1. Topology Control Algorithms Davide Bilò e-mail: davide.bilo@univaq.it

2. What is Topology Control? Mechanisms and algorithms to conserve energy in ad- hoc radio and sensor networks Primary targets of a topology control algorithm abandon long-distance communication links prevent the network from being partitioned Secondary targets each node has “few” neighbors routing path does not have to become non-competitively long topology control algorithms should find a good tradeoff between connectivity and sparsness

3. What does a TC Algorihm do? Let G=(V,E) be an (undirected * ) communication graph where V is the set of devices with |V|=n E contains edge (u,v) iff u and v can communicate directly c(u,v) minimum transmission power at which u has to transmit if it wants to send a msg to v directly (we assume c(u,v)=c(v,u) and c(u,v)  p max ) * As devices are homogeneous, i.e., they have the same characteristics, we can assume that if u can communicate with v directly if u transmits at power p(u), then also v can communicate with u directly if v transmits at power p(v)  p(u). This implies that the devices all have the same maximum transmission power p max. Running the TC algorithm A on all the nodes yields a graph G A =(V,E A ) which is a subgraph of G

4. What is the Directed Communication Graph G in the Euclidean Model? Observe: for every  1, d(u,v)   d(u,v)  iff d(u,v)  d(u,v ). For simplicity, we will assume that  =1 even though everything we will see can be generalized to every  1

5. The (Directed) Communication Graph G in the Euclidean Model is a Unit Disk Graph The transmission range of any node v is the disk centered at v with radius G has bidirectional symmetric link

6. Formal Definition of Unit Disk Graph (UDG) Given a set V of points in the Euclidean plane, the Unit Disk Graph induced by V is the (undirected) graph G=(V,E) where E contains edge (u,v) iff d(u,v)  1

7. What does a TC Algorihm do? Let G=(V,E) be an (undirected * ) communication graph where V is the set of devices with |V|=n E contains edge (u,v) iff u and v can communicate directly c(u,v) minimum transmission power at which u has to transmit if it wants to send a msg to v directly (we assume c(u,v)=c(v,u) and c(u,v)  p max ) * As devices are homogeneous, i.e., they have the same characteristics, we can assume that if u can communicate with v directly if u transmits at power p(u), then also v can communicate with u directly if v transmits at power p(v)  p(u). This implies that the devices all have the same maximum transmission power p max. Running the TC algorithm A on all the nodes yields a graph G A =(V,E A ) which is a subgraph of G

8. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Reason: Asymmetric communications are unpractical

9. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Connectivity is not enough A minimum spanning tree algorithm yields a connected subgraph G MST Not a good topology because close-by nodes in G might end too far in G MST

10. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Spanner: if the shortest path from u to v in G w.r.t. some criteria has cost , then the shortest path from u to v in G A w.r.t. the same criteria has cost f(  ). If f(  ) is bounded from above by a linear function of , then G A is called a spanner Spanner  Connectivity

11. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Spanner: if the shortest path from u to v in G w.r.t. some criteria has cost , then the shortest path from u to v in G A w.r.t. the same criteria has cost f(  ). If f(  ) is bounded from above by a linear function of , then G A is called a spanner Spanner  Connectivity Sparsness: G A is sparse, i.e., |E A |=O(n) Reason: Primary target of a topology control algorithm is to abandon long-distance neighbors Sparsness is not enough as sparse graphs may have high degree

12. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Spanner: if the shortest path from u to v in G w.r.t. some criteria has cost , then the shortest path from u to v in G A w.r.t. the same criteria has cost f(  ). If f(  ) is bounded from above by a linear function of , then G A is called a spanner Sparsness: G A is sparse, i.e., |E A |=O(n) Low Degree: Each node in G A has a constant number of neighbors Spanner  Connectivity Low Degree  Sparsness

13. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Spanner: if the shortest path from u to v in G w.r.t. some criteria has cost , then the shortest path from u to v in G A w.r.t. the same criteria has cost f(  ). If f(  ) is bounded from above by a linear function of , then G A is called a spanner Sparsness: G A is sparse, i.e., |E A |=O(n) Low Degree: Each node in G A has a constant number of neighbors Planarity: G A is planar, i.e., it does not have intersecting edges Reason: we can use geometric routing algorithms on planar graphs Spanner  Connectivity Low Degree  Sparsness

14. Planar Graphs A (geometric) graph is planar if it has no intersecting edges (geometric graphs we consider are graphs whose set of vertices are points on the Euclidean plane, and edges are straight line segments) Example of planar graph Example of non planar graph red edges intersect Intersection point

