1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. Discrete Algs for Mobile Wireless Sys2 Lecture 20  Topic: Part III: Building and Maintaining Network Structures Topology Control  Sources: Li et al. Bahramgiri et al. MIT 6.885 Fall 2008 slides

3. Discrete Algs for Mobile Wireless Sys3 Topology Control  Suppose you have a set of nodes V, where each node v has a Euclidean position and a maximum transmission power.  If all nodes transmit using maximum power, they induce a set of edges E representing the communication links  Starting with connected graph G = (V,E), reduce number of neighbors of each node by reducing transmission power at that node to produce a subset E' of E with G' = (V,E') still connected

4. Discrete Algs for Mobile Wireless Sys4 How to Optimize Power?  Focus on reducing transmission power of each node minimize the maximum transmission power per node

5. Discrete Algs for Mobile Wireless Sys5 Cone-Based Topology Control [Li et al.]  Problem studied is how to reduce power while preserving connectivity  Assumes interference is implicitly handled by reducing node degree and edge set  Claims not to require position of each node and not to be coupled with any radio propagation model  Assumes a MAC layer that provides a reliable broadcast operation; synchronous rounds  Mainly theoretical with some simulations to validate theoretical results

6. Discrete Algs for Mobile Wireless Sys6 Central Idea of Algorithm  Parameter  to algorithm  Each node u starts broadcasting and collecting Acks with minimum power  Node u gradually increases its transmission power stops when reaching maximum power or every cone of degree  contains at least one node (neighbor)

7. Discrete Algs for Mobile Wireless Sys7 Sample Execution >

8. Discrete Algs for Mobile Wireless Sys8 CBTC(  ), Code for Node u N u := Ø // neighbors discovered so far D u := Ø // directions from which Acks have arrived p u := p min while (p u < p max and there exists gap( ,D u )) p u := increase(p u ) bcast(u,p u,("Hello",p u )) wait for Acks N u := N u U {v : v just sent an Ack} D u := D u U {dir u (v) : v just sent an Ack}

9. Discrete Algs for Mobile Wireless Sys9 Some Definitions  Let N  (u) be final set of neighbors discovered by node u  Let E  – = {(u,v) : v is in N  (u) and u is in N  (v)}  Let E  + = {(u,v) : v is in N  (u) or u is in N  (v)}

10. Discrete Algs for Mobile Wireless Sys10 Optimizations  Shrink-back: for nodes which terminate with p = p max, reduce power while keeping coverage.  Asymmetric edge removal: transform E  + into E  –, as long as E  – preserves connectivity  Pairwise edge removal: remove implicit 3- cycles; use node ids to get unique edge weights and remove longest edge

11. Discrete Algs for Mobile Wireless Sys11 Proof of Correctness  INFOCOMM paper proves that when  = 2  /3, then E  + is connected.  PODC paper proves that when  = 2  /3, then E  – is connected and when  = 5  /6, then E  + is connected  Proofs are geometrical in nature, subtle and somewhat long  From MIT course notes, prove something stronger: when  = 2  /3, then E  – is not only connected, but contains the minimum spanning tree

12. Discrete Algs for Mobile Wireless Sys12 About Minimum Spanning Trees  Given a set of points in the plane and all straight- line edges between them, want to find MST of the resulting graph (cost of an edge is its length).  Yao's Lemma: If edge (u,v) is in MST, then v is u's closest neighbor in every  /3 cone centered at u containing v.

13. Discrete Algs for Mobile Wireless Sys13 Using Yao's Lemma  Suppose in contradiction the CBCT output graph G' is missing some edge (u,v) of the MST M of the maximum power graph G  Consider the cone of angle 2  /3 centered at u with v on its bisector; each half of the cone has angle  /3  By construction, CBCT guarantees that there is some neighbor w of u inside this cone in G'.  Since v is not a neighbor of u, but w is, it must be that v is farther away from u than w is.  Contradicts Yao's Lemma! u v w

14. Discrete Algs for Mobile Wireless Sys14 Extension to k-Connectivity [Bahramgiri]  A connected graph is k-connected if removing any set of less than k nodes results in a graph that is still connected.  How can we find minimum power needed by each node to ensure that resulting graph is k-connected?  Use the cone algorithm CBTC with  = 2  /3k.

15. Discrete Algs for Mobile Wireless Sys15 Proof of Correctness  Theorem: If G is k-connected, then G , result of running CBTC(  /k) with  = 2  /3 on G, is also k-connected.  Proof: Suppose in contradiction G  /k is not k-connected: removing nodes v 1, …, v k-1 disconnects G  /k.

16. Discrete Algs for Mobile Wireless Sys16 Proof of Correctness G remove v 1,…,v k-1 G' connected remove v 1,…,v k-1 G 1 disconnected G  /k CBTC(  /k) CBTC(  ) G'  connected by previous result  = 2  /3 show G'  is a subgraph of G 1, and thus G 1 is connected k-connected

17. Discrete Algs for Mobile Wireless Sys17 Proof of Correctness  Suppose in contradiction G'  is not a subgraph of G 1.  So there is some edge (u,v) in G'  but not in G 1.  (u,v) is also not in G  /k, since no edges between non- removed nodes are removed in going from G  /k to G 1  So (u,v) is not needed in G  /k : there are some other nodes w 1, w 2,… that are closer to u than v is, s.t. max angle b/w w i and w i+1 is at most  /k

18. Discrete Algs for Mobile Wireless Sys18 Proof of Correctness G remove v 1,…,v k-1 G' connected remove v 1,…,v k-1 G 1 disconnected G  /k CBTC(  /k) CBTC(  ) G'  connected by previous result  = 2  /3 show G'  is a subgraph of G 1, and thus G 1 is connected k-connected contains (u,v) doesn't contain (u,v)

19. Discrete Algs for Mobile Wireless Sys19 Proof of Correctness  In worst case, the k nodes removed are consecutive w-nodes.  Resulting empty cone has angle at most (  /k)*k =   So in G' , there is no need to use enough power to reach v; it is sufficient to reach the remaining w-nodes  Thus (u,v) is not an angle of G' , contradiction.

20. Discrete Algs for Mobile Wireless Sys20 Proof of Correctness u u u u u

21. Discrete Algs for Mobile Wireless Sys21 Central Idea of Algorithm  Parameter a to algorithm  Each node u starts broadcasting and collecting Acks with minimum power  Node u gradually increases its transmission power stops when reaching maximum power or there is no cone of degree a with no neighbors