1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 06 Dec 2005 08th Lecture Christian Schindelhauer schindel@upb.de

2. Algorithms for Radio Networks 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Topology Control in Wireless Networks  Topology control: establish and maintain links  Routing is based on the network topology  Geometric spanners as network topologies

3. Algorithms for Radio Networks 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Geometric Spanners  G is a c-spanner: for every pair of nodes u,v there exists a path P in G such that ||P|| ≤ c · ||u,v|| P = (v 1,...,v n ), ||P|| :=  ||v i - v i+1 ||  the constant c is the stretch factor P

4. Algorithms for Radio Networks 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Weak Spanner and Power Spanner  Weak Spanner –weak c-spanner: for every pair of nodes u,v exists a path connecting u and v inside the disk C(u, c · ||u,v||)  Power Spanner –(c,d)-power spanner: for every pair of nodes u,v exists a path P such that |P| ≤ c · |P opt | |P| = Energy(P) =  ||v i - v i+1 || d v u P opt P

5. Algorithms for Radio Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Spanner, Weak Spanner, Power Spanner  Every c-Spanner is a weak c-Spanner (Exercise 10)  Every c-Spanner is a (c d,d)-Power Spanner (Exercise 11)  Every weak c-Spanner is a (c’,d)-Power Spanner for d  2  There are weak Spanners that are no Spanners (e.g. the Koch Curve is no c-Spanner but a weak 1-Spanner)  There are Power Spanners that are no Weak Spanners Spanner Weak Spanner Power Spanner X X

6. Algorithms for Radio Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Yao-Graph nearest neighbor in each sector Spanner ⊇ SparsY Sparsified Yao-Graph use only the shortest ingoing edges weak- & power-Spanner, constant in-degree ⊇ SymmY Symmetric Yao-Graph only symmetric edges not a spanner, nor weak spanner, nor power-spanner The Yao-Family

7. Algorithms for Radio Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer From directional to omnidirectional communication...  Directional communication  Easy range assignment  Spanner constructions: Yao, SparsY, SymmY  Omnidirectional communication  Finding a min. range assignment is NP-hard and needs global knowledge  Spanner construction: Hierarchical Layer Graph Khepera mini robot with infrared communication module

8. Algorithms for Radio Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer L0L0 The Hierarchical Layer Graph (HLG)  Basic Ideas: –many short edges on lower layers  energy efficiency –few long edges on higher layers  connectivity  layers = range classes, assigned to power levels L1L1 L1L1 L2L2 L2L2

9. Algorithms for Radio Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Formal definition of the HLG  The HLG contains w+1 layers L 0,...,L w.  The lowest layer L 0 contains all nodes, the highest layer only one node v*. The nodes of layer i+1 are also contained in layer i. {v*} = V(L w ) ...  V(L 1 )  V(L 0 ) = V  In each layer the nodes have a minimum distance:  u,v  V(L i ): |u-v| ≥ r i r i is the domination radius for layer i. r 0 < min |u,v|. r i :=  i · r 0.  All nodes in the next lower layer must be covered by this distance (each node has a dominator within a distance of r i+1 ):  u  V(L i )  v  V(L i+1 ): |u-v| ≤ r i+1  The edge set of layer i contains all edges connecting layer-i nodes that have a maximum distance E(L i ) := {(u,v) | u,v  V(L i )  |u,v| ≤  · r i }  · r i is the publication radius for layer i.   ≥  > 1.

10. Algorithms for Radio Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Construction of the HL Graph  Every node on layer i is dominated by some node in layer i+1 –including self-domination  No nodes may dominate each other  Edges are inserted in the publication radius of each node L 1 node L 0 node L 1 domination radius L 1 publication radius L 1 edge

11. Algorithms for Radio Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Radii of the HL Graph  definition based on parameters  and   r 0 := minimal node distance, rank := highest layer  domination radius for layer i: no other nodes with rank > i within this radius  publication radius for layer i: edges to nodes with rank = i  · r 0  · r 1 r0r0 r1r1 r2r2  · r 2 r i :=  i · r 0  · r i

12. Algorithms for Radio Networks 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Radii and Edges of the HL Graph  L i-1 publication radius ≥ L i domination radius:  ≥  > 1  Layer-i edges are established in between  · r 0 r0r0 r1r1 L 0 /L 1 edge L 0 edge L 1 edge L 0 node L 1 node layer-0 domination radius layer-1 domination radius

13. Algorithms for Radio Networks 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Properties of the HL Graph  The HL Graph is a c-Spanner, if  > 2  / (  -1)  The interference number of the HLG is bounded by O(g(V)) g(V) = Diversity of the node set V g(V) = O(log n) for nodes in random positions with high probability  A c-Spanner contains a path system with load O(g(V) · C*) C* = congestion of the congestion-optimal path system  The HLG contains a path system P with congestion O(g(V) 2 · C*) i.e. P approximates the congestion-optimal path system by a factor of O(log 2 n) for nodes in general position *) Meyer auf der Heide et al. “Congestion, Energy and Dilation in Radio Networks, TOCS 2004  Theorem 8*  Theorem 9*  Lemma 9*  Theorem 10*