15. Properties G A should have Symmetry: G A is symmetric, i.e., u is a neighbor of v in G A iff v is a neighbor of u in G A Connectivity: there is a (direct) path from node u to node v in G A iff there is a (direct) path from u to v in G Spanner: if the shortest path from u to v in G w.r.t. some criteria has cost , then the shortest path from u to v in G A w.r.t. the same criteria has cost f(  ). If f(  ) is bounded from above by a linear function of , then G A is called a spanner Sparsness: G A is sparse, i.e., |E A |=O(n) Low Degree: Each node in G A has a constant number of neighbors Planarity: G A is planar, i.e., it does not have intersecting edges Spanner  Connectivity Low Degree  Sparsness

16. Which TC Algorihm do we need? We do not need a global centralized algorithm for sure no central authority in ad-hoc radio and sensor networks What about a distributed algorithm? better than the centralized one not of practical use in case of mobile devices We do need a local algorithm each node is allowed to exchange msg’s with its neighbors a few times and then must decide which links it wants to keep

17. Topology Control Algorithms for UDG nodes know their coordinates (for instance, nodes use GPS) Minimum Spanning Tree distributed but not local symmetry, connectivity, low degree, and planarity Delaunay Triangulation distributed but not local symmetry, energy-spanner, low degree, and planarity Gabriel Graph local symmetry, energy-spanner, sparsness, and planarity nodes can sense signal strength and can perceive from which direction a signal arrives Cone-based local symmetry, energy-spanner, sparsness, and planarity (an optional distributed (but not local) second phase) satisfies low degree.

18. Limitations of the Euclidean Model signal attenuation is uniform, that is, the Euclidean plane is flat and free of blocking objects Radio propagation is as in vacuum v u u d(v,u)=d(v,u ) transmission range of v

19. If we add obstacles… Euclidean Model does not work in realistic environments v u transmission range of v obstacle u d(v,u)=d(v,u )

20. Algorithm XTC R. Wattenhofer and A. Zollinger, XTC : A Practical Topology Control Algorithm for Ad-Hoc Networks, 4th International Workshop on Algorithms for Wireless, Mobile, Ad Hoc and Sensor Networks, 2004 download link: http://www.dcg.ethz.ch/members/roger.html

21. Algorithm XTC works in every environment (i.e., every undirected graph G ) nodes do not need to know their coordinates nodes do not need to perceive which direction a signal comes from it is local and fast (every node communicates with its neighborhood twice) the system can be asynchronous uniform non-anonymous satisfies symmetry connectivity low degree planarity energy-spanner (in random UDG’s) in UDG’s correctness efficiency

22. Algorithm XTC Three main steps: 1. neighbor ordering 2. neighbor order exchange 3. edge selection ( N u neighborhood of u )

23. Algorithm XTC Algorithm XTC (description for node u) 1. (neighbor ordering) establish total order < u over u ’s neighbors in G v< u w means that u prefers link (u,v) more than link (u,w), i.e., link (u,v) is of higher quality than link (u,w) (for instance, v< u w  c(u,v)  c(u,w) )

24. Algorithm XTC Algorithm XTC (description for node u) 1. (neighbor ordering) establish total order < u over u ’s neighbors in G 2. ( neighbor order exchange) broadcast < u to each neighbor in G and receive orders < v from all neighbors v ’s 3. (edge selection ( N u neighborhood of u )) N u,Ñ u :=  while ( < u contains unprocessed neighbors) v:= least unprocessed neighbor in < u if (  w  N u  Ñ u s.t. w< v u) then Ñ u :=Ñ u  {v} else N u :=N u  {v} i.e., w< u v

25. Graph Yielded After Execution of Algorithm XTC N u is the set of neighbors of u computed by algorithm XTC G XTC =(V,E XTC ) where E XTC ={(u,v)|  u:v  N u }

26. G XTC is Symmetric Theorem (Symmetry): G XTC is symmetric, i.e., a node u includes v in N u iff v includes u in N v. Proof: Assume u includes v in Ñ u. We show that v includes u in Ñ v. u includes v in Ñ u because  w  N u  Ñ u with w < u v and w < v u. When v processes u, w  N v  Ñ v. Thus, v includes u in Ñ v. From now on, we will tacitly assume that G XTC is symmetric

27. Some Assumptions Weak Assumption (WA): Neighbor orders are based on function c, i.e.,  u, w< u v   c(u,w)  c(u,v) Strong Assumption (SA): Every edge (u,v) has a weight l(u,v)=(c(u,v),min{id(u) *,id(v)},max{id(u),id(v )}). Neighbor orders are based on the lexicographic order ** of edge weights, i.e.,  u, w< u v  l(u,w) < l(u,v) * id(w) is the identifier of node w. Nodes have distinct identifiers. ** ( , ,  )<( , ,  )  (  <  ) or ((  =  ) and (  <  )) or ((  =  ) and (  =  ) and (  <  ))

28. G XTC Satisfies Connectivity Theorem (Connectivity): Under SA, two nodes u and v are connected in G XTC iff they are connected in G. Corollary: Under SA, G XTC is connected iff G is connected.

