46. Problem: Protocol Design u • Cost function c and constant >0 s.t. for any two points v and w • c(v,w) 2 [(1-) ¢ ||v w||, (1+) ¢ ||v w||] • c(v,w) = c(w,v)

  47. Realistic wireless model v u w transmission range interference range

  48. Realistic wireless model Design of algorithms much more complicated! s v t Estimate of density difficult without physical carrier sensing

  49. Physical carrier sensing v u w direct influence no influence

  50. Future problems • Overlay networks are hot topic! • Problems: • Local self-stabilization • Robustness against adversarial behavior • Distributed optimization • Unified network model (???)

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