14. Algorithms for Radio Networks 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Degree, Interference Number and Density in a Layer of the HL-Graph  Lemma –For any finite point set V  R d and a layer L i of a Hierarchical Layer Graph with parameters  ≥  > 1 we have the following 1.For any point u, the number of points in layer L i with |u-v| ≤ c r i is at most (2c+1) d. 2.The degree of the sub-graph L i ist at most (2  +1) d. 3.The interference number of L i is bounded by (2  +1) 2d.  Proof: –Exercise –Ideas: 1.follows by the domination property and a rough bound on the volume of the dominated area 2.follows directly from 1. and the definition of the HL-Graph 3.follows from counting all nodes and degrees in the 2  r i surrounding of a node.

15. Algorithms for Radio Networks 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Interference Number of the HL-Graph  Lemma –The number of layers in a HL-Graph is bounded by O(g(V))  Proof: –We only count layers where at least one edge exists. –In layer L i all edges have an edge length in the intervall [r i,  r i ] –Now consider the diversity g(V) of the node set V –Assume that there exist no two nodes with distance in [2 j,  2 j+1 ) for some integer j. Then the corresponding layers with 2 j ≤ r i < 2 j+1 in the HL-graph do not exist. –If there exists two nodes with distance in [2 j, 2 j+1 ), then there may exist some of the layers L j/(log  )-1, L j/(log  ),..., L (j+(log  ))/ (log  )+ 1. –So, there is a linear relationship between g(V) and the number of layers of the HL- graph, i.e. # layers of HL-graph =  (g(V))  Lemma –The interference number of the HL-Graph is bounded by O(g(V))  Proof –Summing up over all g(V) layers with constant interference in each layer yields a number of O(g(V)) interferences in the HL-Graph.

16. Algorithms for Radio Networks 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Every Weak Spanner Allows Routes with Small Load  Theorem –Let C* be the congestion of the congestion-optimal path system P* for a node set V. –Then, every weak c-spanner N can host a path system P’ such that the induced load l(e) in N for each each is bounded by l(e) ≤ c’ g(V) C* –for the diversity g(V) and a positive constant c’.  Proof idea: –Start with optimal path system P* –Simulate optimal path system by replacing each edge with a path in N within the disk of the edge, receiving P’ in N. –Consider edges of P* with length [2 j,2 j+1 ) –Each of these edges of P* reroutes into P’ in the relative vicinity of distance c 2 j+1 defined by the weak spanner property. increasing the load only by a constant factor compared to the congestion –This adds up for each of the g(V) orders of magnitudes of edge lengths to O(g(V) C*)

17. Algorithms for Radio Networks 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Weak Spanner + Small Interference Number = Good Congestion Approximation  Theorem –Let C* be the congestion of the congestion-optimal path system P* for a node set V. –Then, every weak c-spanner N can host a path system P’ where the congestion C in N is bounded by C ≤ c’ g(V) 2 C* for diversity g(V), and a positive constant c’.  Proof follows by the preceding Lemma –using replacement paths which use only a constant interference number between the set of edges of length [2 j,2 j+1 ) in N  If the interference number of such edges is larger than a certain constant, then the density of points is so large that recursive replacement paths using shorter edges are constructed. –leading to a constant interference number of edges of length [2 j,2 j+1 )  Now consider a edge. –It has load of at most c g(V) C* –It can suffer from the interference of longer and shorter edges –For each [2 j,2 j+1 ) this influence adds at most c g(V) C* –leading to an overall congestion for this edge of at most O(g(V) 2 C*)‚.

18. Algorithms for Radio Networks 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The HL-Graph Approximates Congestion  Theorem –Let P* be the congestion optimal path system for the nodes V with congestion C*. –Then, the Hierarchical Layer Graph contains a path system P with congestion O(g(V) 2 C P* (V)).  Proof –The HL-graph is a weak spanner –The rest follows by the theorem before.  Further features of the HL-graph –small diameter: O(g(V)) –small interference number: O(g(V)) –contains energy approximating path since the HL-graph is a Power-Spanner.

19. Algorithms for Radio Networks 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The SparsY-Graph approximates Congestion  Theorem –For directed communication the SparsY-Graph contains a path system P which approximates the congestion optimal path system P* by the congestion: O(g(V) 2 C P* (V)).  Proof idea –Generalize the concept of weak spanner approximation of the load to directed communication –Leads to the analogous results. –Combine with the constant (directed) interference number of the SparsY-Graph  Note: –The SparsY-Graph contains also an O(1)-energy approximating path since SparsY is also a Power-Spanner –But: SparsY-Graphs can contain very long paths.

20. 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention End of 8th lecture Next lecture:Mi 14 Dec 2005, 4pm, F1.110 Next exercise class: Tu 20 Dec 2005, 1.15 pm, F2.211 or Th 15 Dec 2005, 1.15 pm, F1.110