CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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

2. 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 Discrete Algs for Mobile Wireless Sys

3. 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 Discrete Algs for Mobile Wireless Sys

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

5. 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 Discrete Algs for Mobile Wireless Sys

6. 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) Discrete Algs for Mobile Wireless Sys

7. Sample Execution << see CBTCex.pdf >> Discrete Algs for Mobile Wireless Sys

8. CBTC(), Code for Node u Nu := Ø // neighbors discovered so far Du := Ø // directions from which Acks have arrived pu := pmin while (pu < pmax and there exists gap(,Du)) pu := increase(pu) bcast(u,pu,("Hello",pu)) wait for Acks Nu := Nu U {v : v just sent an Ack} Du := Du U {diru(v) : v just sent an Ack} Discrete Algs for Mobile Wireless Sys

9. 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)} Discrete Algs for Mobile Wireless Sys

10. Optimizations • Shrink-back: for nodes which terminate with p = pmax, 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 Discrete Algs for Mobile Wireless Sys

11. 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 Discrete Algs for Mobile Wireless Sys

12. 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. Discrete Algs for Mobile Wireless Sys

13. 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! v w u Discrete Algs for Mobile Wireless Sys

14. 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. Discrete Algs for Mobile Wireless Sys

15. 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 v1, …, vk-1 disconnects G/k. Discrete Algs for Mobile Wireless Sys

16. CBTC(/k) G/k remove v1,…,vk-1 remove v1,…,vk-1 G1 disconnected G' connected show G' is a subgraph of G1, and thus G1 is connected CBTC() G' connected by previous result Proof of Correctness  = 2/3 G k-connected Discrete Algs for Mobile Wireless Sys

17. Proof of Correctness • Suppose in contradiction G' is not a subgraph of G1. • So there is some edge (u,v) in G' but not in G1. • (u,v) is also not in G/k, since no edges between non-removed nodes are removed in going from G/k to G1 • So (u,v) is not needed in G/k: there are some other nodes w1, w2,… that are closer to u than v is, s.t. max angle b/w wi and wi+1 is at most /k Discrete Algs for Mobile Wireless Sys

18. CBTC(/k) G/k remove v1,…,vk-1 remove v1,…,vk-1 G1 disconnected G' connected show G' is a subgraph of G1, and thus G1 is connected CBTC() G' connected by previous result Proof of Correctness  = 2/3 G doesn't contain (u,v) k-connected doesn't contain (u,v) contains (u,v) Discrete Algs for Mobile Wireless Sys

19. 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. Discrete Algs for Mobile Wireless Sys

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