29. G XTC Satisfies Connectivity Theorem (Connectivity): Under SA, two nodes u and v are connected in G XTC iff they are connected in G. Proof: If u and v are connected in G XTC, then they are connected in G. (because G XTC is a subgraph of G ) So we have to prove that Claim: if u and v are connected in G, then they are connected in G XTC. We prove Claim by contradiction, i.e., we assume that there exist u and v which are connected in G but not in G XTC. We use the following scheme: 1. we choose the “right” u and v 2. we show that u and v are connected in G XTC

30. v u G XTC Satisfies Connectivity How to choose the “right” u and v Theorem (Connectivity): Under SA, two nodes u and v are connected in G XTC iff they are connected in G. Proof: … Let Z be the set of all the pair of nodes u and v which are not connected in G XTC but they are connected in G via a direct edge. Is Z  ? YES w t w and t are connected in G but not in G XTC V w : set of nodes connected to w in G XTC V t : set of nodes connected to t in G XTC VwVt=VwVt= VwVw VtVt G

31. G XTC Satisfies Connectivity How to choose the “right” u and v Theorem (Connectivity): Under SA, two nodes u and v are connected in G XTC iff they are connected in G. Proof: … Let Z be the set of all the pair of nodes u and v which are not connected in G XTC but they are connected in G via a direct edge. Is Z  ? YES u and v is the pair of nodes in Z of minimum value l(u,v)

32. G XTC Satisfies Connectivity How to prove that u and v are connected in G XTC Theorem (Connectivity): Under SA, two nodes u and v are connected in G XTC iff they are connected in G. Proof: … What we have shown so far: u and v is the pair of nodes of minimum value l(u,v) among those pair of nodes which are not connected in G XTC but connected in G via a direct edge. u includes v in Ñ u because  w  N u  Ñ u with w< u v, i.e., w< u v AND w< v u l(u,w) < l(u,v) AND l(v,w) < l(u,v) u and w are connected in G XTC AND v and w are connected in G XTC u and v are connected in G XTC

33. G XTC on UDG’s (remember that (u,v) is in G iff c(u,v)=d(u,v)  1 )

34. G XTC on UDG’s has Low Degree Theorem (Low Degree): Under WA, if G is a UDG, then G XTC has degree at most 6. Proof: Let u  V s.t. (u,v),(u,w)  E XTC, i.e., d(u,v),d(u,w)  1. We prove that the angle  /3 by contradiction. So, assume for contradiction that  <  /3. W.l.o.g., assume v< u w, i.e., d(u,v)  d(u,w). Claim: If d(u,v)  d(u,w) and  <  /3, then d(v,w)<d(u,w).  d(v,w)  1  (v,w) is in G. Moreover, v< w u. Thus, u includes w in Ñ u. By Theorem (Symmetry) (u,w)  E XTC. w v  u contradicts (u,w)  E XTC

35. Proof of Claim: If d(u,v)  d(u,w) and  <  /3, then d(v,w)<d(u,w) u w v  A B A=|d(u,v)-d(u,w)cos  | B=d(u,w)sin  d(v,w) 2 =A 2 +B 2 =d(u,v) 2 -2d(u,v)d(u,w)cos  +d(u,w) 2  (1-2cos  )d(u,v) 2 +d(u,w) 2 <d(u,w) 2 (use sin 2  +cos 2  =1 ) (  [0,  /3), cos  >0.5 )

36. v G XTC on UDG’s is Planar Theorem (Planarity): Under WA, if G is UDG, then G XTC is planar. Proof: Let u,v,w,t be any 4 -tuple of distinct nodes forming a quadrangle Q as in figure s.t. d(u,w),d(v,t)  1. The only two intersecting edges of Q may be (u,w) and (v,t). We prove that (u,w)  E XTC or (v,t)  E XTC. (This is almost enough as almost every pair of intersecting edges defines a quadrangle) As the sum of the interior 4 angles of Q is 2 , one of them is  /2. W.l.o.g., assume  /2.  d(u,v),d(w,v)<d(u,w). As d(u,w)  1, then (u,v),(w,v) are in G. Moreover, v< u w and v< w u. When u considers w, v  N u  Ñ u. As v< w u, then u includes w in Ñ u. By Theorem (Symmetry) (u,w)  E XTC. u wt  Q

37. v G XTC on UDG’s is Planar Theorem (Planarity): Under WA, if G is UDG, then G XTC is planar. Proof: … to complete the proof, we should consider the case of three aligned points as in figure. Exercise: Show that (u,w)  E XTC. uw

38. Experimental Results Stretch factor of G XTC w.r.t. energy metric (solid line). Mean values are plotted in black, maximum values in gray. G XTC is an energy-spanner in random UDG’s

39. Experimental Results Node degree of G XTC (solid line). Node degree of G (dotted line). Mean values are plotted in black, maximum values in gray. G XTC has very low degree in random UDG’s … but we already knew it!!!

40. A comparison with the Gabriel Graph